Fractal Dimension for Fractal Structures: With Applications to Finance [1st ed.] 978-3-030-16644-1;978-3-030-16645-8

Table of contents :
Front Matter ....Pages i-xvii
Mathematical Background (Manuel Fernández-Martínez, Juan Luis García Guirao, Miguel Ángel Sánchez-Granero, Juan Evangelista Trinidad Segovia)....Pages 1-48
Box Dimension Type Models (Manuel Fernández-Martínez, Juan Luis García Guirao, Miguel Ángel Sánchez-Granero, Juan Evangelista Trinidad Segovia)....Pages 49-83
A Middle Definition Between Hausdorff and Box Dimensions (Manuel Fernández-Martínez, Juan Luis García Guirao, Miguel Ángel Sánchez-Granero, Juan Evangelista Trinidad Segovia)....Pages 85-147
Hausdorff Dimension Type Models for Fractal Structures (Manuel Fernández-Martínez, Juan Luis García Guirao, Miguel Ángel Sánchez-Granero, Juan Evangelista Trinidad Segovia)....Pages 149-195
Back Matter ....Pages 197-204

Citation preview

19 Manuel Fernández-Martínez Juan Luis García Guirao Miguel Ángel Sánchez-Granero Juan Evangelista Trinidad Segovia

Fractal Dimension for Fractal Structures With Applications to Finance

Se MA

SEMA SIMAI Springer Series Volume 19

Editor-in-Chief Luca Formaggia, MOX–Department of Mathematics, Politecnico di Milano, Milano, Italy Pablo Pedregal, ETSI Industriales, University of Castilla–La Mancha, Ciudad Real, Spain Series Editors Mats G. Larson, Department of Mathematics, Umeå University, Umeå, Sweden Tere Martínez-Seara Alonso, Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain Carlos Parés, Facultad de Ciencias, Universidad de Málaga, Málaga, Spain Lorenzo Pareschi, Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Ferrara, Italy Andrea Tosin, Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Torino, Italy Elena Vázquez-Cendón, Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, A Coruña, Spain Jorge P. Zubelli, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil Paolo Zunino, Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

As of 2013, the SIMAI Springer Series opens to SEMA in order to publish a joint series (SEMA SIMAI Springer Series) aiming to publish advanced textbooks, research-level monographs and collected works that focus on applications of mathematics to social and industrial problems, including biology, medicine, engineering, environment and finance. Mathematical and numerical modeling is playing a crucial role in the solution of the complex and interrelated problems faced nowadays not only by researchers operating in the field of basic sciences, but also in more directly applied and industrial sectors. This series is meant to host selected contributions focusing on the relevance of mathematics in real life applications and to provide useful reference material to students, academic and industrial researchers at an international level. Interdisciplinary contributions, showing a fruitful collaboration of mathematicians with researchers of other fields to address complex applications, are welcomed in this series. THE SERIES IS INDEXED IN SCOPUS

More information about this series at http://www.springer.com/series/10532

Manuel Fernández-Martínez Juan Luis García Guirao Miguel Ángel Sánchez-Granero Juan Evangelista Trinidad Segovia •

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Fractal Dimension for Fractal Structures With Applications to Finance

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Manuel Fernández-Martínez Department of Sciences and Computation University Centre of Defence at Spanish Air Force Academy Santiago de la Ribera, Murcia, Spain

Juan Luis García Guirao Departamento de Matemática Aplicada y Estadística Universidad Politécnica de Cartagena Cartagena, Murcia, Spain

Miguel Ángel Sánchez-Granero Departamento de Matemáticas Universidad de Almería La Cañada de San Urbano, Almería, Spain

Juan Evangelista Trinidad Segovia Departamento de Ciencias Económicas y Empresariales Universidad de Almería La Cañada de San Urbano, Almería, Spain

ISSN 2199-3041 ISSN 2199-305X (electronic) SEMA SIMAI Springer Series ISBN 978-3-030-16644-1 ISBN 978-3-030-16645-8 (eBook) https://doi.org/10.1007/978-3-030-16645-8 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The word fractal, which derives from the Latin term frangere (meaning “to break”), has become a very important concept in mathematics since Benoît Mandelbrot first introduced it in the late 1970s [55]. Indeed, both the study and the analysis of fractal patterns have increasingly grown in importance in recent years due to the large range of applications in diverse scientific fields where fractals have been identified, including economics, physics, and statistics (cf. [29, 33]). Additionally, there has been a particular interest in the application of fractals to the social sciences (cf. [19] and references therein). Interestingly, fractal dimension constitutes the key tool applied in these areas to study fractals since it is their main invariant and yields useful information regarding the complexity and the irregularities that they present when explored in sufficient detail. Indeed, fractal dimension theory has been applied in various fields of science, such as dynamical systems [32], diagnosis of diseases (including osteoporosis [73] and cancer [12]), ecology [1], earthquakes [44], detection of eyes in human face images [53], and analysis of the human retina [51], to name but a few. Usually, both Hausdorff and box dimensions are used. These notions can be defined on any metric space and while the former is “better” from a theoretical viewpoint, since its definition is based on a measure, the latter becomes more appropriate for applications since it is easier to calculate or estimate. Anyway, almost all empirical applications of fractal dimension have been carried out on Euclidean spaces via the box dimension. The idea of defining measures by coverings of certain subsets was first introduced by Carathéodory in 1914 (cf. [20]). Later, Hausdorff applied that approach to define the measures that now bear his name and proved that the middle third Cantor 2 set has positive and finite measure with dimension equal to log log3 (c.f. [29, 42]). Subsequently, several properties and technical aspects of Hausdorff measures and dimensions were explored by Besicovitch [16], Besicovitch and his students [17], Falconer [27, 29], Feder [33], and Rogers [70].

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On the other hand, it seems that the origins of the so-called box-counting dimensions go back to the 1920s, when they were first considered by the pioneers of Hausdorff measure and dimension. Nevertheless, they were rejected at first glance for being less appropriate from a mathematical viewpoint. Bouligand adapted the Minkowski content to nonintegral dimensions [18] and the classical definition of box dimension was first contributed by Pontrjagin and Schnirelman (cf. [68]). The popularity of box dimension is mainly due to the possibility of its effective calculation and empirical estimation. It is also known as Kolmogorov entropy, entropy dimension, capacity dimension, metric dimension, information dimension, logarithmic density, etc. [29]. Several topics on fractal theory have been formalized from both theoretical and applied viewpoints by means of fractal structures, first sketched in [14] and formally defined later in [3] to characterize non-Archimedean quasi-metrization. A fractal structure is a countable collection of coverings of a set which better approaches the whole space as deeper stages, called levels, are explored. As such, fractal structures provide a perfect place where new models of fractal dimension can be developed, as the definition of standard box dimension highlights. It is also worth noting that fractal structures connect diverse topics on topology, including transitive quasi-uniformities, non-Archimedean quasi-metrization, metrization, topological and fractal dimensions, self-similar sets, and space-filling curves (cf. [74]). Self-similar sets constitute a class of fractals which can always be endowed with a fractal structure naturally (first introduced in [11]). As such, they can be further analyzed from the viewpoint of fractal structures. Accordingly, this book contains several Moran type results allowing calculation of the fractal dimension of attractors by equations only involving the similarity ratios associated with the corresponding iterated function systems. In this book, we develop a theory of fractal dimension for fractal structures with novel applications to computation, artificial intelligence, and econophysics. This theory extends the classical models of fractal dimension, namely, Hausdorff and box dimensions, to the more general context of fractal structures. We develop new models of fractal dimension for fractal structures and provide useful expressions to calculate them. We also prove some connections between each definition of fractal dimension and the classical dimensions. While the box dimension is generalized in the Euclidean context by fractal dimensions I, II, and III, the Hausdorff dimension is also generalized from the viewpoint of fractal structures by fractal dimensions V and VI. Interestingly, fractal dimension IV constitutes an intermediate model between box and Hausdorff dimensions especially appropriate for empirical applications of fractal dimension. Let us summarize the content of each chapter in this book: In Chap. 1, we recall notations, concepts, and results that are necessary to develop our theory of fractal dimension for fractal structures. The focus is on quasi-pseudometrics, fractal structures, iterated function systems, and classical fractal dimensions.

Preface

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In Chap. 2, we formally introduce fractal dimension I and II models to calculate the fractal dimension of a set with respect to a fractal structure. This novel theory extends box dimension to the more general context of fractal structures and allows a wide range of fractal structures to calculate the fractal dimension of any subset. It is worth mentioning that the new definitions can be calculated in contexts where box dimension lacks sense or may not be calculated. As such, fractal dimensions I and II can be used in any space admitting a fractal structure as easily as box dimension in empirical applications. We illustrate how to apply fractal dimension I in non-Euclidean contexts. In fact, we define a fractal structure on the domain of words and show how to use fractal dimension I to explore fractal patterns of languages generated by regular expressions, calculate the efficiency of an encoding language, and estimate the number of nodes of a given depth in a search tree. In Chap. 3, we provide a new model to calculate the fractal dimension of a set with respect to a fractal structure generalizing box dimension for Euclidean subsets. This has been defined as an appropriate discretization of both Hausdorff measure and dimension. Some theoretical connections among the new fractal dimension III and the classical ones are provided. Additionally, it has been proved a Moran type theorem to calculate the fractal dimension III of attractors whose iterated function systems are not required to satisfy the open set condition. Novel applications of that fractal dimension to econophysics are developed in the context of asymmetric topology. More specifically, we apply fractal dimension III to study curves (not necessarily continuous) and, in particular, time series. Interestingly, we prove that fractal dimension III is strongly related to the self-similarity (or Hurst) exponent of random processes in the sense that each quantity is the inverse of the other. The Hurst exponent is a consolidated tool in finance to explore long-memory in series of stock prices. In this chapter, we illustrate how fractal dimension III leads to efficient algorithms allowing the detection of long-range dependence in financial time series. In addition, the self-similarity exponent of fractional Brownian motions and Lévy stable processes is properly studied via such approaches. Finally, in Chap. 4, we study how to generalize the Hausdorff dimension by three new models of fractal dimension for fractal structures: two of them are discretizations of the Hausdorff dimension (fractal dimensions IV and V), whereas fractal dimension VI constitutes a continuous approach for the Hausdorff dimension in the context of fractal structures. The new definitions are connected with fractal dimensions I, II, and III and classical dimensions. We also highlight that the analytic construction of fractal dimension VI is based on a measure, as happens with the Hausdorff dimension. Interestingly, the Hausdorff dimension is generalized by fractal dimensions V and VI in the context of Euclidean subsets equipped with their natural fractal structures. The fractal dimensions IV, V, and VI of strict self-similar

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sets lying under the open set condition are calculated by a Moran type equation only involving the similarity ratios associated with the corresponding iterated function systems. Santiago de la Ribera, Spain Cartagena, Spain Almería, Spain Almería, Spain

Manuel Fernández-Martínez Juan Luis García Guirao Miguel Ángel Sánchez-Granero Juan Evangelista Trinidad Segovia

Contents

1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Mathematical Foundations . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Basics on Set Theory and Point-Set Topology 1.1.2 Mappings and Limits . . . . . . . . . . . . . . . . . . 1.1.3 Foundations on Measure Theory . . . . . . . . . . 1.2 Hausdorff Measure and Dimension . . . . . . . . . . . . . . . 1.3 Box Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Iterated Function Systems . . . . . . . . . . . . . . . . . . . . . . 1.5 Quasi-pseudometrics . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Fractal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Self-Similar Processes . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Random Functions and Their Increments. Self-Affinity Properties . . . . . . . . . . . . . . . . . 1.7.2 Fractional Brownian Motions . . . . . . . . . . . . 1.7.3 Stable Processes and Fractional Lévy Stable Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Box Dimension Type Models . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Natural Fractal Structure on Euclidean Subsets . . . . . . 2.3 Generalizing Box Dimension by Fractal Dimension I . . . . . 2.4 Theoretical Properties of Fractal Dimension I . . . . . . . . . . . 2.5 Linking Fractal Dimension I to Box Dimension . . . . . . . . . 2.6 Fractal Dimension I for IFS-Attractors . . . . . . . . . . . . . . . . 2.7 Dependence of Fractal Dimension I on the Fractal Structure 2.8 A Further Step: Fractal Dimension II . . . . . . . . . . . . . . . . . 2.9 Linking Fractal Dimensions I and II via the Semimetric Associated with a Fractal Structure . . . . . . . . . . . . . . . . . . 2.10 Theoretical Properties of Fractal Dimension II . . . . . . . . . . 2.11 Dependence of Fractal Dimension II on Both a Fractal Structure and a Distance . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.12 2.13 2.14 2.15

Linking Fractal Dimension II to Box Dimension . . . . . . . . Generalizing Fractal Dimension I by Fractal Dimension II . Fractal Dimension II for IFS-Attractors . . . . . . . . . . . . . . . Applications to the Domain of Words . . . . . . . . . . . . . . . . 2.15.1 Introducing the Domain of Words . . . . . . . . . . . . 2.15.2 Quasi-metrics, Fractal Structures, and the Domain of Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15.3 Fractal Dimension of Languages Generated by Regular Expressions . . . . . . . . . . . . . . . . . . . 2.15.4 The Efficiency of the BCD Encoding System . . . . 2.15.5 An Empirical Application to Search Trees . . . . . . 2.16 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 A Middle Definition Between Hausdorff and Box Dimensions . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Analytical Construction of a New Fractal Dimension . . . . . . 3.3 Defining Fractal Dimension III . . . . . . . . . . . . . . . . . . . . . . 3.4 Linking Fractal Dimension III to Some Fractal Dimensions . . 3.5 How to Calculate the Effective Fractal Dimension III . . . . . . 3.6 Measure Properties of Hsn;3 . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Linking Fractal Dimension III to Fractal Dimensions I and II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Theoretical Properties of Fractal Dimension III . . . . . . . . . . . 3.9 An Additional Connection with Box Dimension . . . . . . . . . . 3.10 Fractal Dimension III for IFS-Attractors . . . . . . . . . . . . . . . . 3.11 A New Fractal Dimension for Curves . . . . . . . . . . . . . . . . . 3.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11.2 The Fractal Dimension of a Curve . . . . . . . . . . . . . 3.11.3 Theoretical Properties of the New Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11.4 How to Construct Space-Filling Curves via Fractal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11.5 The Fractal Dimension of the Hilbert’s Curve . . . . 3.11.6 A Curve Filling the Sierpiński’s Gasket . . . . . . . . . 3.11.7 The Fractal Dimension of a Modified Hilbert’s Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Applying Fractal Dimension to Deal with Random Processes 3.12.1 Introducing FD Algorithms to Calculate the SelfSimilarity Exponent of Random Processes . . . . . . . 3.12.2 Testing the Accuracy of FD Algorithms . . . . . . . . 3.12.3 Exploring Long-Memory in Stock Market Indices . 3.12.4 The Self-Similarity Exponent of Actual US Stocks . 3.12.5 Next Step: FD4 Algorithm . . . . . . . . . . . . . . . . . .

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3.12.6 3.12.7 3.12.8

Generalizing GM2 and FD Algorithms . . . . . . . An Open Gate to Multifractality . . . . . . . . . . . . Some Notes on the New FD4 Algorithm and Its Implementation . . . . . . . . . . . . . . . . . . . . . . . . 3.12.9 Testing the Accuracy of FD4 Algorithm . . . . . . 3.12.10 A Historic Study Regarding the Self-Similarity Exponent of S&P 500 Stocks . . . . . . . . . . . . . . 3.13 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Hausdorff Dimension Type Models for Fractal Structures . . . . . . 4.1 Improving the Accuracy of Fractal Dimension III . . . . . . . . . . 4.2 Hausdorff-Type Dimensions for Fractal Structures . . . . . . . . . 4.3 Linking Fractal Dimensions V and VI . . . . . . . . . . . . . . . . . . 4.4 Additional Connections Among Fractal Dimensions III, IV, and V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Measure Properties of Hs6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Generalizing Hausdorff Dimension . . . . . . . . . . . . . . . . . . . . . 4.7 Fractal Dimensions for IFS-Attractors . . . . . . . . . . . . . . . . . . 4.8 How to Calculate the Hausdorff Dimension in Empirical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Calculating the Hausdorff Dimension . . . . . . . . . . . . 4.8.2 Calculation of Hsn;m ðFÞ for n  m  nmax . . . . . . . . . 4.8.3 Calculation of Hs ðFÞ . . . . . . . . . . . . . . . . . . . . . . . 4.8.4 Calculation of dim4C ðFÞ . . . . . . . . . . . . . . . . . . . . . . 4.8.5 Empirical Experimentation . . . . . . . . . . . . . . . . . . . 4.8.6 Generating Collections of Sets with a Fixed Hausdorff Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.7 Training the Algorithm . . . . . . . . . . . . . . . . . . . . . . 4.8.8 Cross-Validation of the SVM . . . . . . . . . . . . . . . . . 4.8.9 Final Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.10 Testing the Accuracy of Algorithm 4.8.1 by External Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

About the Authors

Manuel Fernández-Martínez holds an international Ph.D. in Mathematics from UCLA. His research interests include fractal structures, fractal dimension, self-similar sets, computational applications of fractal dimension, and self-similar processes and their applications to finance. Juan Luis García Guirao is a Full Professor of Applied Mathematics at the Technical University of Cartagena, Spain. In 2011, at the age of 33, he became Spain’s youngest Full Professor of Mathematics. He is the author of more than 100 research papers and Editor in Chief of Applied Mathematics and Nonlinear Sciences, and his work has been recognized with the 2017 NSP award for researchers younger than 40 years old and with the 2017 JDEA best paper award (http://www.jlguirao.es/). Miguel Ángel Sánchez-Granero is an Associate Professor at the Department of Mathematics, University of Almería, Spain. He is the author of a number of publications in international journals on asymmetric topology, self-similarity, fractal structures, and fractal dimension with application to financial series. Juan Evangelista Trinidad Segovia, Ph.D. is a Full Professor of Finances at the Department of Economics and Business, University of Almería, Spain. His research interests include financial modeling, portfolio selection, CAPM and, most recently, applications of statistical mechanics to financial markets. He has published over 20 papers in peer-reviewed journals.

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Nomenclature

An ðFÞ dimB ðFÞ Bsd ðFÞ Bs C Card ðFÞ Cd ðFÞ dim5C ðFÞ dim4C ðFÞ dimðaÞ diam ðCn Þ diam ðF; Cn Þ dim2C ðFÞ E½X £ F H Hds ðFÞ

The collection of all elements in level n of C that intersect F, i.e., fA 2 Cn : A \ F 6¼ £g (c.f. Fractal dimension approach 2.3.1) The box(-counting) dimension of a (non-empty) bounded Euclidean subset F (c.f. Definition 1.3.1) The sum of the s-powers of the diameters of all the subsets of any d-cover of F consisting of open balls (c.f. Proposition 1.2.12) The measure given by letting d ! 0 in Bsd ðFÞ for all subset F (c.f. Proposition 1.2.12) The (standard) middle third Cantor set The cardinal number of a set F The class of all d-covers of F (c.f. Definition 1.2.1) The fractal dimension V of a subset F of a GF-space ðX; qÞ endowed with a metric (c.f. Definition 4.2.1) The fractal dimension IV of a subset F of a GF-space ðX; qÞ endowed with a metric (c.f. Definition 4.2.1) The fractal dimension of (the parametrization of) a curve a (c.f. Definition 3.11.3) The diameter of level n of C, i.e., supfdiam ðAÞ : A 2 Cn g (c.f. Definition 2.5.1) The diameter of F in level n of C, i.e., supfdiam ðAÞ : A 2 An ðFÞg (c.f. Definition 2.5.1) The fractal dimension II of a subset F of a GF-space ðX; CÞ endowed with a distance q (c.f. Definition 2.8.2) The mean of a random variable X (c.f. Sect. 3.12) The empty set An iterated function system (c.f. Definition 1.4.1) The Hutchinson’s operator (c.f. Definition 1.4.9) The sum of the s-powers of the diameters of all the subsets of any d-cover of F (c.f. Eq. 1.1)

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Hs dH Hsn ðFÞ Hs ðFÞ Hs3 ðFÞ HðXÞ K dimB ðFÞ Ad;6 ðFÞ An;3 ðFÞ An;4 ðFÞ An;5 ðFÞ B Hs4 ðFÞ Hs5 ðFÞ Hs6 ðFÞ Hsd;6 ðFÞ Hsn;3 ðFÞ Hsn;4 ðFÞ Hsn;5 ðFÞ L Nn ðFÞ Oðf Þ PðXÞ Nd ðFÞ

Nomenclature

The s-dimensional Hausdorff measure (c.f. Eq. 1.2) Hausdorff metric (c.f. Definition 1.4.3) The sum of the s-powers of all the elements in level n of a fractal structure C that intersect F (c.f. Sect. 3.2) The limit as n ! 1 of Hsn ðFÞ (c.f. Sect. 3.2) The limit as n ! 1 of Hsn;3 ðFÞ (c.f. Sect. 3.3) The hyperspace of X (c.f. Definition 1.4.2) An IFS-attractor (c.f. Eq. 1.17) The lower box(-counting) dimension of a (non-empty) bounded Euclidean subset F (c.f. Definition 1.3.1) The collection of d-covers of F by elements of a fractal structure C, each of them lying in some level of C (c.f. Definition 4.2.1) The union of all the families Am ðFÞ, where m  n (c.f. Sect. 3.2) The collection of finite coverings of F by elements of a fractal structure C, each of them lying in some level of a fractal structure deeper than n (c.f. Definition 4.2.1) The collection of coverings of F by elements of a fractal structure C, each of them lying in some level of a fractal structure deeper than n (c.f. Definition 4.2.1) The language of binary-coded decimal numbers (c.f. Sect. 2.15.4) The limit as n ! 1 of Hsn;4 ðFÞ (c.f. Definition 4.2.1) The limit as n ! 1 of Hsn;5 ðFÞ (c.f. Definition 4.2.1) The limit as d ! 0 of Hsd;6 ðFÞ (c.f. Definition 4.2.3) The infimum of the sum of the s-powers of the diameters of the elements in each subfamily Ad;6 ðFÞ of a fractal structure C (c.f. Definition 4.2.1) The infimum of all the quantities Hsm ðFÞ, where m  n (c.f. Sect. 3.3) The infimum of the sum of the s-powers of the diameters of the elements in each subfamily An;4 ðFÞ of a fractal structure C (c.f. Definition 4.2.1) The infimum of the sum of the s-powers of the diameters of the elements in each subfamily An;5 ðFÞ of a fractal structure C (c.f. Definition 4.2.1) A language of the domain of words. Mathematically, a subset of R1 (c.f. Sect. 2.15.2) The number of elements in level n of C that intersect F, i.e., Nn ðFÞ ¼ Card ðAn ðFÞÞ (c.f. fractal dimension approach 2.3.1) The order of a sequence f of positive real numbers (c.f. Definition 2. 13.1) The class of all subsets of a given set X (c.f. Sect. 3.6) One of the equivalent quantities to calculate the box dimension of any Euclidean subset (c.f. Theorem 1.3.3)

Nomenclature

x# xY dim6C ðFÞ R R1 Y dim3C ðFÞ dimB ðFÞ dim1C ðFÞ V u di¼1 ffn gn2N Ad lðxÞ mx ðXÞ xuy ROOT

xvii

The collection of all finite or infinite words starting from x 2 R or being a prefix of x (c.f. Sect. 2.15.2) The collection of all the prefixes of x 2 R (c.f. Sect. 2.15.2) The fractal dimension VI of a subset F of a GF-space ðX; qÞ endowed with a metric (c.f. Definition 4.2.3) A finite non-empty alphabet (c.f. Sect. 2.15.2) The collection of all finite or infinite words from R, i.e., S n N n2N R [ R (c.f. Sect. 2.15.2) The prefix order on R1 (c.f. Sect. 2.15.2) The fractal dimension III of a subset F of a GF-space ðX; CÞ endowed with a metric (c.f. Sect. 3.3) The upper box(-counting) dimension of a (non-empty) bounded Euclidean subset F (c.f. Definition 1.3.1) The fractal dimension I of a subset F of a GF-space ðX; CÞ (c.f. Definition 2.3.2) A feasible open set (in the sense of the open set condition) (c.f. Definition 2.14.1) The golden ratio (c.f. Sect. 2.15.3 ) The finite d-ary Cartesian product (c.f. Example 2.8.4) The Fibonacci sequence (c.f. Sect. 2.15.3) The d-body of a compact set A (c.f. Definition 1.4.4) The length of a word x 2 R1 (c.f. Sect. 2.15.2) The (absolute) s-moment of a random variable X (c.f. Sect. 3.12) The common prefix of any two words x; y 2 R1 (c.f. Sect. 2.15.2) Root node of a search tree (c.f. Sect. 2.15.5)

Chapter 1

Mathematical Background

Abstract The main purpose of this chapter is to recall some definitions, results, and notations that are useful to develop a new theory of fractal dimension for fractal structures. In this way, we will be focused on quasi-pseudometrics, fractal structures, iterated function systems, and box-counting and Hausdorff dimension topics.

1.1 Mathematical Foundations Next, we shall recall some concepts and notations from set theory as well as the basics on point-set topology.

1.1.1 Basics on Set Theory and Point-Set Topology First, it is worth pointing out that several results proved in this work lie in the context of Euclidean spaces. Thus, by Rd we shall denote the d−dimensional Euclidean space, as usual. In particular, R refers to the real numbers (also the real distance line) and R2 denotes the Euclidean plane. We recall that dthe Euclidean |xi − yi |2 )1/2 , where (also metric) on Rd is defined as d(x, y) = |x − y| = ( i=1 x = (x1 , . . . , xd ), y = (y1 , . . . , yd ) are two points in Rd . Thus, the sum of points in Rd and the product of a real constant by a point in Rd will be carried out component by a component. By default, sets will be subsets of Rd , denoted by capital letters. By x ∈ E, we shall understand that E contains the point x (also that x belongs to E). Further, E ⊂ F means that E is a subset of F. A way to describe subsets is by giving what conditions its points satisfy, {x : properties satisfied by x}. Special mention should be paid to the following sets: ∅ refers to the empty set, namely, the set which contains no points, Z denotes the set of integers, N refers to the natural numbers (equivalently, the positive integers), and Q will denote the rationals. It is worth mentioning that the superscript + makes reference to the positive points of a set. For instance, R+ = {x ∈ R : x > 0}

© Springer Nature Switzerland AG 2019 M. Fernández-Martínez et al., Fractal Dimension for Fractal Structures, SEMA SIMAI Springer Series 19, https://doi.org/10.1007/978-3-030-16645-8_1

1

2

1 Mathematical Background

and the notation R+ 0 refers to the subset of real numbers consisting of those which are nonnegative (namely, positive or zero). Let x ∈ X ⊂ Rd and ε > 0. Thus, B(x, ε) = {y ∈ X : |x − y| < ε} denotes the (open) ball centered in x ∈ X with radius ε > 0. The corresponding closed ball is denoted by B(x, ε) = {y ∈ X : |x − y| ≤ ε}. In particular, a ball in the Euclidean plane (endowed with the Euclidean distance) is a disk, whereas in R is just an interval. Regarding real intervals, we shall denote them as usual. In this way, let a, b ∈ R : a < b. Then [a, b] = {x ∈ R : a ≤ x ≤ b} denotes the closed interval with endpoints a, b and (a, b) = {x ∈ R : a < x < b} is the corresponding open interval. In addition, [a, b) = {x ∈ R : a ≤ x < b} and (a, b] = {x ∈ R : a < x ≤ b} are half-open intervals. Let δ > 0. By a δ−cube of Rd , we shall understand the set {y ∈ Rd : |xi − yi | ≤ δ/2, i = 1, . . . , d}. Accordingly, a δ−cube in R is an interval and in R2 is a square (with respect to the Euclidean distance). Let A be a subset of X and δ > 0. The δ−body (also the δ−parallel body) of A is the set Aδ = {x ∈ X : |x − y| ≤ δ, y ∈ A}. Next, we recall the basic set operations. Let A, B be two subsets of X . Thus, A ∪ B denotes the union of these sets and A ∩ B refers to their intersection. They are mathematically described by A ∪ B = {x ∈ X : x ∈ A or x ∈ B} and A ∩ B = {x ∈ X : x ∈ A and x ∈ B}, as well. These notions could be extended to an arbitrary collection of sets {Aλ }λ∈ . In fact, ∪λ∈ Aλ contains all the points lying in some set Aλ and ∩λ∈ Aλ consists of all the points which are in all the sets Aλ . By a disjoint collection of sets, we shall understand a collection whose sets are pairwise disjoint, namely, the intersection involving any pair of sets in that collection equals the empty set. The difference between two sets will be denoted by A \ B, namely, the set consisting of all the points in A that do not belong to B. In this way, the complement of a subset A is written by Rd \ A. Let A, B be two subsets. We define A × B = {(a, b) : a ∈ A, b ∈ B} the Cartesian product of A and B. In particular, if A ⊂ Rn and B ⊂ Rm , then A × B ⊂ Rn+m . Other set operations we shall work with include the sum of sets and the product of a set by a scalar. More specifically, given A, B ⊂ Rd and λ ∈ R, we define A + B = {a + b : a ∈ A, b ∈ B} and λA = {λa : a ∈ A}, respectively. Let A be an infinite set. We say that A is countable provided that all its elements can be denoted throughout natural indices. In other words, if there exists a bijection ϕ : N → A such that for all a ∈ A, there exists i ∈ N such that ϕ(i) = a. Otherwise, A is named an uncountable set. For instance, Q is countable, whereas R is uncountable. Let A be a subset of R. By the supremum of A, sup A, we shall understand the least number k such that a ≤ k for all a ∈ A provided that such a number exists. Otherwise, we shall set sup A = +∞. Also, the infimum of A, inf A, is the greatest number l such that l ≤ a for all a ∈ A. If that number does not exist, we set inf A = −∞. It is worth mentioning that these quantities, supremum, and infimum of a set are not required to belong to that set. Let A be a (non-empty) subset of Rd . The diameter of A is given by diam (A) = sup{|x − y| : x, y ∈ A}, that is, the greatest distance between pair of points of A.

1.1 Mathematical Foundations

3

Hence, A is said to be bounded provided that diam (A) < ∞, or equivalently, if there exists a large enough ball containing A. Next, we deal with the convergence of sequences. Let {xn }n∈N (also {xn }n≥1 ) be a sequence in Rd . We say that {xn }n∈N converges to x ∈ Rd as n → ∞, limn→∞ xn = x, provided that for all ε > 0, there exists a natural number n 0 such that |xn − x| < ε for all n ≥ n 0 . Equivalently, if limn→∞ |xn − x| = 0. Such a point x ∈ Rd is called the limit of the sequence {xn }n∈N , and it is also written xn → x (as n → ∞). Both concepts of “open” and “closed” (previously used concerning balls in Rd ) are stated next for general sets. Let A be a subset of Rd . We say that A is open provided that for each a ∈ A, there exists ε > 0 such that B(x, ε) ⊂ A. On the other hand, A is said to be a closed set if each sequence of points of A, {xn }n∈N : xn ∈ A for every n ∈ N, it holds that limn→∞ xn = x ∈ Rd . Both Rd and the empty set ∅ are closed and open sets simultaneously. A characterization of open (resp. closed) sets in terms of their complements is as follows. A subset A of Rd is open (resp. closed), if and only if, its complement Rd \ A is closed (resp. open). It is worth pointing out that the intersection of a finite number of open sets is open. Moreover, the union of any collection of open sets is open. On the other hand, it holds that the union of a finite number of closed sets is closed and the intersection of any collection of closed sets is closed. Let x ∈ Rd and A be a subset of Rd . We say that A is a neighborhood of x if there exists ε > 0 such that B(x, ε) ⊂ A. The closure of a set A, A, is the smallest closed set that contains A. Equivalently, the intersection of all the closed sets containing A. A set is closed, if and only if, equals its closure. The largest open set containing a set A is called its interior. In other words, the interior of A, which we shall denote by A◦ , is the union of all the open sets containing A. A is open provided that A = A◦ . Clearly, A◦ ⊆ A ⊆ A. Additionally, the boundary of A, is given by ∂ A = A \ A◦ . Thus, closed sets can be characterized in terms of their boundaries. More specifically, A is closed if contains its boundary and open provided that it contains no boundary points. Let B be a subset of A. We shall understand that B is dense in A if we can write B ⊂ A ⊂ B. In other words, B is dense in A if for each point a ∈ A, there exist points of B arbitrarily close to a. Let F be a set. A cover (also covering) of F is any collection of subsets {Uλ }λ∈ such that F ⊆ ∪λ∈ Uλ . We say that F is compact if every covering of F by open sets admits a finite subcovering. Let F be a subset of Rd . Then F is compact, if and only if, F is closed and bounded. It is noteworthy that the intersection of any collection of compact sets is compact. Let {An }n∈N be a decreasing sequence of compact sets, ∞ Ai = ∅. Additionally, if there exists an open set namely, A1 ⊃ A2 ⊃ . . .. Then ∩i=1 k ∞ Ai ⊂ V . V containing ∩i=1 Ai , then there exists k ∈ N such that ∩i=1 A connected set is, roughly speaking, a single piece set. Technically, we say that A ⊂ Rd is connected provided that there do not exist open sets U, V such that A ⊂ U ∪ V with A ∩ U, A ∩ V being non-empty and disjoint. Let x ∈ A. By the connected component of x, we shall understand the largest connected subset of A containing x. Thus, a set is said to be totally disconnected provided that the connected component of every point only contains such a point. In other words, A

4

1 Mathematical Background

is totally disconnected if for all x, y ∈ A, there exist open sets U, V : x ∈ U, y ∈ V and A ⊂ U ∩ V . The class of Borel sets, B, is the smallest collection of Euclidean subsets satisfying the two following conditions: (i) If A is open (resp. closed), then A ∈ B. (ii) The finite/countable intersection (resp. union) of Borel sets is also a Borel set.

1.1.2 Mappings and Limits Let X, Y be two sets. A map (also mapping, function, or transformation) f : X → Y is a rule that maps every x ∈ X to a unique f (x) ∈ Y . In this context, X is called the domain of f and Y is named its codomain. If A is a subset of X , then f (A) will refer the image of A via f , namely, f (A) = { f (x) : x ∈ A}. Let B be a subset of Y . Then f −1 (B) = {x ∈ X : f (x) ∈ B} is the pre-image (also inverse image) of B via f . A function f is said to be injective provided that the identity f (x) = f (y) implies x = y. Equivalently, x = y implies f (x) = f (y). f is said to be onto (or surjective, as well) if each y ∈ Y can be expressed as f (x) : x ∈ X . Any mapping f : X → f (X ) is trivially surjective. Assume that f is both injective and surjective. Then f is called a bijection. In this case, it can be defined the inverse map f −1 : Y → X by f −1 (y) as the unique x ∈ X such that f (x) = y. Further, it holds that f ( f −1 (y)) = y for all y ∈ Y and f −1 ( f (x)) = x for each x ∈ X . Let f : X → Y and g : Y → Z be a pair of functions. Their composition is denoted as g ◦ f : X → Z and defined by (g ◦ f )(x) = g( f (x)) for all x ∈ X . This can be naturally extended to any finite number of mappings. Next, we define some transformations of Rd with geometric meaning. A map f : Rd → Rd is called an isometry (also a congruence) provided that it satisfies the condition | f (x) − f (y)| = |x − y| for all x, y ∈ Rd . Equivalently, if it maintains distances. It is worth mentioning that isometries also keep angles and map sets to geometrically congruent ones. Isometries include the following families of maps as particular cases: translations, which can be expressed in the form f (x) = x + c, reflections, which assigns each point to its mirror image in a (d − 1)−dimensional space, and rotations (centered in d), satisfying the condition | f (x) − d| = |x − d| for all x ∈ Rd . In this way, the identity map ( f (x) = x for all x) stands as a rotation centered in d = 0. By a rigid motion, we shall understand a isometry obtained as a composition only involving translations and rotations. A similarity is a transformation f : Rd → Rd satisfying the equality | f (x) − f (y)| = κ · |x − y| for all x, y ∈ Rd . It is worth pointing out that each similarity maps sets to geometrically similar ones. A map f : Rd → Rd is said to be linear provided that the two following properties are satisfied:

1.1 Mathematical Foundations

5

(i) f (x + y) = f (x) + f (y) for every x, y ∈ Rd . (ii) f (αx) = α · f (x) for each α ∈ R and all x ∈ Rd . It is well known that linear transformations are equivalent to matrices. Thus, by a non-singular map, we shall understand a linear transformation under the condition f (x) = 0, if and only if, x = 0. Let g be a (non-singular) linear map and α ∈ Rd . By an affinity (also an affine transformation), we shall understand a transformation f : Rd → Rd where f (x) = α + g(x) for each x ∈ Rd . Notice that the contracting (resp. expanding) effect of affinities is not the same in each direction. Let T denote any of the following families of transformations: similarities, affinities, rigid motions, and isometries. Hence, (i) (ii) (iii) (iv)

for every f, g ∈ T , it holds that f ◦ g ∈ T ; Id ∈ T , where Id refers to the identity map; for all f ∈ T , f −1 ∈ T ; and f ◦ (g ◦ h) = ( f ◦ g) ◦ h for all f, g, h ∈ T .

In other words, T is a group endowed with the composition of functions in T . We say that f : X → Y is a Hölder function of exponent α provided that | f (x) − f (y)| ≤ κ · |x − y|α for all x, y ∈ X , where κ is a constant. In particular, if α = 1, then f is named a Lipschitz function. Also, f is called a bi-Lipschitz mapping if the next chain of inequalities holds: κ1 · |x − y| ≤ | f (x) − f (y)| ≤ κ2 · |x − y| for every x, y ∈ X , where 0 < κ1 ≤ κ2 < ∞. Next, we deal with the basics on limits and continuity of maps. Let f : X → Y be a function, where X is a subset of Rn and Y is a subset of Rm . Further, let α ∈ X . We shall understand that f (x) tends to L (also has limit L , converges to L or even goes to L ) as x converges to x0 , lim x→x0 f (x) = L (also f (x) → L as x → x0 ), provided that for all ε > 0, there exists δ > 0 such that |x − x0 | < δ implies | f (x) − L | < ε. Let f : X → R be a function. It is said that lim x→x0 f (x) = ±∞ (also f (x) → ±∞ as x → x0 ) if for a certain constant M , there exists δ > 0 such that the condition |x − x0 | < δ leads to M < f (x). Regarding the theory of fractal dimension, we are focused on the values that certain functions reach for small positive values of x. Let f : R+ → R be a function. If f (x) increases as x decreases, there exists lim x→0 f (x) and takes finite values or +∞. Similarly, if f (x) decreases as x decreases, there exists lim x→0 f (x) with a finite value or −∞. Since f (x) may vary widely for small values of x, it holds that lim x→0 f (x) does not need to exist. The lower/upper limits of f allow to tackle with such situations. More specifically, lim x→0 f (x) = lim x→0 inf{ f (x) : 0 < x < r }. It is worth noting that lim x→0 f (x) always exists since the set inf{ f (x) : 0 < x < r } equals −∞ for every r > 0 or increases as r decreases. We also have lim x→0 f (x) = lim x→0 sup{ f (x) : 0 < x < r }. Similarly, the upper limit of f exists lying in R ∪ {±∞}. In fact, both lower and upper limits provide information concerning the fluctuations of f as x is closer to 0. Thus, lim x→0 f (x) exists provided that lim x→0 f (x) = lim x→0 f (x) and equals that common quantity. Notice that the lower/upper limit of f : X ⊂ Rn → R as x → x0 ∈ X can be defined in similarly. The notation f (x) ∼ x s means that the function f (approximately) follows a power law of exponent s for small values of x.

6

1 Mathematical Background

Let f : X → Y be a function. We say that f is continuous at a given point a provided that f (a) = lim x→a f (x). Thus, f is said to be continuous (on X ) if f is continuous at every point x ∈ X . We shall understand that f is a homeomorphism (and hence, the two sets X, Y are homeomorphic) provided that f is a continuous bijection whose inverse map f −1 : Y → X is also continuous. f (x) A map f : R → R is differentiable at a point x if the limit lim h→0 f (x+h)− h  exists and is finite. In such a case, its value, denoted as f (x), is known as the derivative of the function f at x. Further, a map f is said to be continuously differentiable provided that f is differentiable with its differential being continuous. In this book, logarithms will be considered with respect to base e by default. Otherwise, it will be explicitly stated.

1.1.3 Foundations on Measure Theory In this subsection, we are focused on measures on Euclidean subsets. To properly define the concept of a measure on a set, first, we shall recall what we understand by a σ −algebra on such a set. In fact, a collection A of subsets of a set X is named a σ −algebra on X provided that the three conditions stated next are satisfied: (i) ∅, X ∈ A . (ii) For each A ∈ A , it holds that X \ A ∈ A . (iii) If {An }n∈N is a finite/countable sequence of subsets of X , then ∪n∈N An ∈ A . As a consequence of the definition of a σ −algebra on a set X , we have that if {An }n∈N is a finite/countable sequence of subsets of Rd , then ∩n∈N An ∈ A , and in addition, A ∩ B, A ∪ B, A \ B ∈ A for each A, B ∈ A . Let A be a σ −algebra on Rd . By a measure on Rd , we shall understand a set function (i.e., a function whose domain is a collection of sets) μ : A → R+ 0 ∪ {∞} under the following conditions: (i) μ(∅) = 0. (ii) μ is monotonic, i.e., for any two subsets A, B of Rd with A ⊆ B, it holds that μ(A) ≤ μ(B). (iii) μ is countably subadditive, namely, if {An }n∈N  is a countable (resp. finite) sequence of subsets of Rd , then μ(∪n∈N An ) ≤ ∞ n=1 μ(An ). (iv) In particular, if the (finite/countable) sequence {A n }n∈N consists of pairwise disjoint Borel subsets of Rd , then μ(∪n∈N An ) = ∞ n=1 μ(An ). Hence, for each A ∈ A , μ(A) is understood as the measure of the set A. It is worth mentioning that condition (iii) for a measure allows that some subsets A j may overlap. Notice also that our definition of a measure allows to verify that condition (iv) also stands for classes of sets other than just Borel sets. For instance, that equality is also satisfied for sets consisting of continuous images (i.e., images of sets by continuous functions) of Borel sets.

1.1 Mathematical Foundations

7

It is worth noting that the definition of a measure previously stated corresponds to what is usually understood as an outer measure on Rd with respect to the Borel sets are measurable. The support of a measure μ on a set X is the smallest closed set S such that μ(Rd \ S ) = 0. Thus, the support of any measure is always closed and it holds that x ∈ S , if and only if, μ(B(x, ε)) > 0 for every ε > 0. Accordingly, μ is a measure on X provided that X contains S , the support of μ. Let F be a bounded subset of Rd . We say that a measure μ on Rd is a mass distribution on F provided that F contains the support of μ and μ(F) ∈ (0, ∞). In this way, μ(A) gives the mass of any subset A of F. Thus, we can consider a finite mass distribution on A ⊂ X and “someway” extend it to the whole X , so we have a mass distribution on X . Hence, we get a measure on X .

1.2 Hausdorff Measure and Dimension In this section, we sketch the definitions of both Hausdorff measure and dimension. For additional details concerning them, we encourage the reader to consult [29, Sects. 2.1 and 2.2]. The first mathematician to define a measure based on coverings of certain sets was C. Carathéodory in [20]. Several years later (1919), F. Hausdorff applied that approach to define the measures that now bear his name and proved that the middle third Cantor set C possesses a positive and finite (Hausdorff) measure with dimension 2 (c.f. [42]). Some analytical properties regarding both Hausdorff measure equal to log log 3 and dimension were thoroughly explored by Besicovitch and his coauthors during the twentieth century. The Hausdorff dimension, which is the oldest definition of fractal dimension, satisfies the best analytical properties as a dimension function. That classical fractal dimension could be defined for any subset of a Euclidean (resp. metrizable) space. Moreover, we should mention here that its definition is based on a measure which makes it quite appropriate from a mathematical viewpoint. However, it involves several disadvantages, especially from the point of view of applications, since it can be hard to calculate or to estimate empirically. Thus, the Hausdorff dimension is “better” from a theoretical approach, whereas the box dimension is “better” to deal with a wide range of applications. Next, we recall the analytical construction involving the Hausdorff dimension. First, we introduce the concept of a δ−cover for any subset of a metric space. Definition 1.2.1 Let (X, ρ) be a metric space, F a subset of X , and δ > 0. By a δ−cover of F, we shall understand a countable (or finite) family of subsets {Ui }i∈I such that F ⊆ ∪i∈I Ui , where diam (Ui ) ≤ δ for all i ∈ I . In addition, we shall denote Cδ (F) the class of all δ−covers of F.

8

1 Mathematical Background

Let s ≥ 0. The calculation of the Hausdorff measure of F consists of minimizing the sum of the s−powers of the diameters of all the subsets of any δ−cover of F, where s will equal (a posteriori) the fractal dimension of F. Accordingly, the following quantity must be defined: Hδs (F)

= inf

 

 diam (Ui ) : {Ui }i∈I ∈ Cδ (F) . s

(1.1)

i∈I

It is noteworthy that when δ decreases, then the class Cδ (F) consisting of all δ−covers of F is reduced, so the measure of F increases. Letting δ → 0 in previous Eq. (1.1), the following expression always exists: H s (F) = lim Hδs (F),

(1.2)

δ→0

which is called the s−dimensional Hausdorff measure of F. That limit may reach (and usually equals) the values 0 and ∞. Next, we verify that H s is, in fact, a measure (c.f. [31, Exercise 3.3]). Theorem 1.2.2 The set function H s provided in Eq. (1.2) is a measure.

Proof Let s ≥ 0. The three following hold: • For all 0 < ε ≤ δ, it holds that the empty set ∅ can be covered throughout a ε−diameter single set. Thus, 0 ≤ H s (∅) ≤ εs for all ε > 0, leading to H s (∅) = 0. Hence, H s (∅) = limδ→0 H s (∅) = 0. • Let E, F be two subsets of X such that E ⊆ F. Therefore, for all δ > 0, it is clear that all δ−covers of F are also of E by taking infima over Cδ (F). Therefore, Hδs (E) ≤ Hδs (F) and letting δ → 0, we conclude H s (E) ≤ H s (F). collection of disjoint Borel sets and assume, with• Let {Fi }i∈I be a countable  +∞ Hδs (Fi ) 0, let out loss of generality, that i=1 ε s s {Ui, j }i∈I, j≥1 be a δ−cover of each Fi such that +∞ j=1 diam (Ui, j ) ≤ Hδ (Fi ) + 2i . +∞ Thus, {Ui, j }i, j becomes a δ−cover of ∪i=1 Fi and, in addition, for all ε > 0, Hδs

+∞ +∞  +∞   +∞   s   ε Hδs (Fi ) + i ∪i=1 Fi ≤ diam Ui, j ≤ 2 i=1 j=1 i=1

=ε+

+∞  i=1

Hδs (Fi )

≤ε+

+∞  i=1

H s (Fi ).

1.2 Hausdorff Measure and Dimension

9

Finally, letting δ → 0, we have +∞  +∞   +∞   H s ∪i=1 Fi = lim Hδs ∪i=1 Fi ≤ H s (Fi ). δ→0

i=1

It is worth mentioning that the Hausdorff measure generalizes the classical Lebesgue measure for Euclidean subsets. More specifically, if F is a Borel subset of Rd , then the next equality stands: d

π2 H (F) = cd · vol (F), where cd = d  d  2 · 2 ! d

d

is a constant that gives the volume of a 1−diameter d−dimensional ball. Here, (·)! denotes the factorial function extended to non-integer arguments. Notice also that H 0 (F) = Card (F), namely, the cardinal number of F. Further, H 1 (F) is the length of a smooth curve F, H 2 (F) = π4 · area (F) if F is a smooth surface, H 3 (F) = 43 π · vol (F), and in general, H k (F) = ck · vol k (F), if F is a smooth k−dimensional submanifold of Rd , namely, a k−dimensional surface in the classical sense (k ≤ d). A first scaling property regarding the s−dimensional Hausdorff measure is stated next. Proposition 1.2.3 Let α > 0 and F be a subset of Rd . Then H s (α F) = α s · H s (F), where α F = {α · x : x ∈ F} consists of scaling the set F by a dilation ratio α.

Proof Let {Ui }i∈I ∈ Cδ (F). Then it is clear that {αUi }i∈I is a α · δ−cover of α F. Accordingly,  {diam (αUi )s : {Ui }i∈I ∈ Cδ (F)}  {diam (Ui )s : {Ui }i∈I ∈ Cδ (F)}. = αs ·

s (α F) ≤ Hα·δ

(1.3)

i∈I

Taking infima in Eq. (1.3) over the class Cδ (F) of all δ−covers of F, we have s (α F) ≤ α s · Hδs (F). Hα·δ

(1.4)

10

1 Mathematical Background

Letting δ → 0 in Eq. (1.4), H s (α F) ≤ α s · H s (F). To deal with the opposite inequality, just replace F by α F and α by α1 . In fact, if {Ui }i∈I is a δ−cover of α F, then { α1 Ui }i∈I is a αδ −cover of F. Hence, s 



1 Ui : {Ui }i∈I ∈ Cδ (α F) diam α  1  diam (Ui )s : {Ui }i∈I ∈ Cδ (α F) . = s · α i∈I

s Hδ/α (α F) ≤

(1.5)

Taking infima in Eq. (1.5) over the class of all δ−covers of α F, the following inequality stands: 1 HHs δ/α (F) ≤ s · Hδs (α F), α or equivalently, s α s · Hδ/α (F) ≤ Hδs (α F).

(1.6)

Finally, let δ → 0 in Eq. (1.6) to conclude α s · H s (F) ≤ H s (α F). From Proposition 1.2.3, it holds that H s scales by a factor λs as in the classical sense, the length of a curve is multiplied by α if a dilation ratio equal to α is applied to that curve (s = 1). Similarly, the area of a plane region scales by λ2 , and the volume of a 3−dimensional subset is multiplied by λ3 , as well. Additionally, the following result throws upper bounds regarding the s−dimensional Hausdorff measures (up to certain constants) of subsets under the action of Hölder type mappings we define next. Definition 1.2.4 Let F be a subset of Rd and f : F → Rk a map. We shall understand that f is under the Hölder condition of exponent λ > 0 provided that the following inequality stands: | f (x) − f (y)| ≤ κ · |x − y|λ ,

(1.7)

where κ is named the Hölder constant associated with f . It is worth mentioning that Hölder maps do generalize the concept of Lipschitz mappings. In fact, a Lipschitz map is just a map under the Hölder condition of exponent λ = 1. Recall that every differentiable map with bounded differential is Lipschitz (due to the Mean Value Theorem). Moreover, observe that all maps under the Hölder condition are continuous.

1.2 Hausdorff Measure and Dimension

11

Proposition 1.2.5 Let F be a subset of Rd and f : F → Rk be a map under the Hölder condition of exponent λ > 0. Then for all s ≥ 0, it holds that H s/λ ( f (F)) ≤ κ s/λ · H s (F),

(1.8)

where κ is the Hölder constant associated with f . In particular, if f is Lipschitz, then H s ( f (F)) ≤ κ s · H s (F).

Proof Let {Ui }i∈I ∈ Cδ (F). Thus, we have that { f (F ∩ Ui )}i∈I is a κ · δ λ −cover of f (F) since diam ( f (F ∩ Ui )) ≤ κ · diam (Ui )λ . Therefore,  {diam ( f (F ∩ Ui ))s/λ : {Ui }i∈I ∈ Cδ (F)}  {diam (Ui )s : {Ui }i∈I ∈ Cδ (F)}. ≤ κ s/λ · Taking infima over the class Cδ (F), the previous expression leads to s/λ

Hκ·δλ ( f (F)) ≤ κ s/λ · Hδs (F). Letting δ → 0, we have H s/λ ( f (F)) ≤ κ s/λ · H s (F). To deal with the Lipschitz case, just let λ = 1 in Eq. (1.8). Particular cases of Lipschitz maps, namely, maps under the Hölder condition of exponent λ = 1, are isometries, translations, and rotations. In fact, if f is an isometry, then it holds that | f (x) − f (y)| = |x − y| for all x, y ∈ F, leading to H s ( f (F)) = H s (F). Moreover, it is noteworthy that Hausdorff measure remains invariant by translations (H s (F + c) = H s (F), where F + c = {x + c : x ∈ F}) and rotations, as well. Next step is to deal with the definition of the Hausdorff dimension. In fact, from Eq. (1.1), it is clear that the quantity Hδs (F) is nonincreasing with s for every subset F and each δ ∈ (0, 1). Accordingly, H s (F) is also nonincreasing with s (c.f. Eq. (1.2)). Following the above, let t > s and {Ui }i∈I ∈ Cδ (F). Then it is clear that 

diam (Ui )t = =

 

diam (Ui )t+s−s

diam (Ui )t−s · diam (Ui )s  ≤ δ t−s · diam (Ui )s , where the previous sums have been performed over the class Cδ (F). Accordingly,   {diam (Ui )t : {Ui }i∈I ∈ Cδ (F)} ≤ δ t−s · {diam (Ui )s : {Ui }i∈I ∈ Cδ (F)}.

12

1 Mathematical Background

∞

HHs (F)

Fig. 1.1 Graph representation of H s (F) versus s for a set F. The Hausdorff dimension is the point s ≥ 0 where H s (F) “jumps” from ∞ to 0

0 dimH (F)

s Taking infima over the class Cδ (F) yields Hδt (F) ≤ δ t−s · Hδs (F). Letting δ → 0, we have H t (F) ≤ δ t−s · H s (F). Thus, if H s (F) < ∞ provided that s < t, then H t (F) = 0. Therefore, the point s ≥ 0 where H s (F) “jumps” from ∞ to 0 is named the Hausdorff dimension of F (also known as the Hausdorff–Besicovitch dimension of F), c.f. Fig. 1.1. In addition, the Hausdorff dimension of F can be described in the following terms: dimH (F) = sup{s ≥ 0 : H s (F) = ∞} = inf{s ≥ 0 : H s (F) = 0}, or equivalently,

H s (F) =

∞ if s < dimH (F) 0 if s > dimH (F).

(1.9)

(1.10)

In particular, if s = dimH (F), then H s (F) can reach the values 0, ∞, and even it may occur that H s (F) ∈ (0, ∞). Falconer refers to those Borel sets satisfying the condition H s (F) ∈ (0, ∞) as s−sets (c.f. [29, Sect. 2.2]). Hausdorff dimension satisfies several properties as a fractal dimension function we should be mirrored in when giving rise to new models of fractal dimension. Next, they are listed for a generic dimension function dim. Fractal dimension properties 1.2.6 (i) (ii) (iii) (iv)

Monotonicity: if E ⊆ F, then dim(E) ≤ dim(F). Finite stability: dim(E ∪ F) = max{dim(E), dim(F)}. Countable stability: dim (∪i∈I Fi ) = supi∈I {dim(Fi )} for I countable. Countable sets: if F is countable (or finite), then dim(F) = 0.

1.2 Hausdorff Measure and Dimension

13

(v) Closure dimension property: there exists a subset F of X such that dim(F) = dim(F), where F denotes the closure of F, namely, the smallest closed subset of Rd containing F. (vi) Geometric invariance: if f : Rd → Rd is a similarity, affinity, rotation or translation, then dim(F) = dim( f (F)). (vii) Lipschitz invariance: if f is bi-Lipschitz, then dim(F) = dim( f (F)). (viii) Open sets: if F ⊂ Rn is open, then dim(F) = n. (ix) Smooth manifolds: if F is a m−dimensional manifold, then dim(F) = m. Clearly, (iii) ⇒ (ii) and (vii) ⇒ (vi). As Falconer points out in [29, Chap. 3], all definitions of fractal dimension are monotonic and most of them are stable. However, some of them are not countably stable and can be > 0 for certain countable sets. In addition, we should mention here that all common dimensions are Lipschitz invariant (and hence, geometrically invariant). Moreover, distinct definitions of fractal dimension may throw different information regarding Lipschitz equivalent sets. In this book, though, we shall mainly focus on verifying fractal dimension properties (i)–(v) regarding the fractal dimension models for a fractal structure we shall explore in forthcoming chapters. It is also worth mentioning that a dimension function dim not satisfying the closure dimension property (c.f. Fractal dimension properties 1.2.6(v) means that for all subset F of Rd , it holds that dim(F) = dim(F). Interestingly, if a dimension function does not satisfy the closure dimension property, then some consequences (not desirable at all) will follow as we point out next. Corollary 1.2.7 Let dim : Rd → R+ 0 be a dimension function not satisfying the closure dimension property. The two following stand: (i) There exist countable subsets F of Rd such that dim(F) > 0. (ii) dim is not countably stable.

Proof Let F = Q ∩ [0, 1]. Then (i) F is a countable subset which is also dense on [0, 1]. Hence, dim(F) = dim(F) = 1 which equals the length of the closed unit subinterval. (ii) In addition, we can write F = ∪i∈I Fi , where for all i ∈ I , Fi = {qi } : qi ∈ F are single sets. Thus, dim(Fi ) = 0 for all i ∈ I , and hence, supi∈I {dim(Fi )} = 0, whereas dim(F) = 1 by Corollary 1.2.7(i). Consequently, Corollary 1.2.7 highlights that dimension functions not lying under the closure dimension property are distanced from a Hausdorff dimension behavior. Following the above, next we theoretically justify several properties that hold for Hausdorff dimension. That result will be useful afterward for comparison purposes involving the fractal dimension models for a fractal structure introduced afterward.

14

1 Mathematical Background

Theorem 1.2.8 The following from Fractal dimension properties 1.2.6 are satisfied, in particular, by Hausdorff dimension: (1) Monotonicity: if E ⊆ F, then dimH (E) ≤ dimH (F). (2) Countable stability: if {Fi }i∈I is a countable sequence of sets, then dimH (∪i∈I Fi ) = sup {dimH (Fi )} .

(1.11)

i∈I

(3) Countable sets: if F is countable, then dimH (F) = 0. (4) Closure dimension property: there exists a subset F of Rd such that dimH (F) = dimH (F).

Proof (1) Let E ⊆ F. Since the set function H s is a measure (c.f. Theorem 1.2.2), then we have H s (E) ≤ H s (F) for all s ≥ 0. The result follows. (2) Let {Fi }i∈I be a countable sequence of subsets of X . Then for all j ∈ I , it is clear that F j ⊆ ∪i∈I Fi . Thus, dimH (F j ) ≤ dimH (∪i∈I Fi ) for every j ∈ I since Hausdorff dimension is monotonic (c.f. Theorem 1.2.8). Hence, sup j∈I {dimH (F j )} ≤ dimH (∪i∈I Fi ). Conversely, let s > dimH (Fi ) for all i ∈ I and let us verify s that s > dimH (∪i∈I Fi ). In fact, since ss > dimH (Fi ), then sH (Fi ) = 0 for all s i ∈ I . Hence, H (∪i∈I Fi ) ≤ i∈I H (Fi ) = 0, since H is a measure. Thus, H s (∪i∈I Fi ) = 0 so that s > dimH (∪i∈I Fi ), as expected. (3) Let F be countable. Then we can write F = ∪i∈I Fi , with all the points of F, Fi , being 0−sets, namely, H 0 (Fi ) = Card (Fi ) = 1 for every i ∈ I . Hence, dimH (Fi ) = 0 for all i ∈ I . We conclude that dimH (F) = 0 since Hausdorff dimension is countably stable (c.f. Theorem 1.2.8(3)). (4) Let F = Q ∩ [0, 1]. Since F is countable, then we have dimH (F) = 0 due to Theorem 1.2.8(3)). On the other hand, it holds that F = [0, 1] since F is dense on [0, 1]. Accordingly, dim(F) equals the length of the closed unit subinterval, namely, dim(F) = 1. Observe that the countable stability property satisfied by Hausdorff dimension becomes the key to prove its closure dimension property. It is worth mentioning that the invariance of the Hausdorff dimension of sets that lie under the action of geometric transformations stands by Proposition 1.2.5 regarding Hausdorff measures. More specifically, the following result follows. Proposition 1.2.9 Let F be a subset of Rd and f : F → Rk be a map under the Hölder condition of exponent λ > 0. Then

1.2 Hausdorff Measure and Dimension

dimH ( f (F)) ≤

15

1 · dimH (F). λ

Proof Let s > dimH (F). Then H s (F) = 0. Thus, H s/λ ( f (F)) ≤ κ s/λ · H s (F) = 0, where κ is the Hölder constant associated with f . Hence, we have H s/λ ( f (F)) = 0. Accordingly, it holds that λs ≥ dimH ( f (F)) for all s > dimH (F). The so-called geometric invariance regarding Hausdorff dimension (c.f. Fractal dimension properties 1.2.6(vi)) follows immediately as a consequence of Proposition 1.2.9. In fact, the proof of the next result becomes now straightforward. Proposition 1.2.10 Let F be a subset of Rd and f : F → Rk be a map. The two following hold: (i) If f is Lipschitz, then dimH ( f (F)) ≤ dimH (F). (ii) If f is bi-Lipschitz, then dimH ( f (F)) = dimH (F).

Proof (i) Just apply Proposition 1.2.9 for Hölder exponent λ = 1. (ii) The inequality dimH ( f (F)) ≤ dimH (F) can be proved similarly to Proposition 1.2.10(i). Conversely, let us consider the inverse map f −1 : f (F) → F which is also Lipschitz. Once again, Proposition 1.2.9 for Hölder exponent λ = 1 leads to the result. Proposition 1.2.10 allows to distinguish between sets that are distinct from a Fractal Geometry viewpoint. In fact, if the Hausdorff dimensions of two subsets do not match, then we cannot find out a bi-Lipschitz transformation between them which maps the points of one of the sets into the other. Accordingly, these sets are not bi-Lipschitz equivalent. Moreover, in several cases, Hausdorff dimension throws useful information regarding the topology of certain subsets. More specifically, it holds that Euclidean subsets with Hausdorff dimension < 1 are totally disconnected. Proposition 1.2.11 Let F be a subset of Rd such that dimH (F) < 1. Then F is totally disconnected.

Proof Let x, y ∈ F : x = y and define f : Rn → R+ 0 by f (z) = |x − z|. Thus, it is clear that the map f is Lipschitz since | f (z) − f (ω)| ≤ |ω − z| for every ω, z ∈ F. According to Corollary 1.2.10(i), we have dimH ( f (F)) ≤ dimH (F) < 1. Hence, H 1 (F) = 0, namely, F is a subset of Rd whose length is equal to 0. Thus, it has a dense complement. Therefore, we can write F = {z ∈ F : |z − x| < γ } ∪ {z ∈ F :

16

1 Mathematical Background

|z − x| > γ }, where we have chosen γ ∈ / f (F) : γ ∈ (0, f (y)). In other words, F is totally disconnected since it is contained into two disjoint open sets, each of them containing the points x and y, respectively. Table 1.1 summarizes some theoretical properties satisfied by both box and Hausdorff dimensions. Along this work, that table will be extended by adding those properties satisfied by the new fractal dimension models introduced in forthcoming chapters. It is worth mentioning that Hausdorff dimension of a subset F can be calculated throughout δ−covers of F consisting of open balls. Proposition 1.2.12 Let δ > 0, (X, ρ) be a metric space, and F be a subset of X . In addition, let us consider    diam (Bi )s : {Bi }i∈I ∈ Cδ (F) by spherical balls Bδs (F) = inf i∈I

and define B s (F) = lim Bδs (F). δ→0

Thus, the point s ≥ 0 where B s (F) “jumps” from ∞ to 0 equals the Hausdorff dimension of F.

Proof Indeed, since all the coverings involved in the calculation of Bδs (F) are, in particular, δ−coverings, then by definition of H s (F) (c.f. Eq. (1.1)), it becomes clear that Hδs (F) ≤ Bδs (F) for every δ > 0 and all s ≥ 0. Accordingly, H s (F) ≤ B s (F). Additionally, for all {Ui }i∈I ∈ Cδ (F) and each Ui ∈ {Ui }i∈I , let Bi = B(x, diam (Ui )) : x ∈ Ui a ball centered in Ui of radius diam (Ui ) ≤ δ. Thus, {Bi }i∈I is a δ−cover such that Ui ⊆ Bi for all i ∈ I . Hence,  i∈I

diam (Bi )s ≤

 i∈I

(2 · diam (Ui ))s = 2s ·



diam (Ui )s .

i∈I

s Taking infima, it holds that B2δ (F) ≤ 2s · Hδs (F). Therefore, H s (F) ≤ B s (F) ≤ s s 2 · H (F) by letting δ → 0. Finally, observe that

dimH (F) = sup{s ≥ 0 : B s (F) = ∞} = inf{s ≥ 0 : B s (F) = 0}. In fact, similarly to Theorem 1.2.2, it can be proved that B s (F) is also a measure. Moreover, as well as in Proposition 1.2.12, we have just proved that Hausdorff dimension can be calculated throughout δ−coverings consisting of open balls, it is noteworthy that coverings containing open (resp. closed) sets become also valid for

1.2 Hausdorff Measure and Dimension

17

Table 1.1 This table summarizes some properties as dimension functions satisfied by box and Hausdorff dimensions (c.f. Fractal dimension properties 1.2.6) Monotonicity Finite stability Countable Dimension 0 Closure stability for countable dimension sets property dimB dimH

✓ ✓

✓ ✓

✓

✓

✓

Hausdorff dimension calculation purposes. Interestingly, if F is compact, then for every collection of subsets {Fi }i∈I such that F ⊆ ∪i∈I Fi , we can choose Fi to be open just through the collection of balls B(Fi , ε) : 0 < ε < diam (Fi ), and hence, take a finite subcover for F. Accordingly, we are able to calculate Hausdorff dimension by finite δ−covers, namely, coverings containing a finite number of sets.

1.3 Box Dimension Fractal dimension is one of the main tools considered to deal with fractals since it is a single number which throws useful information regarding their complexity when being explored with enough level of detail. We would like to point out that fractal dimension is usually understood as classical box (also box-counting) dimension. According to Falconer (c.f. [29, Sect. 3.1]), such a dimension is known by several terms including information dimension, Kolmogorov entropy, capacity dimension, entropy dimension, metric dimension, and logarithmic density, among others. The origins of the box dimension go back to the 30s, when that dimension was explored by the pioneers of the Hausdorff dimension. Anyway, the current definition of box dimension was provided in [68]. Though Hausdorff dimension is the basic theoretical reference for a dimension function in Fractal Geometry, the box dimension is mainly used to deal with empirical applications since it is easy to be calculated or estimated for a finite range of scales as it happens with practical applications. It is worth noting that popularity of box dimension is due to the possibility of its effective calculation in Euclidean contexts. In fact, the box dimension can be approximated throughout the slope of a regression line for a log–log graph plotted over a discrete range of scales. Next, we recall the standard definition of box dimension we shall work with hereafter. Definition 1.3.1 Let δ > 0 and F be a non-empty bounded subset of Rd . (i) We shall denote Nδ (F) the smallest number of sets of diameter at most δ that cover F.

18

1 Mathematical Background

(ii) The (lower/upper) box dimension of F is given by the following (lower/ upper) limit: log Nδ (F) dimB (F) = lim . (1.12) δ→0 − log δ Interestingly, it holds that box dimension can be calculated via equivalent definitions of Nδ (F). To highlight that fact, let us introduce, first, the concept of δ−cube in Rd . Definition 1.3.2 Let δ > 0. By a δ−cube in Rd , we shall understand a set of the form [k1 δ, (k1 + 1)δ] × · · · × [kd δ, (kd + 1)δ] : k1 , . . . , kd ∈ Z.

Theorem 1.3.3 Let δ > 0 and F be a non-empty bounded subset of Rd . The following definitions of Nδ (F) are equivalent for box dimension calculation purposes: (i) (ii) (iii) (iv) (v)

The smallest number of sets of diameter at most δ that cover F. The smallest number of closed balls of radius δ that cover F. The smallest number of cubes of side δ that cover F. The number of δ−cubes that intersect F. The largest number of disjoint balls of radius δ with centers in F.

Proof (i) ⇔ (iv). Let Nδ (F) be the smallest number of sets of diameter at most δ that cover F and Mδ (F) the number √ of δ−cubes that intersect F. Since the diameter of each δ−cube in Rd equals δ n, then it becomes clear that Nδ√n (F) ≤ Mδ (F). Conversely, just notice that each subset F of Rd with diam (F) ≤ δ is contained into 3d cubes of side δ. Accordingly, Mδ (F) ≤ 3d · Nδ (F). Both inequalities allows us to guarantee that log Nδ√n (F) log Mδ (F) log Nδ (F) + log 3d ≤ . √ √ ≤ log d − log δ d − log δ − log δ Letting δ → 0, we have lim

δ→0

log Nδ (F) log Mδ (F) log Nδ (F) ≤ lim ≤ lim , δ→0 δ→0 − log δ − log δ − log δ

1.3 Box Dimension

19

and hence the (lower/upper) box dimensions calculated throughout Nδ (F) or Mδ (F) are the same. (i) ⇔ (v). Let Nδ (F) be the smallest number of sets of diameter at most δ that cover F and Lδ (F) be the largest number of disjoint balls of radius δ with centers in F. Let B = {B1 , . . . , BL δ (F) } be a collection consisting of Lδ (F) disjoint balls of radius δ with centers in F. Thus, if x ∈ F, then there exists i = 1, . . . , Lδ (F) such that d(x, Bi ) < δ. In fact, otherwise, B(x, δ) should be added to B. But in that case, B would not be the largest number of such balls, a contradiction. Hence, we can choose Lδ (F) balls concentric with the Bi of radius 2δ. They cover F, so that N4δ (F) ≤ Lδ (F). Conversely, let {B1 , . . . , BL δ (F) } be a set consisting of disjoint balls of radius δ with centers in F and {U1 , . . . , Uk } be a collection consisting of sets with diameter ≤ δ that cover F. Notice that the U j do cover the center of each Bi . Thus, each Bi must contain at least one U j : i = 1, . . . , Lδ (F). Moreover, there must be so many U j as Bi since the Bi are disjoint. Consequently, Lδ (F) ≤ Nδ (F). Accordingly, log Lδ (F) log Nδ (F) log N4δ (F) ≤ ≤ . log 4 − log 4δ − log δ − log δ Letting δ → 0, lim

δ→0

log N4δ (F) log Lδ (F) log Nδ (F) ≤ lim ≤ lim . δ→0 δ→0 log 4 − log 4δ − log δ − log δ

Thus, the (lower/upper) box dimensions calculated by Lδ (F) or Nδ (F) are be equal. (i) ⇔ (ii) and (i) ⇔ (iii) may be dealt with in the same way throughout similar arguments. It is noteworthy that box dimension can be calculated via Nδ (F) as the number of δ = 1/2n −cubes that intersect F with n ∈ N (c.f. Theorem 1.3.3(iv). This remark would become essential to tackle with the arguments appeared in upcoming Sect. 3.2. More generally, the next remark will be useful afterward. Remark 1.3.4 To calculate the (lower/upper) box dimension of any subset F of Rd , let δ → 0 throughout any decreasing sequence {δn }n∈N whose general term satisfies κ · δn ≤ δn+1 for all natural number n and c ∈ (0, 1) being a constant. In particular, we can choose δn = κ n .

Proof In fact, let δk+1 ≤ δ < δk . Thus, log Nδk+1 (F) log Nδk+1 (F) log Nδk+1 (F) log Nδ (F) ≤ = ≤ . δk+1 − log δ − log δk log κ − log δk+1 − log δk+1 + log δk

20

1 Mathematical Background

Letting δ → 0 and k → ∞, we have lim

δ→0

log Nδ (F) log Nδk (F) . ≤ lim k→+∞ − log δk − log δ

To deal with the opposite inequality, just observe that log Nδ (F) = sup δ→0 − log δ lim



log Nδk (F) : {δk }k≥1  0 . k→+∞ − log δk lim

The same arguments above allow to deal with the case of lower limits. Regarding fractal dimension properties 1.2.6, next we shall explore what properties are satisfied by standard box dimension as well as it was carried out for Hausdorff dimension in Theorem 1.2.8. This will allow us to better understand the theoretical behavior of box dimension as a dimension function. Theorem 1.3.5 The following from Fractal dimension properties 1.2.6 are satisfied, in particular, by box dimension: (1) Monotonicity: both lower and upper box dimensions are monotonic. (2) Finite stability: upper box dimension is finitely stable whereas lower box dimension is not. (3) Closure dimension property: neither lower nor upper box dimensions satisfy the closure dimension property. In other words, for all bounded subset F of Rd , it holds that dimB (F) = dimB (F) and dimB (F) = dimB (F). (4) Countable stability: neither lower nor upper box dimensions are countably stable. (5) Zero dimension for countable sets: there exist countable subsets F of Rd such that dimB (F) > 0.

Proof (1) Clearly, if E ⊆ F, then it holds that Nδ (E) ≤ Nδ (F). Hence, log Nδ (F) log Nδ (E) ≤ . − log δ − log δ Taking (lower/upper) limits as δ → 0, the result follows. (2) Let E, F be two subsets. First, since both E, F ⊆ E ∪ F, then it is clear that Nδ (E), Nδ (F) ≤ Nδ (E ∪ F). Thus, multiplying both quantities by a factor −1/ log δ and taking upper limits as δ → 0, we have dimB (E), dimB (F) ≤ dimB (E ∪ F). Therefore, max{dimB (E), dimB (F)} ≤ dimB (E ∪ F). Conversely, notice that Nδ (E ∪ F) ≤ Nδ (E) + Nδ (F). Accordingly,

1.3 Box Dimension

21

dimB (E ∪ F) ≤ dimB (E) + dimB (F). However, this equality is not satisfied by upper box dimensions. (3) Let F be any bounded subset of Rd and {B1 , . . . , Bn } be a finite collection of closed balls with radii δ that cover F, namely, F ⊆ ∪i∈I Bi . Thus, ∪i∈I Bi is a closed set containing F. Since F is the smallest closed set containing F, then ∪i∈I Bi also contains F. Accordingly, if Nδ (F) is chosen to be the smallest number of closed balls of radius δ that cover F (c.f. Theorem 1.3.3(iii)), then that number also allows to cover F. (4)–(5) Both statements follow immediately from Theorem 1.3.5(3) and Corollary 1.2.7. A first theoretical connection involving both Hausdorff and box dimensions is provided next for Euclidean bounded subsets. Theorem 1.3.6 Let F be a non-empty bounded subset of Rd . Then dimH (F) ≤ dimB (F) ≤ dimB (F).

Proof Let Nδ (F) be the smallest number of sets of diameter at most δ that cover F. Since    Nδ (F) · δ s = inf δ s : {Ui }i∈I ∈ Cδ (F) i∈I

with all these δ−covers being finite, then it is clear that Hδs (F) ≤ Nδ (F) · δ s for all δ > 0. Let s ≥ 0 : H s (F) > 1. Thus, s · log δ + log Nδ (F) > 0. Hence, for δ small enough the result follows. In fact, observe that log Nδ (F) = dimB (F) ≤ dimB (F). δ→0 − log δ

s ≤ lim

Additionally, an upper bound regarding the (lower) box dimension of any (bounded) Euclidean subset can be found out in terms of certain coverings consisting of small sets. Theorem 1.3.7 Let F be a non-empty bounded subset of Rd . Assume that F can be covered by Nn (F) sets with diameter ≤ δn , where limn→∞ δn = 0. The three following hold: (i)

log Nn (F) . n→∞ − log δn

dimH (F) ≤ dimB (F) ≤ lim

(1.13)

22

1 Mathematical Background

(ii) Additionally, if there exists a constant κ ∈ (0, 1) such that κ · δn ≤ δn+1 , then the following bound regarding the upper box dimension of F can be reached: log Nn (F) dimB (F) ≤ lim . n→∞ − log δn (iii) If limn→∞ δns · Nn (F) is bounded, then the s−dimensional Hausdorff measure of F remains finite.

Proof Let Nδ (F) be the smallest number of sets of diameter at most δ that cover F. Then (i) dimH (F) ≤ dimB (F) due to Theorem 1.3.6. Moreover, it is clear that Nδn (F) ≤ Nn (F) for all n ∈ N by definition of Nδ (F). Let δ > 0 : δ < δn . Thus, log Nn (F) log Nδ (F) ≤ . − log δ − log δn Letting δ → 0,

log Nδ (F) log Nn (F) ≤ lim . n→∞ − log δn δ→0 − log δ lim

log Nn (F) . − log δn (ii) The additional hypothesis regarding the existence of a constant κ ∈ (0, 1) such that κ · δn ≤ δn+1 allows the calculation of the box dimension of F throughout a decreasing sequence {δn }n∈N due to Remark 1.3.4. Thus, we can write Accordingly, we conclude dimB (F) ≤ limn→∞

dimB (F) = lim

n→∞

log Nδn (F) . − log δn

Since Nδn (F) is the smallest number of sets of diameter ≤ δn that cover F, then it holds that Nδn (F) ≤ Nn (F). This clearly implies that dimB (F) = lim

n→∞

log Nδn (F) log Nn (F) ≤ lim . n→∞ − log δn − log δn

(iii) It is clear that Hδsn (F) ≤ δns · Nn (F). Letting n → ∞, we have H s (F) ≤ limn→∞ δns · Nn (F) < ∞. The following result, known as Mass distribution principle (c.f. [29, Sect. 4.1]) throws another chain of inequalities regarding both Hausdorff and (lower/upper) box dimensions. To deal with, recall that a mass distribution on a subset F of Rd is a measure whose support lies in F and satisfies that μ(F) ∈ (0, ∞). Interestingly, this result will play a relevant role to prove the classical Moran’s Theorem for IFS-attractors.

1.3 Box Dimension

23

Mass distribution principle 1.3.8 Let μ be a mass distribution on F and assume that for a certain s ≥ 0, there exist κ, δ > 0 such that μ(U ) ≤ κ · diam (U )s for all subset U of Rd such that diam (U ) ≤ δ. The two following hold: (i) κ1 · μ(F) ≤ H s (F). (ii) Moreover, for that value of s, s ≤ dimH (F) ≤ dimB (F) ≤ dimB (F).

Proof (i) Let {Ui }i∈I ∈ Cδ (F). Since μ is a mass distribution, it holds that 0 < μ(F) ≤ μ (∪i∈I Ui ) ≤

 i∈I

μ(Ui ) ≤ κ ·



diam (Ui )s .

i∈I

 Accordingly, 0 < μ(F) ≤ κ · i∈I diam (Ui )s . Taking infima over the class Cδ (F) leads to 0 < μ(F) ≤ κ · Hδs (F). Thus, μ(F)/κ ≤ Hδs (F). Finally, let δ → 0 to reach the expected inequality, namely, μ(F)/κ ≤ H s (F). (ii) To tackle with, observe that H s (F) > 0 since μ is a mass distribution. Hence, s ≤ dimH (F). Finally, apply Theorem 1.3.6 to complete the result. It is worth mentioning that the Mass distribution principle 1.3.8 stands, in particular, for μ being a mass distribution on a subset F of Rd .

1.4 Iterated Function Systems Self-similar sets are a kind of fractals that can be always endowed with a natural fractal structure. In this section, we shall recall a standard procedure to construct attractors of iterated function systems. In addition, we shall describe that natural fractal structure for any self-similar set. It is noteworthy that all the notions, results, and properties provided below become essential to tackle with the content provided in upcoming chapters. Let (X, ρ) be a complete metric space and F be a subset of X . By a Lipschitz mapping, we shall understand a map f : F → F satisfying the following condition: ρ( f (x), f (y)) ≤ c · ρ(x, y) for all x, y ∈ F, where c > 0 is known as the Lipschitz constant associated with f . In particular, if c ∈ (0, 1), then f is said to be a contraction and c is its contraction ratio. Thus, it is clear that any contraction is a Lipschitz continuous map. Moreover, if the equality ρ( f (x), f (y)) = c · ρ(x, y) stands for all x, y ∈ F, then ρ is said to be a similarity and c is called as its similarity ratio.

24

1 Mathematical Background

Next, let us recall the standard construction of self-similar sets provided by m Hutchinson (c.f. [45]). Assume that (X, ρ) is a complete metric space and let { f i }i=1 be a finite set of contractions defined on X . If ci is the contraction ratio associated with each f i , then it becomes clear that diam ( f i (F)) ≤ ci · diam (F) for all i = 1, . . . , m. Definition 1.4.1 Let (X, ρ) be a complete metric space, F be a subset of X , m be a collection of contractions defined on F. The scheme and F = { f i }i=1 m (X, { f i }i=1 ) is named an iterated function system (IFS, hereafter). If there is no confusion regarding the whole space X , then we will denote that IFS by F , for short. One of the key goals in this section is to prove that IFSs give rise to unique non-empty compact subsets remaining invariant under the action of the maps f i ∈ F. To deal with, the first step is to properly define a metric on the hyperspace of X which is formally defined next. Definition 1.4.2 The hyperspace of F ⊆ X is the collection containing all the non-empty compact subsets of F, namely, H(F) = {C ⊆ F : ∅ = C compact}. That metric, known as Hausdorff metric, is given as follows. Definition 1.4.3 Let (X, d) be a complete metric space, F be a subset of X , and A, B be any two sets in H(F). The Hausdorff metric dH : F × F → R+ can be defined in the following terms: 



dH (A, B) = max sup inf d(x, y), sup inf d(x, y) . x∈A y∈B

y∈B x∈A

(1.14)

The next result provides an alternative description for Hausdorff metric in terms of the δ−body (also δ−parallel body) of a set in the hyperspace of F. Definition 1.4.4 Let (X, d) be a complete metric space, F be a subset of X , δ > 0, and A be any set in H(F). The δ−body of A is defined by Aδ = {x ∈ F : d(x, a) ≤ δ for some a ∈ A}.

1.4 Iterated Function Systems

25

Proposition 1.4.5 Let (X, d) be a complete metric space, F be a subset of X , and A, B be any two sets in H(F). The two following expressions are equivalent for Hausdorff metric calculation purposes:   (i) dH (A, B) = max supx∈A inf y∈B d(x, y), sup y∈B inf x∈A d(x, y) . (ii) dH (A, B) = inf{δ > 0 : A ⊂ Bδ and B ⊂ Aδ }, where Aδ and Bδ denote the δ−bodies of A and B, respectively.

Proof In fact, let us define δ = inf{ε > 0 : A ⊂ Bε and B ⊂ Aε }. First, since B ⊂ Aδ , then for all x ∈ B, we have that d(x, a) ≤ δ : a ∈ A. Thus, d(x, A) ≤ δ for all x ∈ B, where d(x, A) = inf{d(x, a) : a ∈ A}, as usual. Therefore, it holds that supx∈Bd(x, A) ≤ δ. Similarly, A ⊂ Bδleads to supx∈A d(x, B) ≤ δ. Accordingly, max supx∈A d(x, B), supx∈B d(x, A) ≤ δ. Finally, notice that all the previous implications are equivalences to get the result. Interestingly, Proposition 1.4.5 makes easier to prove that the so-called Hausdorff metric is, in fact, a metric. This fact is explored in detail in the following result. Theorem 1.4.6 The Hausdorff metric, given as in Definition 1.4.3 and characterized in Proposition 1.4.5, satisfies the properties of a metric on the hyperspace of F ⊆ X .

Proof Let us verify that the Hausdorff metric satisfies the three properties of a metric. In fact, • The symmetry property is clear due to Proposition 1.4.5. • Let A, B be two closed subsets of F and assume that dH (A, B) = 0. Hence, we have that either supx∈B d(x, A) = 0 or supx∈A d(x, B) = 0. Let us suppose that supx∈B d(x, A) = 0 . Then d(x, A) = 0 for all x ∈ B and hence, B ⊆ A. Similarly, if supx∈A d(x, B) = 0, then it holds that A ⊆ B. Therefore, A = B. • To tackle with the triangular inequality, let us consider   dH (A, C) = inf ε1 > 0 : A ⊂ Cε1 , C ⊂ Aε1 and   dH (B, C) = inf ε2 > 0 : B ⊂ Cε2 , C ⊂ Bε2 . The result stands if we prove that A ⊂ Bε1 +ε2 and B ⊂ Aε1 +ε2 . First, we have B ⊂ Cε2 ⊂ (Aε1 )ε2 . Next step is to show that (Aε1 )ε2 ⊂ Aε1 +ε2 . In fact, let x ∈ (Aε1 )ε2 . Thus, d(x, Aε1 ) ≤ ε2 , so there exists y ∈ Aε1 such that d(x, y) ≤ ε2 . In addition, since y ∈ Aε1 , then we have d(y, A) ≤ ε1 . Accordingly, it holds that d(x, A) ≤ d(x, y) + d(y, A) ≤ ε1 + ε2 due to the triangular inequality satisfied by the metric d. Similarly, A ⊂ Cε1 ⊂ (Bε2 )ε1 ⊂ Bε1 +ε2 .

26

1 Mathematical Background

The following result is well known as classical Zenor-Morita’s Theorem (c.f. [60]). It guarantees that the hyperspace is complete with respect to the Hausdorff metric. To prove it, the concept of totally bounded set will be useful. Definition 1.4.7 Let (X, d) be a metric space. We shall understand that A ⊂ X is totally bounded provided that for all ε > 0, there exist a1 , . . . , an ∈ A such n B(ai , ε). that A ⊂ ∪i=1

Theorem 1.4.8 (Zenor-Morita) Let (X, d) be a complete metric space. Then (H(X ), dH ) also is.

Proof Let {An }n≥1 be a Cauchy sequence in (H(X ), dH ). The goal here is to prove that {An }n≥1 is convergent in H(X ). With this aim, we shall assume, without loss of generality, that 1 dH (An , An+1 ) ≤ n 2 for each natural number n. In fact, otherwise, we could define another sequence {Bn }n≥1 ⊂ H(X ) as follows. Let ε = 21 . Since {An }n≥1 is a Cauchy sequence, then there exists n 0 ∈ N such that dH (An , Am ) < 21 for every n, m ≥ n 0 . Thus, we set B1 = An 0 . Now, let ε = 212 . Similarly, there exists n 1 ≥ n 0 such that dH (An , Am ) < 1 for all n, m ≥ n 1 . Hence, let B2 = An 1 with n 1 ≥ n 0 . In general, we shall obtain 22 {Bn }n≥1 as a subsequence of {An }n≥1 satisfying that dH (Bn , Bn+1 ) < 21n for each n ∈ N. Further, it holds that the limit of {Bn }n≥1 (if exists) equals limn→∞ An . Next step is to properly define a candidate to be the limit of {An }n≥1 . In this way, let A be the set containing all the points x ∈ X for which there exists a sequence {an }n≥1 such that for all n ∈ N, there exists m ≥ n : an ∈ Am , where limn→∞ an = x and 1 for all n ∈ N. The next statements hold. d(an , an+1 ) ≤ 2n−1 • A = ∅. In fact, let {an }n≥1 be a sequence defined as follows. Let a1 ∈ A1 and a2 ∈ A2 be such that

d(a1 , a2 ) = d(a1 , A2 ) < dH (A1 , A2 ) = max max d(x, A2 ), max d(x, A1 ) . x∈A1

x∈A2

It is worth noting that the distance d(a1 , A2 ) stands by a2 ∈ A2 since A2 is compact. In general, let an ∈ An be such that d(an , an+1 ) ≤ dH (An , An+1 ). Next, we prove that {an }n≥1 is a Cauchy sequence in X . To tackle with, observe that

1.4 Iterated Function Systems

27

d(an , an+h ) ≤ d(an , an+1 ) + · · · + d(an+h−1 , an+h ) ≤ dH (An , An+1 ) + · · · + dH (An+h−1 , An+h ) 1 1 1 ≤ n + n+1 + · · · + n+h−1 2 2 2 h +∞ +∞   1 1 1  1 1 ≤ ≤ = n · = n−1 , n+i−1 n+i−1 i 2 2 2 2 2 i=1 i=1 i=0 which does not depend on h ∈ N. Thus, {an }n≥1 is a Cauchy sequence on X . Then {an }n≥1 is convergent (in X ) due to the completeness of X . Hence, we can write a = limn→∞ an . Accordingly, A = ∅. • limn→∞ An = A. Equivalently, let us verify that limn→∞ dH (A, An ) = 0, namely, for a fixed ε > 0, it holds that dH (A, An ) < ε for all n ≥ n 0 . To deal with, first 1 0 and n ∈ N such that 2n−1 and let bn ∈ An . In addition, let us define a sequence {ak }k≥1 whose general term is given in the following terms: ⎧ ⎪ ⎨ak ∈ Ak ak = an = bn ⎪ ⎩ d(ak , ak+1 ) = d(Ak , ak+1 )

if k < n if k = n if k ≥ n.

Hence, it is clear that d(ak , ak+1 ) ≤ 21k for all k ≥ n. Moreover, since d(an+h , a) can be as small as desired, then we can write d(an , a) ≤ d(an , an+h ) + d(an+h , a)
0. Then there exists n 0 ∈ N such that 2n01−1 < ε. Further, let {an }n≥1 be a sequence in X such that for all n ∈ N, there exists m ≥ n with an ∈ Am , 1 for all natural number n. Analogous argulimn→∞ an = a, and d(an , an+1 ) ≤ 2n−1 1 ments lead to d(an , a) < 2n−1 . Hence, we have A ⊂ (An )

1 2n−1

⊂ (An )

1 2n 0 −1

⊂ (An )ε for all n ≥ n 0 .

• A ∈ H(X ). Indeed, since A is a subset of X , a complete metric space, then we can verify that A is compact, if and only if, A is closed and totally bounded (c.f. Definition 1.4.7). To deal with, let ε > 0 and xn ∈ A such that limn→∞ xn = a. We shall verify that a ∈ A. In fact, since xn ∈ A, then there exists a sequence {bkn }k,n≥1 such that limk→∞ bkn = xn for all n ∈ N, namely,

28

1 Mathematical Background

b31 · · · b32 · · · .. . . . . b1n b2n b3n · · · b11 b12 .. .

b21 b22 .. .

bk1 . . . bk2 . . . .. . . . . bkn . . .

→ → .. .

x1 x2 .. .

→ xn .

n+1 Let an = bn+3 . Hence, we have n+1 n+2 d(an , an+1 ) ≤ d(bn+3 , xn+1 ) + d(xn+1 , xn+2 ) + d(xn+2 , bn+4 ) 1 1 1 1 ≤ n+2 + n + n+3 ≤ n−1 . 2 2 2 2

Moreover, observe that n+1 , xn+1 ) + d(xn+1 , a) d(an , a) ≤ d(bn+3 1 1 1 ≤ n+2 + n < n−1 2 2 2

which goes to 0 as n → ∞. Thus, limn→∞ an = a. Accordingly, a ∈ A. • A is totally bounded. In fact, let {An }n≥1 ⊂ H(X ). Then for all n ∈ N, An is compact so in particular, An is totally bounded. Additionally, we affirm that 1 for all natural number n. In fact, notice that dH (An , A) ≤ 2n−1 dH (An , A) ≤ dH (An , An+h ) + dH (An+h , A) ≤ dH (An , An+1 ) + · · · + dH (An+h−1 , An+h ) + dH (An+h , A) 1 1 ≤ n + · · · + n+h−1 + δ 2  2 1 1 = n · 1 + · · · + h−1 + δ 2 2 +∞ 1  1 1 ≤ n · + δ = n−1 + δ i 2 i=0 2 2 1 . Moreand the arbitrariness of δ > 0 gives the result. Hence, A ⊂ (An ) n−1 2 over, since An is totally bounded, then there exist b1 , . . . , bk ∈ An such that 1 k 1 , where G = {b1 , . . . , bk } ⊂ A n . Therefore, An ⊂ ∪i=1 B(bi , 2n−1 ) = G n−1 2

A ⊂ (An )

1 2n−1

 ⊂ G

1 2n−1

1 2n−1

⊂G

1 2n−2

.

Accordingly, (H(X ), dH ) is complete. Our next goal is to prove that the so-called Hutchinson’s operator defined on the hyperspace H(X ) is a contraction with respect to the Hausdorff metric.

1.4 Iterated Function Systems

29

m Definition 1.4.9 Let (X, d) be a metric space and F = { f i }i=1 be an IFS on X . By the Hutchinson’s operator, we shall understand the map H : H(X ) → m f i (A) for all A ∈ H(X ). H(X ) defined as H (A) = ∪i=1

m Theorem 1.4.10 Let (X, d) be a metric space and F = { f i }i=1 be an IFS on X . Then the Hutchinson’s operator (c.f. Definition 1.4.9) is a contraction on the metric space (H(X ), dH ).

Proof First of all, let ci be the contraction ratio associated with each f i ∈ F . It is clear that d( f i (a), f i (b)) ≤ ci · d(a, b) for all a ∈ A, b ∈ B, and i = 1, . . . , m. Taking infima in B, it holds that d( f i (a), f i (B)) ≤ ci · d(a, B) for each a ∈ A. Moreover, by taking suprema in A, the following expression stands: sup d( f i (a), f i (B)) ≤ ci · sup d(a, B). a∈A

(1.15)

a∈A

Similarly, we also have sup d( f i (A), f i (b)) ≤ ci · sup d(A, b). b∈B

(1.16)

b∈B

Thus, both Eqs. (1.15) and (1.16) yield the next inequality: 







max sup d( f i (a), f i (B)), sup d( f i (A), f i (b)) ≤ ci · max sup d(a, B), sup d(A, b) . a∈A

b∈B

a∈A

Equivalently, dH ( f i (A), f i (B)) ≤ ci · dH (A, B) ≤ c · dH (A, B), where c = maxi=1,...,m ci . Let r = dH (A, B). Then  m  f i (A) cr f i (B) ⊂ ( f i (A))cr ⊂ ∪i=1 for all i = 1, . . . , m. Thus,  m  m f i (B) ⊂ ∪i=1 f i (A) cr . ∪i=1 Similarly,

 m  m f i (A) ⊂ ∪i=1 f i (B) cr . ∪i=1

b∈B

30

1 Mathematical Background

Hence, it becomes clear that dH ( f (A), f (B)) ≤ c · dH (A, B). The following result, called as Banach’s contraction mapping principle (also Banach’s fixed point theorem or contraction mapping theorem) is also well known in theory of metric spaces. It was first stated in 1922 (c.f. [13]) and guarantees that any contraction on a complete metric space always possesses a unique fixed point. Theorem 1.4.11 (Banach’s contraction mapping principle) Let (X, d) be a complete metric space and f be a contraction on X . Then there exists a unique point x0 ∈ X remaining fixed under the action of f , namely, f (x0 ) = x0 . In addition, it holds that limn→∞ f n (x) = x0 for all x ∈ X .

Proof Let x ∈ X , assume that c is the contraction ratio associated with f , and define xn = f n (x) for all n ∈ N. The following statements stand: • limn→∞ f n (x) = x0 holds since {xn }n≥1 is a Cauchy sequence. In fact, for every h ∈ N, we have d(x0 , x h ) ≤ d(x0 , x1 ) + d(x1 , x2 ) + · · · + d(x h−1 , x h )   ≤ d(x0 , x1 ) · 1 + c + c2 + · · · + ch−1 ≤ d(x0 , x1 ) ·

+∞  i=1

Let M =

1 1−c

ci =

1 · d(x0 , x1 ). 1−c

· d(x0 , x1 ). Hence, d(xn , xn+h ) = d( f (xn−1 ), f (xn+h−1 )) ≤ c · d(xn−1 , xn+h−1 ) ≤ cn · d(x0 , x h ) ≤ M · cn

for each h > 0. Since d(xn , xn+h ) does not depend on h, then {xn }n≥1 is a Cauchy sequence. Thus, it is convergent due to the completeness of (X, d). Therefore, let x0 = limn→∞ xn . • x0 is a fixed point of f . Indeed, we have limn→∞ f n+1 (x) = f (x0 ). Observe also that {xn+1 }n≥N is a subsequence of {xn }n∈N which goes to x0 , then the uniqueness of the limit leads to x0 = f (x0 ). • The fixed point x0 is unique. Assume that there exist x, y two distinct fixed points. Hence, d(x, y) = d( f (x), f (y)) ≤ c · d(x, y) < d(x, y), a contradiction. Accordingly, x = y.

1.4 Iterated Function Systems

31

• Finally, note that the property limn→∞ f n (x) = x0 does not depend on the selected point x ∈ X and also lies on the fact that the fixed point x0 is unique. Accordingly, by Theorem 1.4.10, the Hutchinson’s operator H is a contraction on the complete metric space (H(X ), dH ), so the Banach’s fixed point Theorem (c.f. Theorem 1.4.11) guarantees that there exists a unique non-empty compact subset K ⊆ X such that K = H (K ). (1.17) Thus, K is known as the attractor of that IFS, or equivalently, its IFS-attractor. In particular, by a strict self-similar set, we shall understand an attractor of an IFS whose contractions are similarities. In the sequel, by an IFS-attractor, we shall understand a strict self-similar set. The next result, which can be found out in [29, Theorem 9.1], follows as a consequence of Theorems 1.4.8, 1.4.10, and 1.4.11, as well. Theorem 1.4.12 Let (X, d) be a complete metric space, F be a closed subset m be an IFS on F, and ci be the contraction ratio associated of X , F = { f i }i=1 with each f i ∈ F . The three following hold: (1) There exists a unique non-empty compact subset K of X remaining invariant under the action of the similarities f i ∈ F , namely, m f i (K ). K = ∪i=1

(1.18)

(2) In addition, let f : H(X ) → H(X ) be a contraction (with respect to dH ) given as m f i (C) (1.19) f (C) = ∪i=1 for all C ∈ H(X ) and assume that c = maxi=1,...,m ci is its contraction ratio. Then we have k (1.20) K = ∩+∞ k=1 f (E) for any E ∈ H(X ) with K ⊂ E, where f k denotes the kth iterate of f , namely, f 0 (E) = E and f k (E) = f ( f k−1 (F)) for all k ≥ 1. (3) Equation (1.20) also stands for all E ∈ H(X ) such that f i (E) ⊂ E for all i = 1, . . . , m.

Proof (1) In fact, since (X, d) is complete, then we can affirm that (H(X ), dH ) also is, due to Zenor-Morita’s Theorem (c.f. Theorem 1.4.8. In addition, Theorem 1.4.10 yields that the Hutchinson’s operator is a contraction with contraction ratio c = maxi=1,...,m ci . Thus, the Banach’s contraction mapping principle (c.f. The-

32

1 Mathematical Background

orem 1.4.11 guarantees that there exists a unique non-empty compact subset K of X such that K = f (K ) which is named the attractor of the IFS F . It is also worth mentioning that for each k ≥ 1, f i1 ,i2 ,...,ik (K ) = f i1 ◦ f i2 ◦ · · · f ik (K ) is the general term of a sequence consisting of a nest of (non-empty) compact subsets with diameters going to 0. Accordingly, the so-called Cantor’s completeness principle yields that ∩+∞ k=1 f i 1 ,i 2 ,...,i k (K ) becomes a single point. In this way, let us denote xi1 ,i2 ,...,ik = ∩+∞ k=1 f i 1 ,i 2 ,...,i k (K ).

(1.21)

Hence, it holds that K = ∪+∞ k=1 {x i 1 ,i 2 ,...,i k }. k f (E), where E ∈ H(X ) : K ⊂ E. First, we shall prove that (2) Let K  = ∩+∞ k=1 K ⊂ K  . Indeed, since K ⊂ E and K is a self-similar set, then we have K = f (K ) ⊂ f (E). Similarly, K = f 2 (K ) = f ( f (K )) ⊂ f 2 (E). In general, it k holds that K = f k (K ) ⊂ f k (E) for each k ≥ 1. Thus, K ⊂ ∩+∞ k=1 f (E) =  K . Reciprocally, observe that we can write +∞ k K  = ∩+∞ k=1 f (E) = ∩k=1 ∪ Jk f i 1 ,i 2 ,...,i k (E),

(1.22)

where Jk = {(i 1 , . . . , i k ) : i j = 1, . . . , m, j = 1, . . . , k}. Let x ∈ K  and ε > 0. Then there exists n 0 ∈ N such that cn 0 · supe∈E, k∈K d(e, k) < ε, where c = maxi=1,...,m ci . Thus, x ∈ f i1 ,i2 ,...,in0 (E), namely, x = f i1 ,i2 ,...,in0 (y) for some y ∈ E. On the other hand, there exists z ∈ K which can be expressed in the form z = xi1 ,...,in0 ,i1 ,i1 ,... = f i1 ,i2 ,...,in0 (xi1 ,i1 ,... ). Hence, d(x, z) ≤ cn 0 · d(y, xi1 ,i1 ,... ) < ε, namely, d(x, K ) = inf z∈K {d(x, z)} ≤ d(x, K ) < ε for all ε > 0. Accordingly, x ∈ K . (3) Let x ∈ K which can be written as x = f i1 ,i2 ,...,in (xin+1 ,in+2 ,... ). Since f i (E) ⊂ E for all i = 1, . . . , m, then it holds that f i1 ,i2 ,...,in (E) is the general term of a sequence consisting of a nest of non-empty compact subsets with diameters going to 0. Hence, the Cantor’s completeness principle guarantees that the intersection  of all its terms must be a single point, namely, let y = ∩+∞ k=1 f i 1 ,i 2 ,...,i n (E) ⊂ K . Let ε > 0 be fixed but arbitrarily chosen. Then there exists a natural number n 0 such that cn 0 · supe∈E, k∈K d(e, k) < ε, where c = maxi=1,...,m ci . Thus, there exists z ∈ E such that y = f i1 ,i2 ,...,in0 (z) for some z ∈ E. Accordingly, d(x, y) ≤ cn 0 · d(xin+1 ,in+2 ,... , z) ≤ cn 0 ·

sup

d(e, k) < ε

e∈E, k∈K

for all ε > 0. Therefore, d(x, y) = 0, so that K  ⊂ K . The reciprocal may be dealt with similarly to the first half regarding the proof for Theorem 1.4.12(2). The example below describes analytically the Sierpi´nski gasket, a typical example of self-similar set first contributed by W. Sierpi´nski in 1915 (c.f. [80]).

1.4 Iterated Function Systems

33

Example 1.4.13 Let I = {1, 2, 3} be a finite index set and { f i }i∈I be an IFS whose similarities f i : R2 → R2 are given by ⎧ x y if i = 1 ⎨(2, 2) 1 (x, y) + ( , 0) if i = 2 f 1 f i (x, y) = 2 ⎩ 1 1 f 1 (x, y) + ( 4 , 2 ) if i = 3 for all (x, y) ∈ R2 . Thus, the Sierpi´nski gasket is fully determined as the unique non-empty compact subset K satisfying the Hutchinson’s equation K = ∪i∈I f i (K ). Observe that each piece f i (K ) becomes a self-similar copy of the whole gasket.

In 1946, P.A.P. Moran proved a strong result allowing the calculation of both the box and the Hausdorff dimensions for a certain class of Euclidean IFS-attractors throughout the (unique) solution of an equation only involving the similarity ratios associated with each similarity of the IFS (c.f. [59, Theorem III] and [29, Theorem 9.3]). That classical result will be proved afterward for the sake of completeness. To deal with, we shall follow the strategy posed by Falconer which first makes use of the following technical lemma. Lemma 1.4.14 Let {Vi }i∈I be a finite collection consisting of pairwise disjoint open subsets of Rd . Assume that for every i ∈ I , there exists a ball of radius α1r contained in Vi and a ball of radius α2 r containing Vi . Then each ball of d  radius r intersects a number of V i ≤ 2αα21+1 .

Proof Let B be a ball of radius r that intersects V i . Then there exists a ball with radius (2a2 + 1) · r concentric with B. Assume that Card ({V i : B ∩ V i = ∅}) = q. Recall that for each Vi , there exists a ball of radius α1r contained in Vi . If we sum the volumes of these q interior balls, then we have q · (α1r )d ≤ (2α2 + 1)d · r d . Therefore, q ≤ (2α2 + 1)d · α1−d .

Theorem 1.4.15 (Moran, 1946) Let F = { f 1 , . . . , f k } be an Euclidean IFS under the OSC whose IFS-attractor is K and ci be the similarity ratio associated with each similarity f i ∈ F . Then dimH (K ) = dimB (K ) = s, where s is the unique solution of the following expression:

34

1 Mathematical Background k 

cis = 1.

(1.23)

i=1

Additionally, for that value of s, H s (K ) ∈ (0, ∞). In other words, the IFSattractor K is an s−set. k Proof First of all, let s ≥ 0 satisfying Eq. (1.23), namely, i=1 cis = 1. We should mention here that the following notation will be used in this proof: for all subset F of Rd , we shall write Fi1 ,...,ih = f i1 ◦ · · · ◦ f ih (F), and by Jh = {(i 1 , . . . , i h ) : i j = 1, . . . , k, j = 1, . . . , h}. Accordingly, we have K = ∪J h Ki1 ,...,ih just by applying repeatedly the Hutchinson’s equation for the IFS-attractor K , namely, K = k K . Next, let us verify that the collection of coverings {Ki1 ,...,ih : (i 1 , . . . , i h ) ∈ ∪i=1 Jh } yields an upper bound regarding the s−dimensional Hausdorff measure of K . In fact, observe that   diam (Ki1 ,...,ih )s = (ci1 . . . cih )s · diam (K )s Jh

Jh

=

 k 

⎛



cis1 . . . ⎝

i 1 =1

k 

⎞ cish ⎠ · diam (K )s

i h =1

= diam (K ) , s

where we have used that ci1 . . . cih is the similarity ratio associated with the composition of similarities f i1 ◦ · · · ◦ f ih and Eq. (1.23). Thus, for every δ > 0, we can find out a certain h such that diam (Ki1 ,...,ih ) ≤ (maxi=1,...,k ci )h ≤ δ. Therefore, Hδs (K ) ≤ diam (K )s and hence, H s (K ) ≤ diam (K )s . Accordingly, dimH (K ) ≤ s. Conversely, define I = {(i 1 , i 2 , . . .) : i j = 1, . . . , k, j ≥ 1} and Ii1 ,...,ih be the set consisting of all infinite sequences starting by (i 1 , i 2 , . . . , i h ), namely, Ii1 ,...,ih = {(i 1 , i 2 , . . . , i h , qh+1 ) : q j = 1, . . . , k, j ≥ h + 1}. Let us define a mass distribution μ : P(I ) → [0, ∞] by μ(Ii1 ,...,ih ) = (ci1 . . . cih )s . Next, we shall verify the following: k • μ(Ii1 ,...,ih ) = i=1 μ(Ii1 ,...,ih ,i ). In fact, μ(Ii1 ,...,ih ,i ) = (ci1 . . . cih ci )s = cis · k k μ(Ii1 ,...,ih ) which leads to i=1 μ(Ii1 ,...,ih ,i ) = i=1 cis · μ(Ii1 ,...,ih ) = μ(Ii1 ,...,ih ) · k s i=1 ci = μ(Ii 1 ,...,i h ). Accordingly, μ is a mass distribution on P(I ) such that μ(I ) = 1. • μ satisfies the hypothesis appeared in the Mass distribution principle 1.3.8. Let k f i (V ) ⊂ V , and V be the open set provided by the OSC on F . Thus, ∪i=1 k hence, H (V ) = ∪i=1 f i (V ) ⊂ V . Then the decreasing sequence {H k (V )}k≥1 goes to K by Eq. (1.20). Further, Theorem 1.4.12(2) also gives that K ⊂ V so that Ki1 ,...,ih ⊂ V i1 ,...,ih for all (i 1 , . . . , i h ) ∈ Jh . Next, we shall calculate an upper bound for μ(B), where B is a ball of radius r < 1. This will be carried out throughout sets of the form Vi1 ,...,ih having diameters comparable with the diameter of B

1.4 Iterated Function Systems

35

and such that their closures V i1 ,...,ih intersect B ∩ K . With this aim, let us shorten each sequence (i 1 , i 2 , . . .) ∈ I by the first term i h satisfying that

 r·

min ci

i=1,...,k

≤ ci1 ci2 . . . cih ≤ r.

(1.24)

One of the two following cases may occur: (i) By Q we shall denote the finite set containing all the finite sequences of this kind. Accordingly, for all (i 1 , i 2 , . . .) ∈ I , there exists a unique h such that (i 1 , . . . , i h ) ∈ Q. It is worth noting that the collection Vi1 ,...,ih ,1 , . . . , Vi1 ,...,ih ,k is disjoint for a every finite sequence (i 1 , . . . , i h ) since Vi : i = 1, . . . , k are pairwise disjoint. Therefore, the collection {Vi1 ,...,ih : (i 1 , . . . , i h ) ∈ Q} consists of disjoint open sets. Hence, F ⊂ ∪(i1 ,...,ih )∈Q Ki1 ,...,ih ⊂ ∪(i1 ,...,ih )∈Q V i1 ,...,ih . Since V is open, let us assume that V contains a ball of radius α1 , and there exists a ball of radius α2 containing V , as well. Thus, it becomes clear that for every (i 1 , . . . , i h ) ∈ Q, the open set Vi1 ,...,ih contains a ball of radius a1 · ci1 . . . cih and theset Vi1 ,...,ih conis contained in a ball of radius a2 · ci1 . . . cih . Accordingly,  tains a ball of radius mini=1,...,k ci · a1 r (since mini=1,...,k ci · r ≤ ci1 . . . cih by Eq. (1.24) and is contained in a ball of radius a2 r (since ci1 . . . cih ≤ r by the same argument). Let Q1 = {(i 1 , . . . , i h ) ∈ Q : B ∩ V i1 ,...,ih = ∅}. Lemma 1.4.14 guarantees that Card (Q1 ) ≤ q = (2α2 + 1)d · α1−d ·  −d mini=1,...,k ci . Therefore, μ(B) = μ(B ∩ K )   ≤ μ {(i 1 , i 2 , . . .) : xi1 ,i2 ,... ∈ B ∩ K }   ≤ μ ∪(i1 ,...,ih )∈Q 1 Ii1 ,...,ih . In fact, if xi1 ,i2 ,... ∈ B ∩ K ⊂ ∪(i1 ,...,ih )∈Q 1 V i1 ,...,ih , then (i 1 , . . . , i h ) ∈ Q1 for a certain term h. Thus, μ(B) = μ(B ∩ K )   ≤ μ ∪(i1 ,...,ih )∈Q 1 V i1 ,...,ih    μ V i1 ,...,ih = ≤ (i 1 ,...,i h )∈Q 1

=



(i 1 ,...,i h )∈Q 1



μ(Ii1 ,...,ih )

(i 1 ,...,i h )∈Q 1 ,xi1 ,...,ih ∈V i1 ,...,ih

(ci1 . . . cih )s ≤



r s = r s · Card (Q1 ) ≤ r s · q,

(i 1 ,...,i h )∈Q 1

where we have applied Eq. (1.24) to get the last inequality. We conclude that μ(U ) ≤ q · diam (U )s for all subset U of Rd with diam (U ) ≤ δ since U can be contained in a ball with radius diam (U ). Accordingly, the Mass distribution principle 1.3.8 gives 0 < q1 ≤ H s (K ) and hence, s ≤ dimH (K ).

36

1 Mathematical Background

(ii) Assume that Q is a set consisting of infinite sequences such that for all (i 1 , i 2 , . . .) ∈ I thereexists a unique k such that (i 1 , . . . , i k ) ∈ Q. From Eq. (1.23), we have (i1 ,...,ih )∈Q (ci1 . . . cih )s = 1. Moreover, it is also worth mentioning that if Q is selected to satisfy the chain of inequalities in Eq. (1.24), then Card (Q) ≤ q1 = r −s · (mini=1,...,k ci )−s . Then we can affirm that F can be covered throughout q1 sets of diameter r · diam (V ) for every r < 1. To justify that, observe that for each (i 1 , . . . , i h ) ∈ Q, it holds that diam (V i1 ,...,ih ) = ci1 . . . cih · diam (V ) ≤ r · diam (V ). Recall that dimB (K ) ≤ s, where the box dimension of K can be calculated as the smallest number of sets of diameter ≤ δ that cover F (c.f. Definition 1.3.1). Finally, the result follows since dimH (K ) = s. It is worth mentioning that Moran also contributed in [59, Theorem II], a weaker version of the result described above under the assumption that all the similarity ratios are the same. Interestingly, if the Euclidean IFS F appeared in Moran’s theorem 1.4.15 does not lie under the OSC, then we still have dimH (K ) = dimB (K ) ≤ s. This result is due to Falconer and first appeared in [28]. For additional details regarding this, we especially refer the reader to [30, Corollary 3.3]. Next, we recall some definitions, results, and notations that are useful to develop a new theory of fractal dimension for fractal structures. In this way, we will be focused on quasi-pseudometrics, fractal structures, iterated function systems and box-counting and Hausdorff dimension topics.

1.5 Quasi-pseudometrics First, recall that a quasi-pseudometric on a set X is a nonnegative real-valued function ρ defined on X × X such that for all x, y, z ∈ X , the two following conditions are verified: (1) ρ(x, x) = 0. (2) ρ(x, y) ≤ ρ(x, z) + ρ(z, y). In addition, if ρ satisfies also the next one: (3) ρ(x, y) = ρ(y, x) = 0 if and only if x = y, then ρ is called a quasi-metric. In particular, a non-Archimedean quasi-pseudometric is a quasi-pseudometric which verifies also that ρ(x, y) ≤ max{ρ(x, z), ρ(z, y)} for all x, y, z ∈ X . Moreover, we have that each quasi-pseudometric ρ on X generates a quasi-uniformity Uρ on X which has as a base the family of sets of the form {(x, y) ∈ X × X : ρ(x, y) < 1/2n }, n ∈ N. The topology τ (Uρ ) induced by the quasi-uniformity Uρ will be denoted simply by τ (ρ). Therefore, a topological space (X, τ ) is said to be (non-Archimedeanly) quasi-pseudometrizable if there exists

1.5 Quasi-pseudometrics

37

a (non-Archimedean) quasi-pseudometric ρ on X such that τ = τ (ρ). The theory of quasi-uniform spaces is covered in detail in [40]. Let (X, ρ) be a (quasi-)metric space. Then we will denote the diameter of a subset A ⊆ X by diam (A) = sup{ρ(x, y) : x, y ∈ A}, as usual. In addition to that, we will use the expression Bρ (x, ε) to denote the ball of center x ∈ X with respect to the metric (resp. quasi-metric) ρ, and radius ε > 0, namely, Bρ (x, ε) = {y ∈ X : ρ(x, y) < ε}.

1.6 Fractal Structures The concept of fractal structure was first introduced in [3] to characterize nonArchimedeanly quasi-metrizable spaces but it can also be used to study fractals. For example, in [11] it was used to study attractors of iterated function systems. The use of fractal structures provides a powerful tool to introduce new models for a definition of fractal dimension, since it is a natural context in which the concept of fractal dimension can be developed. Moreover, it will allow to calculate the fractal dimension in new spaces and situations.  Recall that a family  of subsets of a space X is called a coveringif X = {A : A ∈ }. Let  be a covering of X . Then we will denote St (x, ) = {A ∈  : x ∈  / A}. Furthermore, if  = {n }n∈N is a countable A}, and Ux = X \ {A ∈  : x ∈ family of coverings of X , then we will denote Uxn = Uxn , Ux = {Uxn }n∈N , and St (x, ) = {St (x, n )}n∈N . Let us present a first approach to define a fractal structure on a set X . Indeed, let 1 and 2 be two coverings of X . Thus, we will write 1 ≺ 2 to denote that 1 is a refinement of 2 , namely, for all A ∈ 1 there exists B ∈ 2 such that A ⊆ B. In addition to that, the notation 1 ≺≺ 2 means that 1 ≺ 2 , and for all B ∈ 2 we have that B = {A ∈ 1 : A ⊆ B}. Thus, a fractal structure on a set X can be defined as a countable family of coverings of X ,  = {n }n∈N , such that n+1 ≺≺ n for all n ∈ N. Next, we present the definition of a fractal structure on a topological space as it was introduced in [3, Definition 3.1]. Definition 1.6.1 Let X be a topological space. (1) A pre-fractal structure on X is a countable family of coverings,  = {n }n∈N , such that Ux is an open neighborhood base of x, for each x ∈ X . (2) Moreover, if n+1 is a refinement of n such that for all x ∈ A with A ∈ n there exists B ∈ n+1 such that x ∈ B ⊆ A, then we will say that  is a fractal structure on X . (3) If  is a (pre-)fractal structure on X , then we will say that (X, ) is a generalized (pre-)fractal space, or simply a (pre-)GF-space. If there is no

38

1 Mathematical Background

doubt about the fractal structure , then we will say that X is a (pre-)GFspace. Covering n is called level n of the fractal structure . Remark 1.6.2 To simplify the theory, the levels of any fractal structure  will not be coverings in the usual sense. Instead of this, we are going to allow that a set can appear more than once in any level of . For instance, 1 = {[0, 1/2], [1/2, 1], [0, 1/2]} may be the first level of a fractal structure defined on the closed unit interval [0, 1]. Recall also that if  is a pre-fractal structure, then any of its levels is a closurepreserving closed covering (see [7, Proposition 2.4]). If  is a fractal structure on X and St (x, ) is a neighborhood base of x for all x ∈ X , then we will call  a starbase fractal structure. Starbase fractal structures are connected to metrizability (see [6, 7]). A fractal structure  is said to be finite if all levels n are finite coverings. A fractal structure  is said to be locally finite if for each level n of the fractal structure  we have that any point x ∈ X belongs to a finite number of elements A ∈ n . Moreover, a fractal structure  is said to be An+1 ⊆ An -Cantor-complete if for each decreasing sequence {An }n∈N (namely,  for all n ∈ N) of subsets of X with An ∈ n , then it holds that n∈N An = ∅. In general, if n has the property P for all n ∈ N and  = {n }n∈N is a fractal structure on X , then we will say that  is a fractal structure with the property P, and that (X, ) is a GF-space with the property P. Interestingly, every self-similar set always admits a fractal structure that can be defined naturally. Such a fractal structure was first sketched by Bandt and Retta (c.f. [14]) and formally defined later (c.f. [11]). m Definition 1.6.3 Let F = { f i }i=1 be an IFS whose IFS-attractor is K . The natural fractal structure on K as a self-similar set is defined as the countable family of coverings  = {n }n∈N whose levels are given by n = { f ω (K ) : ω ∈ I n }, where I = {1, . . . , m}.

Regarding Definition 1.6.3, we shall use the following notation for each n ∈ N and all word ω = ω1 ω2 . . . ωn ∈ I n : f ω = f ω1 ◦ · · · ◦ f ωn (Fig. 1.2). Remark 1.6.4 Another description concerning the levels of the natural fractal structure that any self-similar set can be always endowed with is as follows: 1 = { f i (K ) : i ∈ I } and n+1 = { f i (A) : A ∈ n , i ∈ I } for all n ≥ 2.

1.6 Fractal Structures

39

Fig. 1.2 Sierpi´nski gasket

In Example 1.4.13, we provided an IFS whose attractor is the classical Sierpi´nski gasket (c.f. Fig. 1.2). Next, we describe the natural fractal structure on that strict self-similar set. Example 1.6.5 The natural fractal structure which the Sierpi´nski gasket can be endowed with as a self-similar set can be described as the countable family of coverings  = {n }n∈N , where 1 is the union of three equilateral “triangles” with sides equal to 1/2, 2 consists of the union of 32 equilateral “triangles” whose sides are equal to 1/22 , and in general, n is the union of 3n equilateral “triangles” with sides equal to 1/2n for each n ∈ N (c.f. Fig. 1.3 for a graphical approach regarding the first levels of that fractal structure). It is worth mentioning that this is a finite starbase fractal structure.

The study concerning this natural fractal structure for IFS-attractors can allow a useful insight regarding the topology of these kind of recursive sets (c.f. [11]).

1.7 Self-Similar Processes In this section, we recall some useful concepts and properties about self-similar processes and their increments and provide a rigorous description regarding (fractional) Lévy stable processes and their particular cases: (fractional) Brownian motions and stable processes.

40

1 Mathematical Background

1

2

Fig. 1.3 First two levels of the natural fractal structure on the Sierpi´nski gasket as a self-similar set (c.f. Definition 1.6.3)

1.7.1 Random Functions and Their Increments. Self-Affinity Properties The definitions, properties, and results that we recall next come from the theories of probability and stochastic processes and are essential in order to formalize our mathematical ideas. Some useful references are [29, 57]. Let (X, A , P) be a probability space and let t ∈ [0, ∞) denote time. We say that X = {X (t, ω)}t≥0 is a random process or a random function from [0, ∞) ×  to R, if X (t, ω) is a random variable for all t ≥ 0 and all ω ∈  (ω belongs to a sample space ). We think of X as defining a sample function t → X (t, ω) for all ω ∈ . Thus, the points of  parametrize the functions X : [0, ∞) → R and P is a probability measure on this class of functions. Let X (t, ω) and Y (t, ω) be two random functions. The notation X (t, ω) ∼ Y (t, ω) means that the two preceding random functions have the same finite joint distribution functions. Recall also that (1) A random process X = {X (t, ω)}t≥0 is said to be H −self-similar if for any H > 0, X (at, ω) ∼ a H X (t, ω) for all a > 0 and t ≥ 0. The parameter H is the self-similarity index or exponent. (2) The increments of a random function X (t, ω) are said to be: (i) stationary, if for each a > 0 and t ≥ 0, X (a + t, ω) − X (a, ω) ∼ X (t, ω) − X (0, ω);

1.7 Self-Similar Processes

41

(ii) self-affine with parameter H ≥ 0, if for any h > 0 and any t0 ≥ 0,

X (t0 + τ, ω) − X (t0 , ω) ∼

 1  X (t + hτ, ω) − X (t , ω) . 0 0 hH (1.25)

Note that by [57, Corollary 3.6], we have that if a random function X (t, ω) has self-affine increments with parameter H , then a T H −law as the next one is satisfied: M(T, ω) ∼ T H · M(1, ω),

(1.26)

where its cumulative range is given by M(t, T, ω) =

sup {Y (s, t, ω)} −

s∈[t,t+T ]

inf

s∈[t,t+T ]

{Y (s, t, ω)} ,

(1.27)

where Y (s, t, ω) = X (s, ω) − X (t, ω) and moreover, M(T, ω) = M(0, T, ω). The next remark links cumulative ranges with fractal structures. Remark 1.7.1 Let α : I → R be a sample function of a random process X with stationary increments, where I = [0, 1]. Let  be the natural fractal structure on I and let  be the fractal structure induced by  on α(I ). Then, for each n ∈ N, the collection {diam (A) : A ∈ n } is a sample of the random variable M( 21n , ω) for all natural number n. Next, we include a proof for the following result, since we have not found it in the literature. Lemma 1.7.2 Let X be a H −self-similar random process with stationary increments. Then, X has self-affine increments with parameter H .

Proof First of all, note that X (0, ω) = 0 for all ω ∈ , since X is a H −self-similar process. Further, by combining both the H −self-similarity of X with the stationariness of the increments of X , we have that 1 1 1 {X (t0 + ha, ω) − X (t0 , ω)} ∼ H {X (ha, ω) − X (0, ω)} ∼ H X (ha, ω) H h h h for all t0 ≥ 0 and all h > 0. Furthermore, the next expression is also based on the H −self-similarity of X :

42

1 Mathematical Background

1 1 X (ha, ω) ∼ H h H X (a, ω) ∼ X (a, ω) H h h and now it is clear that X (a, ω) ∼ X (a, ω) − X (0, ω) ∼ X (t0 + a, ω) − X (t0 , ω) for all t0 ≥ 0. Accordingly, the random process X has self-affine increments with exponent H .

1.7.2 Fractional Brownian Motions A Gaussian process with mean 0 and autocovariance function given by R(t1 , t2 ) =

 1  2H |t1 | + |t2 |2H − |t1 − t2 |2H · Var (X (1, ω)) 2

(1.28)

may be constructed, where R(t1 , t2 ) ≡ Cov (X (t1 , ω), X (t2 , ω)) = E [X (t1 , ω) · X (t2 , ω)], with E [·] being the expected value of the corresponding random variable, since the function {|t1 |2H + |t2 |2H − |t1 − t2 |2H : ti ∈ R, i = 1, 2} is positive definite for all H ∈ (0, 1). This provides a random process called a fractional Brownian motion (FBM for short) which will be written as B H = {B H (t, ω) : t ≥ 0}. This kind of motion can be characterized as the unique H -self-similar Gaussian process with stationary increments for H ∈ (0, 1). In particular, if Var (X (1, ω)) = 1 in Eq. (1.28), then B H is said to be a standard FBM, whose integral representation is as follows:  0   t 1 1 1 1 |t − u| H − 2 − |u| H − 2 B(du) + |t − u| H − 2 B(du) , CH −∞ 0 (1.29) ∞ 1 1 1 where C H2 = 0 ((1 + t) H − 2 − t H − 2 )2 dt + 2H and B(du) is a symmetric Gaussian independently scattered random measure. Note that if H = 21 , then the classical Brownian motion (BM onwards), also called the Wiener process, stands as a particular case. Those H −self-similar random processes which have stationary increments are interesting in order to model long-memory phenomena. In this way, note that any H −self-similar process X = {X (t, ω)}t∈R with stationary increments induces a stationary sequence Y = {Y (k, ω) : k ∈ Z} defined as follows: Y (k, ω) = X (k + 1, ω) − X (k, ω) for all k ∈ Z. The stationary sequence associated with the FBM given in Eq. (1.29) is called fractional Gaussian noise (fGn for short). In particular, if Var (Y (k, ω)) = 1 for all k ∈ Z, then the sequence {Y (k, ω) : k ∈ Z} is called a B H (t, ω) =

1.7 Self-Similar Processes

43

standard fractional Gaussian noise. The fGn has some properties which are described next: • if H = 21 , then the fGn is a sequence of i.i.d. Gaussian random variables, and • if H = 21 , then the random variables Y (k, ω) of the fGn are dependent. If H ∈ ( 21 , 1), then it is said that the process X shows long-memory or long-range dependence. On the other hand, if H ∈ (0, 21 ), then it is said that the process X displays short memory. In Fig. 1.4, we provide some examples of 1024 point FBMs. Equivalently, a BM may be defined as a random process B H = {B H (t, ω) : t ≥ 0} satisfying the next conditions (see [29, Sect. 16.1]): (i) B H (0, ω) = 0 with probability 1 (i.e., the process starts at the origin), and B H (t, ω) is a continuous function of time t, (ii) B H (t + h, ω) − B H (t, ω) ∼ N (0, h), that is, 



−u 2 P({B H (t + h, ω) − B H (t, ω) ≤ x}) = √ exp 2h 2π h −∞ 1

x

du,

(1.30) (iii) and the increments B H (t2 , ω) − B H (t1 , ω), . . . , B H (t2m , ω) − B H (t2m−1 , ω) are independent for 0 ≤ t1 ≤ t2 ≤ · · · ≤ t2m . Observe that (i) and (ii) imply (iii). Moreover, B H (t, ω) ∼ N (0, t) for each t ≥ 0. In addition to that, notice that the distribution of the increments B H (t + h, ω) − B H (t, ω) is independent of t, so they are stationary. A Brownian process may be characterized as the unique probability distribution of functions which has independent and stationary increments of finite variance. Thus, in order to obtain sample functions with different features, it is necessary to relax some of these conditions. In this way, note that FBMs and stable processes constitute two usual variants of Brownian processes. On the one hand, recall that FBMs present normally distributed increments which are not independent, while on the other hand, stable processes relax the finite variance condition that leads to discontinuous functions. An alternative description to the expression provided in Eq. (1.29) for FBMs is the next one (see [29, Sect. 16.2]). An FBM of index α (with α ∈ (0, 1)) is a random process B H = {B H (t, ω) : t ≥ 0} that satisfies the next properties: (i) B H (0, ω) = 0 with probability 1, where B H (t, ω) is a continuous function of time t, and (ii) B H (t + h, ω) − B H (t, ω) ∼ N (0, h 2α ), i.e.,

44

1 Mathematical Background

Fig. 1.4 From up to down, the figure above shows three 210 point FBMs with Hurst exponents equal to 0.25, 0.5 (a BM), and 0.75, respectively

1.7 Self-Similar Processes

45

1 P({B H (t + h, ω) − B H (t, ω) ≤ x}) = √ α h 2π



−u 2 du exp 2h 2α −∞ (1.31) x



Note that for α ∈ (0, 1) such a process exists. Further, the previous conditions imply that the increments of B H are stationary, that is, they have probability function independent of t. In particular, for α = 21 , the expression contained in Eq. (1.31) leads to the formula appeared in Eq. (1.30) for BMs. Remark 1.7.3 (c.f. [57, Theorem 3.3]) (Fractional) Brownian motions are random processes with stationary and self-affine increments with parameter H .

1.7.3 Stable Processes and Fractional Lévy Stable Motions Another generalization of BMs leads to stable processes introduced by Lévy. A stable process is a random function X = {X (t, ω)}t≥0 whose increments X (t + h, ω) − X (t, ω) are stationary (the increment distribution only depends on h) and independent, namely, X (t2 , ω) − X (t1 , ω), . . . , X (t2m , ω) − X (t2m−1 , ω) are independent if 0 ≤ t1 ≤ t2 ≤ · · · ≤ t2m . In general, stable processes have infinite variance and are discontinuous with probability 1, except in special cases such as BMs. In general, it is not possible to specify the probability distribution of stable processes directly, so that Fourier transforms are used in order to define such distributions. In this way, the probability distribution of a random variable Y may be described by means of its characteristic function, that is, the Fourier transform E [exp(iuY )] : u ∈ R. Thus, to define a stable process, let us consider an appropriate function ψ : R → R, and let us require that the increments X (t + h, ω) − X (t, ω) satisfy the next condition: E [exp(iu(X (t + h, ω) − X (t, ω)))] = exp(−hψ(u)), where X (t2 , ω) − X (t1 , ω), . . . , X (t2m , ω) − X (t2m−1 , ω) are independent if 0 ≤ t1 ≤ t2 ≤ · · · ≤ t2m . It is clear that the increments are stationary. Stable processes exist for an appropriate choice of ψ. In this way, if ψ(u) = c |u|α with α ∈ (0, 2], then we obtain a stable symmetric process of index α. In terms of random variables, a real-valued random variable X is stable if, for all n ∈ N, there exists a sequence {βn , γn }n∈N such that X 1(n) + · · · + X n(n) ∼ βn X + γn

46

1 Mathematical Background

n where {X i(n) }i=1 are i.i.d. random variables with the same distributions as X . Equivalently, a real-valued random variable is stable if for any a > 0, then there exists b > 0 and c ∈ R such that its characteristic function μ verifies the next condition: (μ(u))a = μ(bu) · exp(icu). A random variable is said to be strictly stable if for any a > 0 there exists b > 0 such that (μ(u))a = μ(bu). For a strictly stable random 1 variable, we have that b(a) = k · a α where α ∈ (0, 2]. This random variable is said to be α-stable. Note that for α ∈ (0, 2), an α−stable random variable verifies that E [|X |γ ] < ∞, if and only if, γ < α. Thus, the second order moment exists, if and only if, α = 2, and in this case, X is a Gaussian random variable (and hence, it has all moments). For instance, a Gaussian variable is α−stable with α = 2. The Cauchy law is α−stable with α = 1. The fractional Lévy stable motion (FLSM for short) becomes a widely used generalization of an FBM to the α-stable case. This process will be denoted as ZαH = {Z αH (t, ω) : t ∈ R}, where

 Z αH (t, ω)

=

0

−∞

 t  1 H − α1 H − α1 |t − u| Z α (du) + − |u| |t − u| H − α Z α (du) 0

(1.32) and Z α is a symmetric Lévy α−stable independently scattered random measure. Note that the integral expression provided in Eq. (1.32) is well defined for H ∈ (0, 1) and α ∈ (0, 2] as a weighted average of the Lévy stable motion (LSM) Z α (u), with the weight given by the kernel K H,α (t, u). This integral can be understood as the next limit in the L p −norm, where p < α: 

t

−∞

K H,α (t, u)Z α (du) = lim

m→∞

m 

c j (Z α (u j ) − Z α(u j−1 )).

j=1

This process is H −self-similar and has stationary increments. Note that the H −selfsimilarity of the random process ZαH is due to the integral representation provided in Eq. (1.32) and also to the d−self-similarity of the kernel K H,α (t, u) with d = H − α1 , when the integrator Z α (du) is α1 −self-similar. Hence, H = d + α1 . In Fig. 1.5, we show a graphical representation of some examples of 1024 point LSMs. Note that from both Eqs. (1.29) and (1.32), it holds that the representation of any FLSM is similar to the representation of any FBM. In fact, any FBM is an FLSM for which α = 2. If we write H = α1 , then we obtain the Lévy α−stable motion which is a generalization of the BM to the α−stable case. Nevertheless, unlike the FBM, the Lévy α−stable motion is not the unique α1 −self-similar Lévy α−stable process with stationary increments (which is only true for α ∈ (0, 1)). The increment process corresponding to the FLSM is called a fractional stable noise (fsn for short). In parallel with the case α = 2, the next properties yield: • if H > α1 , then the fsn presents long-range dependence, • if H = α1 , then the increments of any FLSM are i.i.d. symmetric α-stable variables, and

1.7 Self-Similar Processes

47

Fig. 1.5 From left to right, the figure above shows three 210 point LSMs with self-similarity exponents equal to 0.6, 0.75 (a BM), and 0.8, respectively. Recall that in this case, the parameter α ∈ (0, 2) is equal to H1 ∈ (1/2, 1)

48

1 Mathematical Background

Table 1.2 The table above contains the (fractional) Lévy stable motion and its particular cases (including their corresponding noises) depending on the parameter α and the self-similarity exponent H α ∈ (0, 2) α=2 H = 1/α H = 1/α

FLSM (fsn) LSM (stable noise)

FBM (fGn) BM (white noise)

• if H < α1 , then the fsn has negative dependence. Further, there is no long-range dependence when α ∈ (0, 1], since H ∈ (0, 1). Remark 1.7.4 (Fractional) Lévy stable motions are random processes with stationary and self-affine increments with parameter H .

Proof It suffices with taking into account that any FLSM is an H −self-similar random process with stationary increments. Thus, by applying Lemma 1.7.2, it holds that any FLSM has self-affine increments with parameter H . Table 1.2 summarizes all the self-similar processes and their corresponding noises.

Chapter 2

Box Dimension Type Models

Abstract The main goal of this chapter is to generalize the classical box dimension in the broader context of fractal structures. We state that whether the so-called natural fractal structure (which any Euclidean subset can be always endowed with) is selected, then the box dimension remains as a particular case of the generalized fractal dimension models. That idea allows to consider a wider range of fractal structures to calculate the fractal dimension of a given subset. Interestingly, unlike how it happens with classical box dimension, the new models we provide in this chapter can be further extended to non-Euclidean contexts, where the classical definitions of fractal dimension may lack sense or cannot be calculated. In this chapter, we illustrate this fact in the context of the domain of words. Another advantage of these models of fractal dimension for fractal structures lies in the possibility of their effective calculation or estimation for any space admitting a fractal structure. To calculate these dimensions, we can proceed as easy as to estimate the box dimension in Euclidean applications.

2.1 Introduction Since fractals were first sketched and explored by pioneer B.B. Mandelbrot (1924– 2010) in early 70s [55], research concerning nonlinear patterns has grown increasingly during the last years. In this way, fractals have been applied to diverse fields in science. In that context, the key tool considered for that purpose has been the fractal dimension, mainly understood as the classical box dimension, since it is a single quantity that provides useful information regarding the complexity that a given subset presents for a whole range of scales. In this chapter, new models of fractal dimension are introduced with respect to a fractal structure. To deal with, a look at the box dimension definition allows to state that fractal structures constitute a powerful context where new definitions of fractal dimension could be developed. The first generalized definition of fractal dimension we contribute (denoted as fractal dimension I, herein) only depends on a fractal structure. Another approach for a generalized fractal dimension depends not only on a fractal structure but also on a distance function. In particular, that second model of fractal © Springer Nature Switzerland AG 2019 M. Fernández-Martínez et al., Fractal Dimension for Fractal Structures, SEMA SIMAI Springer Series 19, https://doi.org/10.1007/978-3-030-16645-8_2

49

50

2 Box Dimension Type Models

dimension equals the first one provided that the corresponding generalized fractal space is endowed with a certain semimetric. Thus, that second notion of fractal dimension can be applied whether it becomes necessary to take into account the size of the elements in each level of a fractal structure. It is noteworthy that the fractal dimension approaches for fractal structures explored along this chapter are as easy to calculate as the box dimension on Euclidean spaces. However, such fractal dimensions can be further applied for any space endowed with a fractal structure. And what kind of spaces admit a fractal structure? Exactly those admitting a non-Archimedean quasi-metric. They include metrizable or second countable (T0 −) topological spaces, but also other non-metrizable spaces, such as the domain of words (c.f. Sect. 2.15) or any non-regular second countable space. This fact enables the use of fractal dimension in new contexts and situations where the standard box dimension may lack sense or cannot be applied. In this way, recall that the box dimension can be defined for metric spaces though the definition of box dimension allowing an easier computation is available only for Euclidean subspaces. Nevertheless, the fractal dimension models for fractal structures explored along this chapter remain available to deal with empirical applications on non-Euclidean contexts. In this way, we shall explore in depth the fractal dimension of a regular language defined on the domain of words (c.f. Sect. 2.15). The structure of this chapter is as follows. In Sect. 2.2, we formally introduce the natural fractal structure which any Euclidean subspace can be always endowed with. Next, in Sects. 2.3 and 2.8, the first two notions of fractal dimension for fractal structures are provided as discretizations of the standard box dimension. In addition, some theoretical properties of these novel dimensions are explored in both Sects. 2.4 and 2.10, respectively. These fractal dimensions for fractal structures are theoretically connected among them and also with the box dimension for Euclidean spaces (see Sects. 2.5, 2.9, 2.12, and 2.13). This has been carried out via some conditions on the elements of a fractal structure. Sections 2.7 and 2.11 contribute useful information concerning the behavior of both fractal dimensions I and II with respect to metrics and fractal structures, as well. To end this chapter, Sects. 2.6 and 2.14 contain several formulae to calculate these dimensions for IFS-attractors. Finally, in Sect. 2.15, we explore the fractal dimension I in the context of the domain of words and Sect. 2.16 collects some references that allowed us to develop these fractal dimension models for fractal structures.

2.2 The Natural Fractal Structure on Euclidean Subsets Let F be a subset of Rd and Nδ (F) be the number of δ−cubes that intersect F (c.f. Theorem 1.3.3 (iv)). Further, we shall apply Remark 1.3.4 for δn = 21n . First, we shall define a fractal structure which any Euclidean set can be always endowed with. It is worth mentioning that such a fractal structure satisfies some properties such as locally finite, tiling, starbase, Cantor-complete, and has order n.

2.2 The Natural Fractal Structure on Euclidean Subsets

51

Definition 2.2.1 The natural fractal structure on every Euclidean space Rd is the countable family of coverings  = {n }n∈N whose levels are given by      kd kd + 1 k1 k1 + 1 × ··· × n, : k1 , . . . , kd ∈ Z . , n = 2n 2n 2 2n

Next, we highlight that natural fractal structures can be induced on Euclidean subsets. Remark 2.2.2 natural fractal structures can be always defined on Euclidean subsets from previous Definition 2.2.1. For instance, the natural fractal structure (induced) on the closed unit interval [0, 1] is defined by the family of coverings  = {n }n∈N whose levels are    k k+1 n : k = 0, 1, . . . , 2 − 1 . , n = 2n 2n Note that the natural fractal structure on Rd is just the tiling consisting of 21n −cubes on Rd . Hence, if Nδ (F) is the number of δ−cubes that intersect F (c.f. Theorem 1.3.3 (iv)) and Definition 2.2.1 is also considered, then we can affirm that N 1 (F) equals 2n

the number of elements in level n of the natural fractal structure that intersect a given subset F of Rd . Following the analogy with the Euclidean case, we will define Nn (F) as the number of elements in level n of the fractal structure that intersect F. The previous remark motivates the definition of our first generalized model of fractal dimension for a fractal structure, which we state in the next section.

2.3 Generalizing Box Dimension by Fractal Dimension I Let us introduce our first generalized box type model of fractal dimension. Fractal dimension approach 2.3.1 Let  = {n }n∈N be a fractal structure on X = Rd and assume that the order of the diameters in each level n of  is equal to 2−n . Further, let c = 21 . Hence, by Remark 1.3.4, we can choose δn = 2−n as a suitable decreasing sequence. According to Theorem 1.3.3 (iv), it holds that N2−n (F) is the number of 2−n −cubes that intersect F ⊆ Rd . Moreover, let Nn (F) be the number of elements in level n of  that intersect F, namely, Nn (F) = Card (An (F)), where An (F) = {A ∈ n : A ∩ F = ∅},

52

2 Box Dimension Type Models

and Card (A) being the cardinal number of A. Thus, N2−n (F) = Nn (F). Next step is to explore the logarithmic rate at which Nn (F) increases as n → ∞. This leads to a discrete version of the underlying idea in the box dimension case, where it is analyzed how Nδ (F) increases as δ → 0 (also in a logarithmic scale). Hence, a gradient of the graph involving log 2−n versus log Nn (F) gives a register regarding fractal patterns in F. That idea can be empirically extended to non-Euclidean contexts where the standard box dimension can have no sense or cannot be calculated [38]. Thus, both definition Theorem 1.3.3(iv) and Remark 1.3.4 lead to the next key definition. Definition 2.3.2 Let F be a subset of X ,  be a fractal structure on X , and Nn (F) be the number of elements in level n of that fractal structure that intersect F. The (lower/upper) fractal dimension I of F is defined as the following (lower/upper) limit: dim1 (F) = lim

n→∞

1 · log2 Nn (F). n

The following remark becomes especially appropriate for empirical applications involving the calculation of fractal dimensions. Remark 2.3.3 Fractal dimension I can be estimated in empirical applications as the slope of a regression line comparing level n versus log2 Nn (F) just like with box dimension estimation. An example of application of Remark 2.3.3 can be found out in upcoming Fig. 2.4, where fractal dimension I has been empirically estimated for a board configuration of Othello after a number of moves. The first theoretical result included in this chapter establishes that fractal dimension I generalizes classical box dimension in the context of Euclidean subsets endowed with their natural fractal structures. Theorem 2.3.4 Let F be a subset of a Euclidean space Rd and  be the natural fractal structure on Rd . Then the (lower/upper) fractal dimension I of F equals the (lower/upper) box dimension of F, namely: dimB (F) = dim1 (F).

2.3 Generalizing Box Dimension by Fractal Dimension I

53

Proof Let Nδ (F) be the number of δ−cubes that intersect F (c.f. Theorem 1.3.3 (iv)). Due to Remark 1.3.4, let δ → 0 throughout any decreasing sequence {δn }n∈N whose general term satisfies the condition c · δn ≤ δn+1 , where c ∈ (0, 1) is an appropriate constant. Let δn = 21n . Hence, log Nδ (F) log Nδn (F) = lim n→∞ − log δ − log δn 1 = lim · log2 Nn (F) = dim1 (F), n→∞ n

dimB (F) = lim

δ→0

since all the elements in level n of that fractal structure are

1 −cubes 2n

in Rd .

2.4 Theoretical Properties of Fractal Dimension I Recall that the Hausdorff dimension constitutes the main theoretical model that we should be mirrored in when providing a new definition for fractal dimension. Thus, our next goal is to explore some theoretical properties from those listed in Theorem 1.3.5 for fractal dimension I. Proposition 2.4.1 Let  be a fractal structure on X . The following statements hold. (1) Both lower and upper fractal dimensions I are monotonic. (2) Upper fractal dimension I is finitely stable. (3) There exist a countable subset F of X and a fractal structure  on X such that dim1 (F) > 0. (4) Neither lower nor upper fractal dimensions I are countably stable. (5) There exists a locally finite starbase fractal structure  defined on a certain subset F ⊆ X such that dim1 (F) = dim1 (F).

Proof (1) Let E, F be two subsets of X and assume that E ⊆ F. Thus, Nn (E) ≤ Nn (F) since each element A ∈ n : A ∩ E = ∅ also satisfies that A ∩ F = ∅. Hence, dim1 (E) ≤ dim1 (F). Notice that the same arguments are valid for upper fractal dimension I. (2) Let E, F be two subsets of X . Since fractal dimension I is monotonic (due to Proposition 2.4.1(1)), then max{dim1 (E), dim1 (F)} ≤ dim1 (E ∪ F). Let us focus on the opposite inequality. First, note that

54

2 Box Dimension Type Models

Nn (E ∪ F) ≤ Nn (E) + Nn (F). Moreover, let ε be a positive real number and d1 = dim1 (E). Then there exists n 1 ∈ N such that Nn (E) ≤ 2n(d1 +ε) for all n ≥ n 1 . Similarly, if d2 = dim1 (F), there exists n 2 ∈ N such that Nn (F) ≤ 2n(d2 +ε) for all n ≥ n 2 . Let us assume, without loss of generality, that d1 ≥ d2 and let m = max{n 1 , n 2 }. Hence, Nn (E) + Nn (F) ≤ 2n(d1 +ε)+1 for all n ≥ m. The following inequalities hold: 1 · log2 (Nn (E) + Nn (F)) n 1 ≤ limn→∞ · (n(d1 + ε) + 1) n = d1 + ε

dim1 (E ∪ F) ≤ limn→∞

for all ε > 0. This leads to   dim1 (E ∪ F) ≤ max dim1 (E), dim1 (F) . (3) Let X = [0, 1], F = Q ∩ X , and  be the natural fractal structure on X (as a Euclidean subset) with levels given by  n =

  k k+1 n : k = 0, 1, . . . , 2 − 1 . , 2n 2n

Hence, it becomes clear that Nn (F) = 2n . Accordingly, dim1 (F) = 1. (4) This stands as a corollary from Proposition 2.4.1(3). (5) Let  = {n }n∈N be a fractal structure defined on  X = ([0, 1] × {0}) ∪

1 2n



 × [0, 1] : n ∈ N ,

with levels given by  n =

  k k+1 n × {0} : k = 0, 1, . . . , 2 − 1 , 2n 2n      k k+1 1 n × n, n : k = 0, 1, . . . , 2 − 1, m ∈ N . ∪ 2m 2 2

2.4 Theoretical Properties of Fractal Dimension I

55

Fig. 2.1 Graphical representation of both the space X and the fractal structure  defined to show both Theorem 2.4.1(5) and Proposition 3.8.1(3). Notice that the rectangles marked in red refer to the elements in 1 , whereas the grey marked rectangles refer to all the elements in the second level of . This figure is due to Magdalena Nowak

Figure 2.1 illustrates the set X as well as the first two levels of . Moreover, let

 1 1 , F= × {0} ⊂ X. 2k+1 2k k∈N Hence, F = [0, 1] × {0}, Nn (F) = 2n , and Nn (F) = ∞. Accordingly, dim1 (F) = 1, whereas dim1 (F) = ∞.

2.5 Linking Fractal Dimension I to Box Dimension In Theorem 2.3.4, we proved that fractal dimension I generalizes box dimension on Euclidean subsets. To deal with, we selected the natural fractal structure that any Euclidean subspace can be always endowed with. Next step is to explore how box dimension and fractal dimension I are theoretically connected for any GF-space. First of all, we shall define the diameter of any level of a fractal structure and the diameter of any subset in a level of a fractal structure, as well. Definition 2.5.1 Let  be a fractal structure on a distance space (X, ρ) and F be a subset of X . (1) The diameter of level n of  is given by diam (n ) = sup{diam (A) : A ∈ n }. (2) The diameter of F in level n of  is calculated by the next expression:

56

2 Box Dimension Type Models

diam (F, n ) = sup{diam (A) : A ∈ An (F)}, where An (F) = {A ∈ n : A ∩ F = ∅}. It is worth mentioning that starbase fractal structures give rise to GF-spaces satisfying some desirable (topological) properties. In this way, next we provide a sufficient condition (not too restrictive) on the elements of each level of a fractal structure to guarantee the starbase property. Proposition 2.5.2 Let  be a fractal structure on a compatible metric (resp., quasi-metric) space (X, ρ) and assume that diam (n ) → 0. Then  is starbase.

Proof We shall prove that St (x, ) is a neighborhood base for all x ∈ X . First of all, it is clear that x ∈ Uxn ⊂ St (x, n ) for each x ∈ X and all n ∈ N. On the other hand, let x ∈ X and ε > 0. Since diam (n ) → 0, then there exists n 0 ∈ N such that diam (n ) < ε for all n ≥ n 0 . In addition, let St (x, m ) for m ≥ n 0 . Hence, for all y ∈ St (x, m ), there exists A ∈ m such that x, y ∈ A. Moreover, since diam (m ) < ε, then it holds that ρ(x, y) < ε. Thus, y ∈ Bρ (x, ε). Therefore, there exists m ∈ N such that St (x, m ) ⊂ Bρ (x, ε). Accordingly,  is starbase. Another feasible condition to be satisfied by a fractal structure consists of a geometric decrease involving the sequence of diameters {diam (F, n )}n∈N . That assumption will provide (up to a constant) an upper bound regarding the box dimension of F in terms of its fractal dimension I. Theorem 2.5.3 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , and assume that there exists a constant c ∈ (0, 1) for which the next inequality stands: diam (F, n+1 ) ≤ c · diam (F, n ). The following three hold: (1) dim B (F) ≤ γc · dim1 (F). (2) dim B (F) ≤ γc · dim1 (F). (3) Additionally, if there exist both the fractal dimension I of F and the box dimension of F, then dimB (F) ≤ γc · dim1 (F), where γc is a constant which depends on c.

2.5 Linking Fractal Dimension I to Box Dimension

57

Proof Let Nδ (F) be the smallest number of sets of diameter at most δ that cover F (c.f. Theorem 1.3.3(i)). (1) The geometric decrease involving the sequence of diameters {diam (F, n )}n∈N leads to diam (F, n ) ≤ cn−1 · diam (F, 1 ) : c ∈ (0, 1). Let δn = cn−1 · diam (F, 1 ) be the general term of a decreasing sequence that goes to 0. Hence, log Nδn (F) log Nn (F) ≤ limn→∞ − log δn −n log c 1 log 2 · limn→∞ · log2 Nn (F), =− log c n

dim B (F) = limn→∞

where Remark 1.3.4 has been applied to deal with the first equality. The choice γc = − log 2/ log c completes the proof. (2) Let {δn }n∈N be a decreasing sequence whose general term is δn = cn−1 · diam (F, 1 ). Since limδ→0

log Nδ (F) log Nδn (F) ≤ limn→∞ , − log δ − log δn

a similar argument to the one applied in Theorem 2.5.3(1) leads to the result. (3) This follows as an immediate consequence of the two previous items. In fact, the existence of the box dimension (resp., the fractal dimension I) of F implies that dimB (F) = dim B (F) = dim B (F) (resp., dim1 (F) = dim1 (F) = dim1 (F)).

2.6 Fractal Dimension I for IFS-Attractors It is worth noting that Theorem 2.5.3 can be further extended to deal with IFSattractors since their sequences of diameters {diam (n )}n∈N always decrease geometrically provided that their natural fractal structures as self-similar sets are selected for fractal dimension calculation purposes. Following the above, the next result stands. Corollary 2.6.1 Let (X, F ) be an IFS where X is a complete metric space and K is its IFS-attractor. Moreover, let  be the natural fractal structure on K as a self-similar set. Then

58

2 Box Dimension Type Models

dimB (K ) ≤ γc · dim1 (K ), where c = maxi∈I ci with the ci ’s being the similarity ratios associated with F .

Proof Notice that diam (K , n+1 ) ≤ c · diam (K , n ), where c = maxi∈I ci . Finally, Theorem 2.5.3 gives the result since the sequence of diameters {diam (K , n )}n∈N decreases geometrically. It is worth mentioning that Corollary 2.6.1 throws an estimation of the box dimension of K in terms of its fractal dimension I. Remark 2.6.2 Notice also that Corollary 2.6.1 still remains valid for IFSs consisting of contractions (not necessarily being similarities).

2.7 Dependence of Fractal Dimension I on the Fractal Structure It turns out that fractal dimension I generalizes box dimension for Euclidean subsets (c.f. Theorem 2.3.4). Additionally, under a geometric decrease concerning the sequence of diameters {diam (F, n )}n∈N , an upper bound regarding the box dimension of F stands in terms of its fractal dimension I (c.f. Theorem 2.5.3 and Corollary 2.6.1). Next, we highlight how fractal dimension I depends on the selected fractal structure. Remark 2.7.1 There exists a Euclidean subset C ⊂ R endowed with two distinct fractal structures, say  1 and  2 , such that dim11 (C ) = dim12 (C ).

Proof Let  1 be the natural fractal structure on the middle third Cantor set C . By 2 (c.f. [29, Theorem 2.3.4, we have dim11 (C ) = dim B (C ) and that value equals log log 3 Example 3.3]). On the other hand, let  2 be the natural fractal structure on C as a self-similar set. Hence, dim12 (C ) = 1 since each level n of  2 consists of 2n “subintervals” with lengths equal to 31n . It is worth pointing out that a fractal structure is a kind of uniform structure. In fact, if there is no metric available in the space, the only way to “measure” a subset is by determining which level of the fractal structure contains it. In other words, it becomes quite natural that fractal dimension I depends on a fractal structure as well as box dimension depends on a metric.

2.8 A Further Step: Fractal Dimension II

59

2.8 A Further Step: Fractal Dimension II Recall that fractal dimension I, formally introduced in previous Definition 2.3.2, actually considers all the elements in level n of a fractal structure as having the same “size”, equal to 21n . In addition to a fractal structure, we can define a distance in the space to powerful effect. More specifically, that distance could allow to “measure” the size of the elements in each level of the fractal structure. This is the case of any Euclidean subset, where they can be always considered both the natural fractal structure and the Euclidean metric, as well. Next, we define the more general concept of a distance function. Definition 2.8.1 (c.f. [79]) By a distance function (or a distance, for short), we shall understand a nonnegative map ρ : X × X → R such that ρ(x, x) = 0 for all x ∈ X . Diameters of subsets, coverings, …, etc. with respect to a distance are defined as in the case of a metric. The second model of fractal dimension with respect to a fractal structure is stated in terms of a distance function. Definition 2.8.2 Let  be a fractal structure on a distance space (X, ρ), F be a subset of X , and Nn (F) be the number of elements in level n of  that intersect F. The (lower/upper) fractal dimension II of F is defined as the following (lower/upper) limit: dim2 (F) = lim

n→∞

log Nn (F) , − log diam (F, n )

where diam (F, n ) is the diameter of F in level n of , as provided in Definition 2.5.1(2). We recall that a fractal structure is finite if all its levels are finite coverings of the whole space. In general, the levels of a fractal structure do not have to be finite coverings though in such a case, the calculation of Nn (F) becomes easier. Next, we recall what kind of spaces admit a finite fractal structure. Remark 2.8.3 A topological space X is second countable, if and only, there exists a finite fractal structure on X (c.f. [7, Theorem 4.3]), and also that a topological space is metrizable, if and only if, there exists a starbase fractal structure on X (c.f. [6, 7]). In addition, a space is separable and metrizable,

60

2 Box Dimension Type Models

if and only if, there exists a finite starbase fractal structure on X (c.f. [8, Theorem 5.7] or [4, Lemma 4.19]). As it was stated above, the size of each element involved in the calculation of fractal dimension II is measured by a distance function. Observe that, at least at a first glance, diam (n ) may be considered for fractal dimension II calculation purposes instead of diam (F, n ) (c.f. Definition 2.8.2). Nevertheless, certain disadvantages follow. More specifically, let  be a finite fractal structure on a distance space (X, ρ) and assume that ρ is non-bounded. In that case, it holds that diam (n ) = ∞, and hence, it does not throw useful information regarding the fractal dimension of F. Additional details concerning this fact are provided next. Example 2.8.4 Let  be a finite fractal structure defined on Rd with levels given as follows: 

d   k i=1

i , 2n

   ki + 1 n n : ki = −n2 , . . . , n2 − 1 ∪ Rd \ (−n, n)d , n 2

d where i=1 refers to the finite d−ary Cartesian product. Thus, for any bounded subset F of Rd , it holds that diam (n ) = ∞, where that diameter has been calculated with respect to the Euclidean distance. Nevertheless, there still exists n 0 ∈ N such that diam (F, n ) < ∞ for all n ≥ n 0 .

2.9 Linking Fractal Dimensions I and II via the Semimetric Associated with a Fractal Structure Observe that fractal dimensions I and II are equal provided that any fractal structure such that diam (F, n ) = 21n is selected to deal with the calculations (c.f. Definitions 2.3.2 and 2.8.2). Next, we go beyond by exploring several conditions on the elements in each level of a fractal structure to reach the equality between both fractal dimensions. In fact, next we provide a first connection between fractal dimensions I and II via the semimetric associated with a fractal structure. To deal with, first we shall define the concepts of a semimetric on a topological space (c.f. [41, Definition 9.5]) and a semimetric associated with a starbase fractal structure (c.f. [11, Theorem 6.4]), as well.

2.9 Linking Fractal Dimensions I and II via the Semimetric …

61

Definition 2.9.1 (i) A semimetric on a topological space X is a nonnegative map ρ : X × X → R satisfying the three following conditions: (1) ρ(x, y) = 0, if and only if, x = y. (2) ρ is symmetric, namely, ρ(x, y) = ρ(y, x) for all x, y ∈ X . (3) The family {Bρ (x, ε) : ε > 0} is a neighborhood base for all x ∈ X . Equivalently, the topology induced by the semimetric ρ yields the starting topology. (ii) Let  be a starbase fractal structure on X . The semimetric associated with  is defined as the nonnegative map ρ : X × X → R given by ⎧ ⎨ 0 if x = y ρ(x, y) = 21n if y ∈ St (x, n ) \ St (x, n+1 ) ⎩ 1 if y ∈ / St (x, 1 ).

(2.1)

It is worth pointing out that Eq. (2.1) implies Bρ (x, 21n ) = St (x, n+1 ) for all n ∈ N and each x ∈ X . Moreover, since  is starbase, we can affirm that the topology induced by the semimetric ρ matches with the topology induced by the fractal structure. Next, we contribute a condition concerning the levels of a (starbase) fractal structure to reach the equality between fractal dimensions I and II. Theorem 2.9.2 Let  be a starbase fractal structure on (X, ρ), where ρ is the semimetric associated with the fractal structure , and F be a subset of X . Moreover, let us assume that for all n ∈ N there exists x ∈ F such that St (x, n ) = St (x, n+1 ). The following three hold. (1) dim1 (F) = dim2 (F). (2) dim1 (F) = dim2 (F). (3) Additionally, if there exists either the fractal dimension I of F or the fractal dimension II of F, then dim1 (F) = dim2 (F).

Proof (1) By hypothesis, there exist x ∈ F and A ∈ n : x ∈ A with A  St (x, n+1 ). Thus, diam (A) = 21n so diam (F, n ) = 21n . As such, dim2 (F) = limn→∞

log Nn (F) 1 = limn→∞ · log2 Nn (F) = dim1 (F). − log diam (F, n ) n

(2) The case of lower limits may be dealt with similarly.

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2 Box Dimension Type Models

(3) This holds as a consequence from the two results proved above. In fact, let us assume, without loss of generality, that there exists dim1 (F). This yields dim1 (F) = dim1 (F) = dim1 (F). Hence, Theorem 2.9.2(1) leads to dim1 (F) = dim2 (F), whereas dim1 (F) = dim2 (F) by Theorem 2.9.2(2). Therefore, there exists dim2 (F) and equals dim1 (F). Additionally, it can be proved that fractal dimension II generalizes both fractal dimension I and box dimension in the context of Euclidean subsets endowed with their natural fractal structures. That result, which extends former Theorem 2.3.4, is stated next. Theorem 2.9.3 Let  be the natural fractal structure on Rd and F be a subset of Rd . Then the (lower/upper) box dimension of F equals both the (lower/upper) fractal dimension I of F and the (lower/upper) fractal dimension II of F, namely: dimB (F) = dim1 (F) = dim2 (F).

Proof First, dimB (F) = dim1 (F) by Theorem 2.3.4. Further, since diam (A) = √ d/2n for all A ∈ n , then we have log Nn (F) 1 = lim · log2 Nn (F) = dim1 (F). n→∞ − log diam (F, n ) n→∞ n

dim2 (F) = lim

2.10 Theoretical Properties of Fractal Dimension II In this section, we explore the behavior of fractal dimension II as a dimension function, similarly to Proposition 2.4.1 for fractal dimension I, Theorem 1.3.5 for box dimension, and Theorem 1.2.8 for Hausdorff dimension. Proposition 2.10.1 Let  be a fractal structure on a distance space (X, ρ). The following statements hold. (1) Both lower and upper fractal dimensions II are monotonic. (2) Neither lower nor upper fractal dimensions II are finitely stable. (3) There exist a countable subset F of X and a fractal structure  on X such that dim2 (F) > 0. (4) Neither lower nor upper fractal dimensions II are countably stable. (5) There exists a locally finite starbase fractal structure  defined on a certain subset F ⊆ X such that dim2 (F) = dim2 (F).

2.10 Theoretical Properties of Fractal Dimension II

63

Table 2.1 The table summarizes some properties as dimension functions satisfied by box dimension and fractal dimensions I and II (c.f. Theorem 1.3.5 and Propositions 2.4.1 and 2.10.1) Monotonicity Finite stability Countable Countable sets Closure dimension stability property dimB dim1 dim2

✓ ✓ ✓

✓ ✓

Proof (1) It is clear that both dim2 and dim2 are monotonic. (2) Let C ⊂ [0, 1] be the middle third Cantor set and  1 be the natural fractal structure on C as a self-similar set. In addition, let C2 = [2, 3] and  2 be a fractal structure on C2 with levels given by  2,n =

  k k+1 2n+1 2n+1 2n :k=2 , ,2 + 1, . . . , 3 · 2 − 1 . 22n 22n

Let C = C ∪ C2 endowed with the fractal structure  = {n }n∈N , where 4 2 > 1, dim2 (C ) = log , and n = 1,n ∪ 2,n . Thus, dim2 (C) = log log 3 log 3 2 dim (C2 ) = 1. (3)–(5) Recall that fractal dimension II generalizes fractal dimension I in the sense of Theorem 2.9.2. Thus, any counterexample valid to justify statements (3), (4), and (5) in Proposition 2.4.1 for fractal dimension I also remains valid to deal with the case of fractal dimension II. Upcoming Table 2.1 summarizes all the information concerning the behavior of the box dimension type models introduced along this chapter. It is worth mentioning that unlike (lower/upper) box dimension, both fractal dimensions I and II satisfy the closure dimension property. This could be understood as an advantage of these models with respect to classical box dimension. On the other hand, neither lower nor upper fractal dimensions II are finitely stable unlike both fractal dimension I and box dimension definitions.

2.11 Dependence of Fractal Dimension II on Both a Fractal Structure and a Distance In Remark 2.7.1, we highlighted the dependence of fractal dimension I on the selected fractal structure. In this section, we point out the additional dependence of fractal dimension II on a distance. In particular, we shall justify why the fractal dimension II of the middle third Cantor set (endowed with its natural fractal structure as a self-similar set) equals its box dimension.

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2 Box Dimension Type Models

Remark 2.11.1 Let C denote the middle third Cantor set and  be the natural fractal structure on C as a self-similar set. In Remark 2.7.1, we proved that the fractal dimension I of C does not equal its box dimension. More specifically, 2 , whereas dim1 (C ) = 1. It is worth noting that these we have dimB (C ) = log log 3 fractal dimensions have been calculated with respect to distinct fractal structures. In fact, the natural fractal structure (induced) on C as a Euclidean subset is always chosen for box dimension calculation purposes. Nevertheless, if the natural fractal structure on C as a self-similar set is considered to calculate its fractal dimension II, then we still have dimB (C ) = dim2 (C ) =

log 2 . log 3

Proof Indeed, log 2n log 2 = dimB (C ), = n→∞ − log 3−n log 3

dim2 (C ) = lim

since level n of  consists of 2n “subintervals” with diameters equal to 1/3n . Even more, though the value obtained in Remark 2.7.1 for dim1 (C ) may seem counterintuitive at a first glance, it still becomes possible to justify it through its fractal dimension II. Once again, the key reason lies in the fact that fractal dimension I only depends on the selected fractal structure. This is emphasized along the next remark. Remark 2.11.2 Fractal dimension I only depends on a fractal structure whereas fractal dimension II also depends on a distance.

Proof To highlight that difference, we shall construct a family of spaces which are the same from the viewpoint of fractal structures. To deal with, let us consider slight modifications from the middle third Cantor set C we shall denote by Ci . Assume that their similarity ratios are ci ∈ [ 13 , 21 ) for each of the two similarities that yield Ci . These sets can be defined as the attractors of the IFS { f 1i , f 2i }, where the similarities f 1i , f 2i : R → R are given by f 1i (x) = ci · x and f 2i (x) = (1 − ci ) + ci · x, for each x ∈ R. In addition, let  i be the natural fractal structure on each space Ci as a selfsimilar set. Then diam (Ci , n ) = cin for all n ∈ N and hence, easy calculations lead to (upcoming Theorem 2.14.3 can be also applied with this aim) dimB (Ci ) = dim2i (Ci ) = −

log 2 → 1 = dim1 (C ), log ci

if ci → 21 , where  is the natural fractal structure on C as a self-similar set.

2.12 Linking Fractal Dimension II to Box Dimension

65

2.12 Linking Fractal Dimension II to Box Dimension In this section, we find out an upper bound for both the Hausdorff and the box dimensions of any subset F in terms of its fractal dimension II. To prove this kind of results we shall assume that diam (F, n ) → 0. Theorem 2.12.1 Let  be a fractal structure on a distance space (X, ρ), F be a subset of X , and let us assume that diam (F, n ) → 0. The following three hold: (1) dimH (F) ≤ dim B (F) ≤ dim2 (F). (2) In addition, if there exist both the box dimension and the fractal dimension II of F, then dimH (F) ≤ dimB (F) ≤ dim2 (F). (3) If there exists a constant c > 0 such that diam (F, n ) ≤ c · diam (F, n+1 ), then dim B (F) ≤ dim2 (F).

Proof To calculate the box dimension of F, let Nδ (F) be the smallest number of sets of diameter at most δ that cover F (c.f. Theorem 1.3.3(i)). First, observe that F ⊆ ∪{A : A ∈ An (F)} = ∪{A ∈ n : A ∩ F = ∅}. It is also clear that diam (A) ≤ diam (F, n ) for all A ∈ An (F). Accordingly, it holds that F can be covered throughout Nn (F) sets with diameters being at most δn . In addition, let us denote δn = diam (F, n ), for short. (1) First, note that Eq. (1.13) (c.f. Theorem 1.3.7) leads to dimH (F) ≤ dim B (F) ≤ limn→∞

log Nn (F) = dim2 (F). − log δn

(2) This stands as a consequence of Theorem 2.12.1(1). (3) Let c ∈ (0, 1] be a constant such that δn ≤ c · δn+1 . This implies that δn  0, a contradiction. Thus, c > 1. Hence, there exists d ∈ (0, 1) such that d · δn ≤ δn+1 (just take d = 1/c). Accordingly, Remark 1.3.4 leads to dim B (F) = limn→∞

log Nδn (F) log Nn (F) ≤ limn→∞ = dim2 (F). − log δn − log δn

Notice that Theorem 2.12.1 allows to achieve an upper bound for the box dimension of IFS-attractors throughout their fractal dimension II values. That result is stated next.

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2 Box Dimension Type Models

Corollary 2.12.2 Let F be a Euclidean IFS whose IFS-attractor is K . In addition, let F be a subset of K and  be the natural fractal structure on K as a self-similar set. The following three hold: (1) dim B (F) ≤ dim2 (F). (2) If there exist both the box dimension and the fractal dimension II of F, then dimB (F) ≤ dim2 (F). (3) Assume that f i is a bi-Lipschitz function for some i ∈ I . Then dim B (F) ≤ dim2 (F). In particular, this stands for strict self-similar sets.

Proof (1) Since K is the IFS-attractor for the Euclidean IFS F , then it is clear that diam (F, n ) → 0 for all F ⊆ K . Hence, Theorem 2.12.1(1) gives the result. (2) This follows immediately from Corollary 2.12.2(1), since the existence of the box dimension (resp., the fractal dimension II) of F, implies that dimB (F) = dim B (F) (and the same equality holds for dim2 (F)). (3) Let f i be a bi-Lipschitz contraction. Thus, there exist constants ci,1 > 0 and 0 < ci,2 < 1 such that ci,1 · ρ(x, y) ≤ ρ( f i (x), f i (y)) ≤ ci,2 · ρ(x, y) for all x, y ∈ K , where ρ denotes the Euclidean distance. On the other hand, let A ∈ An (F) be such that diam (A) = diam (F, n ). By definition of supremum, it holds that for all ε > 0 there exist x, y ∈ A such that diam (A) − ε < ρ(x, y). Let B = f i (A) ∈ n+1 . Hence, ci,1 · (diam (A) − ε) < ci,1 · ρ(x, y) ≤ diam ( f i (A)) = diam (B), which leads to ci,1 · diam (A) ≤ diam (B) = diam ( f i (A)). Accordingly, ci,1 · diam (F, n ) ≤ diam (B) ≤ diam (F, n+1 ). Finally, Theorem 2.12.1(3) gives the proof. Our next goal is to explore which properties underlying the natural fractal structure on any Euclidean space (c.f. Definition 2.2.1) could allow to generalize Theorem 2.9.3. With this aim, observe that given a scale δ > 0, it is satisfied that any Euclidean subspace F of Rd such that diam (F) ≤ δ intersects at most to 3d δ−cubes. In this way, a similar property in the broader context of fractal structures would lead to an additional connection between fractal dimension II and box dimension. Such a property, we shall refer to as κ−condition, is stated next.

2.12 Linking Fractal Dimension II to Box Dimension

67

Definition 2.1 Let  be a fractal structure on a metrizable space X and F be a subset of X . We shall understand that  is under the κ−condition if there exists a natural number κ such that for all n ∈ N, every subset A of X such that diam (A) ≤ diam (F, n ) intersects at most to κ elements in level n of . Given the definition of the κ−condition, the next result follows. Theorem 2.12.3 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , and assume that diam (F, n ) → 0. Moreover, suppose that  is under the κ−condition. The following three hold. (1) dim B (F) ≤ dim2 (F) ≤ dim2 (F) ≤ dim B (F). (2) If there exists dimB (F), then dimB (F) = dim2 (F). (3) If there exists a constant c ∈ (0, 1) such that c · diam (F, n ) ≤ diam (F, n+1 ), then dim B (F) = dim2 (F) and dim B (F) = dim2 (F).

Proof To calculate the box dimension of F, let Nδ (F) be the smallest number of sets of diameter at most δ that cover F (c.f. Theorem 1.3.3(i)). Moreover, let δn = diam (F, n ), for short. (1) Theorem 2.12.1 (1) gives dim B (F) ≤ dim2 (F). Moreover, the κ-condition leads to Nn (F) ≤ k · Nδn (F). Hence, dim2 (F) = limn→∞

log Nn (F) log Nδn (F) ≤ limn→∞ = dim B (F). − log δn − log δn

(2.2)

(2) Notice that Remark 1.3.4 leads to dim B (F) = limn→∞

log Nδn (F) . − log δn

Hence, dim B (F) ≤ dim2 (F) since Nδn (F) ≤ Nn (F). The opposite inequality is due to Eq. (2.2). The case of lower limits may be dealt with similarly. The following counterexample points out that κ−condition on  becomes necessary to guarantee the equality between fractal dimension II and box dimension.

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2 Box Dimension Type Models

Counterexample 2.12.4 There exists a Euclidean IFS F whose IFS-attractor K , endowed with its natural fractal structure as a self-similar set, satisfies that dimB (K ) = dim2 (K ).

Proof Let I = {1, . . . , 8} be a finite index set and (R2 , F ) be a Euclidean IFS whose associated attractor is K = [0, 1] × [0, 1]. Further, define the contractions f i : R2 → R2 as follows:  f i (x, y) =

( x2 , 4y ) + (0, i−1 ) if i = 1, 2, 3, 4 4 ) if i = 5, 6, 7, 8. ( x2 , 4y ) + ( 21 , i−5 4

In addition, let  be the natural fractal structure on K as a self-similar set. First, notice that the self-maps f i are not similarities but affinities and all of them have the same contraction ratio, namely, ci = 1/2. It is also clear that dimB (K ) = 2. On the other hand, there are 8n rectangles in level n of  whose dimensions are 1 × 212n . Hence, 2n  1 + 22n diam (A) = diam (K , n ) = 24n for all A ∈ n . Next, we calculate the fractal dimension II of K . dim2 (K ) = lim

n→∞

log Nn (K ) 3n log 2 3n log 2 = lim = 3. 2n = lim n→∞ n log 2 − log diam (K , n ) n→∞ − 21 log 1+2 24n

We also provide lower bounds for the ratios between diam (K , n ) and the sides of each 21n × 212n −rectangle: 

1+22n 24n 1 22n

=



 1+

22n

>2

n

1+22n 24n 1 2n

 =

1+

1 1 ≥ n. 2n 2 2

 2n intersects at Accordingly, each subset A ⊂ K whose diameter is at most 1+2 24n n+1 most to 3 · 2 elements in level n of . Since that quantity depends on each n ∈ N, then the κ−condition is not satisfied.

2.13 Generalizing Fractal Dimension I by Fractal Dimension II

69

2.13 Generalizing Fractal Dimension I by Fractal Dimension II In this section, we provide some conditions regarding the elements in each level of a fractal structure to reach the equality between fractal dimensions I and II. To deal with, first we recall when two sequences of positive real numbers are said to be of the same order. Definition 2.13.1 Let f, g : N → R be two sequences of positive real numbers. It is said that f and g are of the same order, O( f ) = O(g), if and only if, the following condition holds: lim

n→∞

f (n) ∈ (0, ∞). g(n)

Thus, whether it is assumed that all the elements in each family An (F) = {A ∈ n : A ∩ F = ∅} have a diameter of order 21n , then it can be proved that fractal dimension II equals fractal dimension I. Theorem 2.13.2 Let  be a fractal structure on a distance space (X, ρ), F be a subset of X , and assume that diam (A) = diam (F, n ) for all A ∈ An (F). If O(diam (F, n )) = O( 21n ), then the (lower/upper) fractal dimension I of F equals the (lower/upper) fractal dimension II of F, namely, dim1 (F) = dim2 (F).

Proof Let ε > 0 be fixed but arbitrarily chosen. If α = dim1 (F), then there exist n 0 ∈ N and a strictly increasing mapping σ : N → N such that     log2 Nσ (n) (F)  − α  ≤ ε  σ (n)

(2.3)

for all σ (n) ≥ n 0 . Further, it is clear (by induction in n ∈ N) that σ (n) ≥ n for all n ∈ N. Thus, Eq. (2.3) leads to the following chain of inequalities: 2σ (n) (α−ε) ≤ Nσ (n) (F) ≤ 2σ (n) (α+ε)

(2.4)

for all σ (n) ≥ n 0 . On the other hand, for any two sequences of positive real numbers f, g : N → R such that O( f ) = O(g), it is satisfied that for all ε1 > 0 there exists n 1 ∈ N such that

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2 Box Dimension Type Models

    f (n)  ≤ ε1  − k   g(n) for all n ≥ n 1 . Accordingly, log f (n) ≤ log g(n) + log(k + ε1 )

(2.5)

for all n ≥ n 1 . Since O(diam (F, n )) = O( 21n ), then let f (n) = diam (F, n ) and g(n) = 21n in Eq. (2.5). This leads to 1 1 ≤ − log diam (F, n ) n log 2 − log(k + ε1 ) for all n ≥ n 1 . Further, since k − holds similarly:

f (n) g(n)

(2.6)

≤ ε for all n ≥ n 1 , then the following inequality

1 1 ≥ − log diam (F, n ) n log 2 − log(k − ε1 ) for all n ≥ n 1 . Accordingly, both Eqs. (2.4) and (2.6) allow to reach the following expression: σ (n) (α + ε) log 2 log Nσ (n) (F) ≤ − log diam (F, σ (n) ) σ (n) log 2 − log(k + ε1 )

(2.7)

for all σ (n) ≥ N = max{n 0 , n 1 }. Equivalently, σ (n) (α − ε) log 2 log Nσ (n) (F) ≥ − log diam (F, σ (n) ) σ (n) log 2 − log(k − ε1 )

(2.8)

for all σ (n) ≥ N . From Eqs. (2.7) and (2.8), we have the following chain of inequalities: log Nσ (n) (F) α − ε ≤ lim sup ≤α+ε n→∞ σ (n)≥n − log diam (F, σ (n) ) for all σ (n) ≥ N . Finally, the arbitrariness of ε gives α = limn→∞

log Nn (F) , − log diam (F, n )

i.e., dim1 (F) = dim2 (F). The case of lower limits may be dealt with similarly.

2.14 Fractal Dimension II for IFS-Attractors

71

2.14 Fractal Dimension II for IFS-Attractors Fractal dimension II provides an upper bound concerning the box dimension of any Euclidean IFS-attractor (c.f. Corollary 2.12.2). Going beyond, it is even possible to reach that equality under certain conditions on the corresponding IFS. More specifically, this kind of result stands provided that the elements in each level of the fractal structure do not overlap “too much”. And due to the shape of the elements in the natural fractal structure which any IFS-attractor can be endowed with, this restriction will rely on the similarities of the IFS. In this context, the so-called open set condition (OSC) (c.f. [45, Sect. 5.2] and [52]) plays a key role. Definition 2.14.1 Let F be an IFS and K be its IFS-attractor. (i) We understand that F is under the OSC, if and only if, there exists a (nonempty) bounded open subset V ⊆ X such that ∪i∈I f i (V ) ⊂ V , where f i (V ) ∩ f j (V ) = ∅ for all i = j. (ii) Additionally, if V ∩ K = ∅, then F is said to satisfy the strong open set condition (SOSC). Schief proved that both the OSC and the SOSC are equivalent for Euclidean IFSs (c.f. [78, Theorem 2.2]). The following technical lemma allows to prove the equality between the box dimension and the fractal dimension II of any Euclidean IFS-attractor endowed with its natural fractal structure as a self-similar set. Lemma 2.14.2 Let F be a Euclidean IFS under the OSC whose IFS-attractor is K . Moreover, assume that each self-map f i ∈ F is injective. Then there exist ε > 0 and x ∈ K such that for all n ∈ N and all ω, u ∈ I n : ω = u, the following equality stands: f ω (B(x, ε)) ∩ f u (B(x, ε)) = ∅.

Proof First, since K ⊂ Rd is a Euclidean IFS-attractor, then the OSC is equivalent to the SOSC, due to [78, Theorem 2.2]. Accordingly, there exists a nonempty bounded open subset V ⊂ Rd such that i∈I f i (V ) ⊂ V , where f i (V ) ∩ f j (V ) = ∅ for all i, j ∈ I : i = j. Additionally, we can choose x ∈ V ∩ K ⊂ V since V ∩ K = ∅. Moreover, there exists ε > 0 such that B = B(x, ε) ⊂ V since V is open. Further, it is also satisfied that f i (B) ⊂ V for all i ∈ I . The result will be proved by induction in n, the length of the I n −words. To deal with, let ωn = i n i n−1 · · · i 1 ∈ I n , and denote f ωn (F) = f in ◦ f in−1 ◦ · · · ◦ f i1 (F) for all F ⊂ Rd .

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2 Box Dimension Type Models

Let n = 1. Thus, we can choose ω1 = i 1 and u 1 = j1 ∈ I . Accordingly, f ω1 (B) ∩ f u 1 (B) ⊂ f ω1 (V ) ∩ f u 1 (V ) = ∅ since F is under the OSC. Assume that the induction hypothesis is satisfied for n ∈ N which means that f ωn (B) ∩ f u n (B) = ∅ for ωn , u n ∈ I n : ωn = u n . Let us prove that condition for n + 1. In fact, let ωn+1 , u n+1 ∈ I n+1 . The two following cases can be distinguished. (i) Assume that i n+1 = jn+1 , so ωn+1 = i n+1 i n · · · i 1 and u n+1 = i n+1 jn · · · j1 , where ωn+1 , u n+1 ∈ I n+1 . Hence, by the injectivity of f in+1 and the induction hypothesis, we can conclude that f ωn+1 (B) ∩ f u n+1 (B) = f in+1 ( f ωn (B)) ∩ f in+1 ( f u n (B)) = ∅. (ii) Suppose that ωn+1 = u n+1 , so ωn+1 = i n+1 i n · · · i 1 and u n+1 = jn+1 jn · · · j1 , where i n+1 = jn+1 . Thus, the following expression becomes now clear: f ωn+1 (B) ∩ f u n+1 (B) = f in+1 ( f ωn (B)) ∩ f jn+1 ( f u n (B)) ⊂ f in+1 (V ) ∩ f jn+1 (V ) = ∅, where we have applied that f ωn (B) ⊂ V for all ωn ∈ I n . This is clear for words with length equal to 1, since it is obvious that fi1 (B) ⊂ V . Assume that f ωn (B) ⊂ V for all ωn ∈ I n . Then f ωn+1 (B) = f in+1 ( f ωn (B)) ⊂ f in+1 (V ) ⊂ V . Accordingly, under the OSC, the box dimension of any IFS-attractor equals its Hausdorff dimension, and that common value can be easily calculated from Eq. (4.19). The next result we formulate guarantees the equality between the box dimension and the fractal dimension II of IFS-attractors lying under the OSC. Indeed, the calculation of these dimensions follows immediately from the number of similarities in the IFS and their common similarity ratio, as in [59, Theorem II]. Theorem 2.14.3 Let F = { f 1 , . . . , f m } be a Euclidean IFS under the OSC whose IFS-attractor is K and let  be the natural fractal structure on K as a self-similar set. Moreover, assume that all the similarities f i ∈ F have a common similarity ratio c ∈ (0, 1). Then dimB (K ) = dim2 (K ) = −

log m . log c

(2.9)

2.14 Fractal Dimension II for IFS-Attractors

73

Proof To calculate the box dimension of K , let Nδ (K ) be the largest number of disjoint balls of radius δ with centers in F (c.f. Theorem 1.3.3(v)). Moreover, let δn = diam (K , n ). First of all, observe that δn = cn · diam (K ) since K is a strict self-similar set. Additionally, Lemma 2.14.2 implies that there are so many disjoint balls centered in K and having radii equal to εn = cn · ε : ε > 0 as the number of elements in I n . Thus, since Nεn (K ) is the largest number of such balls, then the number of elements in I n is at most Nεn (K ), namely, Nn (K ) ≤ Nεn (K ). Further, we affirm that there exists k > 0 such that diam (K , n ) = k · εn . Indeed, it stands for k = diam (K )/ε. Hence, dim2 (K ) ≤ limn→∞

log Nεn (K ) log Nεn (K ) = limn→∞ = dim B (K ). (2.10) − log k · εn − log εn

Accordingly, the following chain of inequalities holds: dim B (K ) ≤ dim2 (K ) ≤ dim2 (K ) ≤ dim B (K ), where the first inequality is due to Corollary 2.12.2(1) and Eq. (2.10) gives the last one. It is also worth noting that the existence of the box dimension of K implies the existence of the fractal dimension II of K , so dimB (K ) = dim2 (K ). Notice also that Moran’s Theorem 1.4.14 gives the last equality in Eq. (2.9). In fact, 

cis = m · cs = 1,

i∈I

and hence, s = − log m/ log c. We would like to point out that the hypothesis consisting of equal similarity ratios in Theorem 2.14.3 is necessary. Recall that Counterexample 2.12.4 implies that all the contractions involved in Theorem 2.14.3 must be similarities. Further, the following counterexample justifies why all the similarity ratios must be equal. Counterexample 2.14.4 There exists a Euclidean IFS F under the OSC whose IFS-attractor K , endowed with its natural fractal structure as a selfsimilar set, satisfies that dimB (K ) < dim2 (K ).

Proof Let F = { f 1 , f 2 } be a Euclidean IFS with similarities f 1 , f 2 : R → R defined by  x if i = 1 f i (x) = 2x+3 if i = 2. 4 It is clear that their associated contraction ratios are c1 = 1/2 and c2 = 1/4, respectively. Moreover, it holds that K is a strict self-similar set. It is also possible to

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2 Box Dimension Type Models

justify that F is under the OSC. In fact, let V = (0, 1) ⊂ R. Thus, Moran’s Theorem allows to affirm that the box dimension of K equals the solution of the equation 2−s + 4−s = 1. Hence, √ 1+ 5 1 · log . dimB (K ) = log 2 2 Finally, observe that there are 2n “subintervals” of [0, 1] in level n of the fractal structure , where the diameter of the largest of them equals 21n . Accordingly, dimB (K ) < dim2 (K ) = 1.

2.15 Applications to the Domain of Words Fractal structures can be applied to study the fractal behavior of non-Euclidean spaces. In forthcoming sections, we shall show how to extend the use of the classical box dimension to new contexts and empirical applications. To deal with, the domain of words is endowed with a fractal structure. This allows us to use fractal dimension I for fractal pattern detection in languages generated by regular expressions, to determine the efficiency of an encoding system, and also to estimate the number of nodes of a given depth in a search tree.

2.15.1 Introducing the Domain of Words Both the study and the analysis of fractal patterns have grown increasingly in the last years due to the wide range of applications of fractals to diverse scientific fields. One of the main tools considered for that purpose is the fractal dimension, since it is a single quantity which throws useful information about the complexity of sets. Moreover, the introduction of fractal structures as well as a fractal dimension for them allow the study of fractal patterns in novel topics and applications where the classical fractal dimensions cannot be calculated or may lack sense. In recent years, there has been a growing interest in the application of quasimetrics, and more generally, tools from asymmetric topology in computer sciences (c.f. , e.g., [50, 54, 63, 69, 71, 72]). In particular, the study regarding the domain of words first appeared when modeling streams of information in Kahn’s model of parallel computation [46, 58]. Along upcoming sections, we use the fractal structure induced by a nonArchimedean quasi-metric on the domain of words (c.f. Sect. 2.15.2) to understand the fractal dimension of a language. This provides an example of how fractal dimension models for fractal structures could be calculated in a non-Euclidean context. More specifically, in Sect. 2.15.3, we show how to use fractal dimension to study fractal patterns of languages generated by regular expressions, how to calculate the

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efficiency of an encoding language (c.f. Sect. 2.15.4), and finally, in Sect. 2.15.5, how to estimate the number of nodes of a given depth in a search tree. For illustration purposes, we have written a computer program to play Othello. The fractal dimension I is used therein to estimate the time that the software needs to find out a move depending on the number of moves it can analyze throughout a look-forward algorithm. It is worth mentioning that all the examples included along Sects. 2.15.3–2.15.5 allow to show the power of fractal structures and fractal dimension models for them to explore fractal patterns in non-Euclidean contexts. In this way, we encourage the reader to use them as a starting point to construct new applications.

2.15.2 Quasi-metrics, Fractal Structures, and the Domain of Words This section contains the mathematical background necessary to formally introduce the domain of words. The domain of words appears when modeling streams of information in Kahn’s model of parallel computation [46, 58]. Let  be a finite nonempty alphabet and  ∞ denote the collection of all finite (∪n∈N  n ) or infinite ( N ) sequences from , called words, namely,  n ∪ N. ∞ = n∈N

In particular, let ε denote the empty word. The prefix order  is defined on  ∞ as usual, by x  y, if and only if, x is a prefix of y. Moreover, for each x ∈  ∞ , let l(x) be the length of x, where l(ε) = 0, and for all x, y ∈  ∞ , x  y refers to the common prefix of x and y. A (non-Archimedean) quasi-metric ρ can be defined on  ∞ as follows [81]:  0 if x  y; ρ(x, y) = (2.11) 2−l(xy) otherwise. It is noteworthy that any (non-Archimedean) quasi-pseudometric ρ : X × X → R defined on a topological space X induces a fractal structure which can be described as the countable family of coverings  = {n }n∈N , whose levels are given by

  1 n = Bρ −1 x, n : x ∈ X , 2 for all n ∈ N (c.f. [3]). In particular, if X =  ∞ , then the (non-Archimedean) quasimetric ρ defined in Eq. (2.11) gives rise to a fractal structure  whose levels are as follows: (2.12) n = {ω# : ω ∈  n } ∪ {ω : ω ∈  k , k < n},

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2 Box Dimension Type Models

where for all ω ∈  n , ω = {u ∈  k : u  ω, k ≤ n} is the collection consisting of all the prefixes of ω and ω# = {ωu : u ∈  ∞ } ∪ ω is the collection of all finite or infinite words starting from ω or being a prefix of ω, instead. Further, for each word ω ∈  n , we have ω# = Bρ −1 (ω, 21n ) and for each ω ∈  k : k < n, it holds that ω = Bρ −1 (ω, 21n ). Hence, a language L lies as a subspace of  ∞ and it is usually described throughout a formal grammar. For instance, languages generated by regular expressions could be explored in further applications. In fact, since we have a fractal structure defined on the domain of words, then it is possible to calculate the fractal dimension of any language. Nevertheless, the box dimension has no sense in this context, so cannot be applied for fractal dimension calculation purposes.

2.15.3 Fractal Dimension of Languages Generated by Regular Expressions Next, we explore the fractal dimension of a language generated by a regular expression for illustration purposes. To deal with, let us consider the regular expression (00 + 1)+ which is constructed by concatenating consecutively (at least once) 00 or 1. The main goal here is to calculate the fractal dimension of the language L ⊂  ∞ generated by that regular expression. First, notice that L can be described as the following set: L = {1, 00, 11, 100, 001, 111, 0000, 0011, 1001, 1100, 1111, . . .}.

(2.13)

Next step is to consider the (non-Archimedean) quasi-metric ρ as provided in Eq. (2.11). Moreover, let  be the fractal structure induced by ρ (c.f. Eq. (2.12)). Observe that the first levels of that fractal structure are 1 = {1# , 00# }. 2 = {10# , 11# , 00# , 1 }. 3 = {000# , 001# , 100# , 111# , 110# , 00 , 11 , 1 }. ... To calculate the fractal dimension of the language L , notice that N1 (L ) = 2 N2 (L ) = 3 + 1 N3 (L ) = 5 + 3 N4 (L ) = 8 + 6 N5 (L ) = 13 + 11 ...

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77

where the first term in each sum is the number of elements in the appropriate level of the form ω# , and the second one refers to the number of elements in each level of the form ω . The following expression follows for all natural number n: Nn (L ) = f n+2 +

n 

fi ,

i=2

where { f n }n∈N denotes the Fibonacci’s sequence. k Recall that f 1 = f 2 = 1 and f n = f i = 0 provided that k < 2, then f n−1 + f n−2 for all n ≥ 3. If we assume that i=2 Nn (L ) = 2 · ( f n+2 − 1), since the Fibonacci’s sequence satisfies the next property: n 

f i = f n+2 − 1,

i=1

for all n ≥ 2. Additionally, it holds that fn = where ϕ =

√ 1+ 5 2

ϕn − β n , ϕ−β

is the golden ratio and β =

√ 1− 5 . 2

Therefore,

1 1 · log2 Nn (L ) = lim · log2 f n+2 n→∞ n n 1 = lim · log2 (ϕ n+2 − β n+2 ) n→∞ n  n+2  1 β = lim · log2 1 − + (n + 2) · log2 ϕ n→∞ n ϕ

dim1 (L ) = lim

n→∞

(2.14)

= log2 ϕ. The previous result points out that the fractal dimension I of L is connected with the golden ratio. That result provides valuable information regarding the complexity of the language L as well as its evolution. Indeed, Eq. (2.14) implies that Nn (L )  ϕ n for all natural number n, which is equivalent to Nn+1 (L )  ϕ · Nn (L ). Roughly speaking, it holds that for any word of length n, there are about ϕ words of length n + 1. Moreover, notice that dim1 ( ∞ ) = 1, so Nn+1 ( ∞ )  2 · Nn ( ∞ ) for all n ∈ N. Thus, fractal dimension I provides a register about both the evolution and the complexity of the language L with respect to the domain of words  ∞ where it has been defined. Interestingly, it is also possible to consider a simplified fractal structure  on  ∞ to obtain information regarding the complexity of the language L , as the following remark highlights.

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2 Box Dimension Type Models

Remark 2.15.1 Let  = {n }n∈N be the fractal structure defined by n = {ω# : ω ∈  n } and L be the language generated by the regular expression (00 + 1)+ as described in Eq. (2.13). In this case, the number of elements in level n of  that intersect L is Nn (L ) = f n+2 . Hence, dim1 (L ) = dim1 (L ) = log2 ϕ. Accordingly, the fractal dimension I calculated with respect to the fractal structure  throws the same information about the language L as the fractal dimension I calculated with respect to its simplified version .

2.15.4 The Efficiency of the BCD Encoding System The Binary-Coded Decimal (BCD) is a device which allows to encode decimal numbers. To deal with, it replaces each decimal digit by its corresponding binary sequence. Despite it has been used with less frequency in applications, it is noteworthy that such a system still becomes useful in computer and electronic systems only consisting of digital logic without microprocessors to display decimal numbers. A key advantage of BCD is that it allows faster decimal calculations. However, it is not an efficient encoding system since it uses more space than a simple binary representation. The BCD system stores each decimal digit from 0 to 9 via 4 bits containing its binary sequence. Accordingly, the set {0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001} generates the language B of BCD numbers. For instance, observe that 1100 ∈ / B, since it does not correspond to the codification of any decimal digit. Moreover, notice that the number 215 (in base 10) is equivalent to the BCD expression 001000010101. However, its binary representation is 11010111 which needs less space to be stored. Our main goal here is to show that fractal dimension I could allow to properly describe the computational efficiency of an encoding language with respect to the original one. This has been illustrated throughout the BCD encoding system. Next, let us provide some conclusions regarding the encoding goodness of the BCD system. Indeed, let  be the fractal structure defined in Eq. (2.12). The simplified version of that fractal structure,  (c.f. Remark 2.15.1), could be applied for that purpose, too. Anyway, the following equality holds:

2.15 Applications to the Domain of Words

79

⎧ 2 · 10i ⎪ ⎪ ⎪ ⎨3 · 10i Nn (B) = ⎪ 4 · 10i ⎪ ⎪ ⎩ 6 · 10i leading to dim1 (B) = dim1 (B) =

if n if n if n if n

= 4i = 4i + 1 = 4i + 2 = 4 i + 3,

√ log 10 4 = log2 10. log 16

(2.15)

√ n Equation (2.15) implies that Nn (B)  10 4 , or equivalently, Nn+1 (B)  4 10 · Nn (B). Thus, given a real number with n decimal digits, then 4n binary digits are required to encode it. This allows to represent 10n different BCD numbers, whereas 4n binary digits allow to represent 24n different numbers in a binary representation. Hence, n4 n 10 4 Nn (B) 10 = n = . (2.16) ∞ Nn ( ) 2 16 Equation (2.16) allows to “measure” the efficiency of the BCD system. For instance, take a decimal number with 10 decimal digits. Since it needs n = 40 bits to be encoded, then it follows that N40 (B) = N40 ( ∞ )



10 16

10 

1010 = 0.01, 1012

which means that BCD only encodes the 1% of all the numbers that could be encoded in binary. In other words, there is a lack of the order of 62.5% for each encoded digit.

2.15.5 An Empirical Application to Search Trees In this section, we explore how to calculate fractal dimension I in non-Euclidean contexts with respect to both fractal structures provided in Eq. (2.12) and Remark 2.15.1, respectively. Although fractal dimension I could be calculated for languages described by regular expressions, which may contain infinite length words, it is also possible to calculate the effective fractal dimension I of languages only containing a finite number of words. It is worth mentioning that in this case, though, fractal dimension I is only appropriate for a certain range of scales. Since there is always a maximum length for any word in this context, it holds that dim1 (L ) = dim1 (L ) = 0, where L refers to the language. This is the reason for which we shall calculate the fractal dimension I only for a limited range of levels. Thus, we shall calculate the slope of a regression line comparing n versus log2 Nn (L ), where n denotes the corresponding level of the fractal structure. This is similar to deal with the estimation of the box dimension in Euclidean applications.

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2 Box Dimension Type Models

Fig. 2.2 Search tree as a language

Let us consider a tree. The arguments provided next could be also carried out for a graph with slight modifications. Each node of the tree is named as follows. ROOT will refer to the root node whose children are denoted by a, b, c, . . . , etc. Let us consider any of those nodes, for instance, the node a. Next step is to name the children of such a node as a, b, c, . . . and so on. Thus, the word ab means that we go from ROOT to its children a, and hence, to the children of a, named as b. This allows the tree to be understood as a language (see Fig. 2.2). Next, we illustrate this empirical application by the search tree of the board game Othello, though it is noteworthy that this approach is quite general and can be further considered to deal with search trees appeared in other applications from Artificial Intelligence. Let ROOT be the configuration of the board at the beginning of the game. Observe that the first player can choose among four possible moves we shall label by a, b, c, d. Similarly, the second player has 3 possible moves. However, notice that the choice of the second player does not depend on the move of the first player due to the symmetry of the game, and so on (see Fig. 2.3). In this context, the maximum number of possible moves is less than 64 (the size of the board), so the alphabet is finite. Therefore, the language L consists of the collection of all the words corresponding to all the possible configurations of the board after any number of moves. Next, let us analyze the fractal dimension I of L . First, the ROOT node has been considered and its fractal dimension I in the range of 1–10 moves has been found to be equal to 2.5. Then a random configuration of the board after 10 moves has been chosen, and it follows that its fractal dimension I in the range of 1–8 moves is equal to 2.9. Finally, for a random configuration after 18 moves and in the range of 1–8 moves, the corresponding fractal dimension I was found to be equal to 3.3. The regression coefficients were found to be 0.998, 1, and 1. Since all of them are quite close to 1, it becomes clear that there is a fractal pattern in the search tree.

2.15 Applications to the Domain of Words

81

Fig. 2.3 Board configurations of Othello. The first figure at the top from left illustrates the node ROOT, namely, the configuration of the board at the beginning of the game. Next to it at the first row, the four possible moves of the first player (with black pieces) are displayed. They represent the four possible children of ROOT. Once the first player has moved, then only three possible moves remain available for the second player, namely, the three children of the configuration ROOT–a (the two figures at the bottom)

Let us explain how the fractal dimension I can be explored. Assume that we want to write a computer program to play Othello being based on the following algorithm. Given a configuration of the board, we look ahead for k moves and then we use an evaluation function for each one of the possible future configurations after k moves. Then we use a max − min approach to choose which one is the best move. If we want to provide an answer within, for instance, 1 s, which one is the maximum value of k that could be used? To answer that question, let us estimate the average time of the evaluation function for one node. This allows to know how many nodes could be evaluated in 1 s. Then we can estimate the number of nodes that we shall find if we look ahead for k moves. Indeed, such a quantity is equal to d k , where d denotes the fractal dimension I of the board configuration. Therefore, we can estimate the greatest k that can be used in our algorithm. It is noteworthy that this example provides a standard strategy that can be

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2 Box Dimension Type Models

Fig. 2.4 Fractal dimension I of a random board configuration after 10 moves using 8 levels of the fractal structure. In this case, the regression coefficient is equal to 0.99, so there is a fractal pattern. The slope of a regression line comparing n versus log2 Nn (L ) gives the fractal dimension I of this board configuration

further applied to deal with the full search tree or a pruned search (for example, an α − β pruned search), as well. Accordingly, next step is to estimate d. To deal with, first, we have to determine the number of levels of the fractal structure that will be used for fractal dimension I calculation purposes. In this case, 8 or 10 levels seems to be a good choice. Then the fractal dimension I could be calculated for some random nodes along the search tree (as we carried out previously, but throughout more nodes to reach a better approach). This leads to an overall value of d. Another option is to calculate the fractal dimension I of some random nodes after k moves and then calculate their mean with k ranging from 0 to 60 with a chosen step. This option becomes more appropriate if the fractal dimension I is found to be more or less stable in all the nodes after k moves, but not so stable for different values of k. Thus, if the fractal dimension I becomes stable for almost all k, then the first option should be considered instead. In this case, for an Othello board game and for illustration purposes, we used 5 levels (since the regression coefficient is very close to 1) to calculate the fractal dimension I of 100 random nodes after 6 moves (6–11). A geometric mean equal to 2.96 and a geometric standard deviation equal to 1.036 (a mean of 2.96 and a standard deviation of 0.103) were obtained. Additionally, the fractal dimension I of 100 random nodes after 26 moves threw a geometric mean equal to 3.42 and a geometric standard deviation equal to 1.037 (a mean of 3.42 and a standard deviation of 0.125). Table 2.2 contains a detailed study regarding the fractal dimension I Table 2.2 The table contains the fractal dimension I and the geometric standard deviation of different depths of board configurations for 100 random nodes

Depth

dim1

geo. std.

ROOT (1–6) (10–15) (20–25) (30–35) (40–45) (50–55)

2.34 3.13 3.19 2.80 2.79 2.21

1.00 1.04 1.04 1.06 1.06 1.06

2.15 Applications to the Domain of Words

83

(including the geometric standard deviation) of different depth ranges of the board configuration for 100 random nodes. Given this, it seems that we are in the second case commented above: the fractal dimension becomes quite stable for nodes after k moves but it changes slightly for different k.

2.16 Notes and References F. G. Arenas and M.A. Sánchez-Granero introduced the concept of fractal structure in [3] to characterize non-Archimedean quasi-metrizable spaces and also proved some topological equivalences throughout fractal structures. Fractal dimensions I and II were first introduced by M. Fernández-Martínez and M.A. Sánchez-Granero (2014) to provide generalized definitions of fractal dimension in the broader context of fractal structures. These models of fractal dimension were explored in [35], where some of their properties and results were also contributed therein. It is worth mentioning that the natural fractal structure which any Euclidean subspace can be always endowed with first appeared in [35, Definition 3.1]. This idea allowed the authors to extend the range of applications of fractal dimensions even to non-Euclidean contexts, where the classical models of fractal dimension cannot be applied (c.f. [38]). Recall that Urysohn proved that X is separable and metrizable, if and only if, X is regular and second countable (c.f. [83]). Moreover, F. G. Arenas and M.A. SánchezGranero showed that a topological space X is second countable, if and only, there exists a finite fractal structure on X (c.f. [7, Theorem 4.3]), and also that a topological space is metrizable, if and only if, there exists a starbase fractal structure on X (c.f. [6, Theorem 4.2] or [7, Theorem 3.18]). In addition, a space is separable and metrizable, if and only if, there exists a finite starbase fractal structure on X (c.f. [8, Theorem 5.7] or [4, Lemma 4.19]). These results were applied in Remark 2.8.3. On the other hand, celebrated Theorem 1.4.14 was contributed by Moran (1946) to calculate the dimension of self-similar sets constructed as generalizations of the classical Cantor’s ternary set (c.f. [59, Theorems I–III]). Further, while the OSC was first provided by Hutchinson in his 1981 anthological paper (c.f. [45, Sect. 5.2]), such a key concept was strengthened by Lalley, who defined the SOSC several years later (1988) (c.f. [52]). Finally, it is noteworthy that the equivalence between the SOSC and the OSC for Euclidean IFS-attractors was proved by Schief (1994) in [78, Theorem 2.2].

Chapter 3

A Middle Definition Between Hausdorff and Box Dimensions

Abstract In this chapter, we explore a new model to calculate the fractal dimension of a subset with respect to a fractal structure. The new definition we provide presents better analytical properties than box dimension and can be calculated with easiness. It is worth mentioning that such a fractal dimension will be formulated as a discretization of Hausdorff dimension. Interestingly, we shall prove that it equals box dimension for Euclidean subsets endowed with their natural fractal structures. Therefore, it becomes a middle definition of fractal dimension which inherits some of the advantages of classical Hausdorff dimension and can be also calculated in empirical applications.

3.1 Introduction In Chap. 2, two novel definitions of fractal dimension for fractal structures have been explored from both theoretical and applied viewpoints. Recall that fractal dimension I allows a larger collection of fractal structures than box dimension for calculation purposes. In fact, if the natural fractal structure which any Euclidean subset can be endowed with (c.f. Definition 2.2.1) is fixed, then the classical box dimension remains as a particular case (c.f. Theorem 2.2.4). On the other hand, though fractal dimension II allows that different diameter sets may appear in any level of a fractal structure, it does not actually distinguish among different diameter sets (c.f. Remark 2.11.2). Recall that we have to count the number of elements in each level of a fractal structure that intersects a given subset F to calculate its fractal dimensions I and II. Then we have to weigh these quantities by a discrete scale: either a fixed quantity for each level (in the case of fractal dimension I) or the “largest” diameter of all the elements in each family An (F) = {A ∈ n : A ∩ F = ∅} (in the case of fractal dimension II). Both ideas lead to suitable discretizations regarding the classical box dimension. Nevertheless, the Hausdorff dimension still constitutes the most accurate model to calculate the fractal dimension in metrizable spaces. Thus, our main goal in this chapter is to analytically construct a new definition of fractal dimension for a fractal structure similarly to classical Hausdorff dimension. © Springer Nature Switzerland AG 2019 M. Fernández-Martínez et al., Fractal Dimension for Fractal Structures, SEMA SIMAI Springer Series 19, https://doi.org/10.1007/978-3-030-16645-8_3

85

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3 A Middle Definition Between Hausdorff and Box Dimensions

3.2 Analytical Construction of a New Fractal Dimension Let  be a fractal structure on a metric space (X, ρ) and F be a subset of X . We shall “measure” the size of each element in any collection An (F) by its diameter. Moreover, let s be a nonnegative real number. Notice that the sum of the s−powers of the diameters of all the elements in each family An (F) allows to study the irregularities that F presents provided that it is explored by a whole range of scales. Hence, let us define the following expression for each natural number n: Hns (F) =

 {diam (A)s : A ∈ An (F)},

(3.1)

as well as its asymptotic behavior, H s (F) = lim Hns (F).

(3.2)

n→∞

Let t be another nonnegative real number and consider Eq. (3.1). Then 

diam (A)t ≤ diam (F, n )t−s ·



diam (A)s ,

(3.3)

where A ∈ An (F). Notice that Eq. (3.3) is equivalent to Hnt (F) ≤ Hns (F) · diam (F, n )t−s .

(3.4)

Letting n → ∞, we have H t (F) ≤ H s (F) · lim diam (F, n )t−s . n→∞

Thus, if H s (F) < ∞ and diam (F, n ) → 0 for all t > s, then H t (F) = 0. Accordingly, under the hypothesis diam (F, n ) → 0 (recall that such a condition regarding the elements in each level of  makes that fractal structure to be starbase, c.f. Proposition 2.5.2), this new fractal dimension for any subset with respect to a fractal structure can be described as the critical point where H s (F) “jumps” from ∞ to 0. Formally, sup{s ≥ 0 : H s (F) = ∞} = inf{s ≥ 0 : H s (F) = 0}, provided that diam (F, n ) → 0. Going beyond, that hypothesis, which is only a natural constraint regarding the size of the elements in each level of the involved fractal structure, becomes necessary as the next counterexample highlights (c.f. Fig. 3.1).

3.2 Analytical Construction of a New Fractal Dimension

87

∞ ...

Hs (F)

0 0

2

s

∞

H3s (F)

0 0

3

dimΓ (F)

Fig. 3.1 Graph representation of s ≥ 0 versus H s (F) for a subset F of X such that diam (F, n )  0 as n → ∞ (c.f. Counterexample 3.2.1) and plot of s versus H3s (F) via Definition 3.3.1 of fractal dimension (bottom)

Counterexample 3.2.1 There exist a fractal structure  on a metric space (X, ρ) and a subset F of X with diam (F, n )  0, satisfying that sup{s ≥ 0 : H s (F) = ∞} = inf{s ≥ 0 : H s (F) = 0}.

Proof Let F = [0, 1] × [0, 1] and  be the natural fractal structure (induced) on the unit square as a Euclidean subset but adding F itself to each level of . Let us apply both Eqs. (3.1) and (3.2) to calculate the fractal dimension of F. Thus, we shall obtain a plot of s versus H s (F) similar to that one appeared in Fig. 3.1. In fact, we have

88

3 A Middle Definition Between Hausdorff and Box Dimensions

√ diam (F, n ) = 2 for all n ∈ N which leads to diam (F, n )  0. In addition, the following expression holds: Hns (F)

 =2 · 1+ s 2

Hence,

 H s (F) =

1 2n(s−2)

 .

∞ if s < 2 s 2 2 if s > 2.

However, unlike it happens with Hausdorff measure (which always exists for all subsets of X ), the set function Hns (F) described in Eq. (3.1) is not monotonic in n ∈ N. This implies that H s (F) does not exist in general. Accordingly, it becomes necessary to consider lower/upper limits in Eq. (3.2) again. Interestingly, the problem regarding the existence of the limit in Eq. (3.2) can be avoided if each family An (F) is properly replaced by the following covering of F by elements of a certain level of  (deeper than n), instead: An,3 (F) = ∪{Am (F) : m ≥ n}.

(3.5)

It is worth pointing out that if the families An,3 (F) are considered to calculate the fractal dimension of F, then the arguments carried out above still remain valid. Our new approach for a fractal dimension function with respect to a fractal structure is formally stated in the following section.

3.3 Defining Fractal Dimension III Next, we provide the key definition of fractal dimension for fractal structures we shall explore in this chapter. Definition 3.3.1 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , and assume that diam (F, n ) → 0. In addition, consider s (F) = inf{Hms (F) : m ≥ n}, where Hn,3

Hns (F) =

 {diam (A)s : A ∈ An (F)}, and

s H3s (F) = lim Hn,3 (F). n→∞

The fractal dimension III of F is defined as the unique critical point

(3.6)

3.3 Defining Fractal Dimension III

89

dim3 (F) = sup{s ≥ 0 : H3s (F) = ∞} = inf{s ≥ 0 : H3s (F) = 0}. (3.7) s It is worth noting that the sequence {Hn,3 (F)}n∈N (c.f. Eq. (3.6)) can be also described throughout any of the equivalent expressions provided in the next remark.

s Remark 3.3.2 (Equivalent Definitions of Hn,3 ) Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , and assume that diam (F, n ) → 0. s (F) for all natural The following expressions are equivalent to calculate Hn,3 number n: s (1) inf{H m (F) : m ≥ n}. (2) inf{ A∈A m (F) diam (A)s : m ≥ n}. (3) inf{ A∈B diam (A)s : B ∈ An,3 (F)}, where An,3 (F) has been defined in Eq. (3.5).

From Definition 3.3.1, it holds that H3s (F) can be described in terms of dim3 (F):  H3s (F) =

∞ if s < dim3 (F) 0 if s > dim3 (F),

(3.8)

provided that diam (F, n ) → 0. The next remark becomes quite useful for fractal dimension III calculation purposes, since it highlights that it is not necessary to consider lower/upper limits to define H3s (F). Remark 3.3.3 Let  be a fractal structure on a metric space (X, ρ), F be a s (F) is the general subset of X , and assume that diam (F, n ) → 0. Since Hn,3 term of a monotonic increasing sequence in n ∈ N, then the fractal dimension III of F always exists.

3.4 Linking Fractal Dimension III to Some Fractal Dimensions In this section, we contribute some results which theoretically connect fractal dimension III with fractal dimension II (explored in Chap. 2) as well as the classical definitions of fractal dimension, i.e., both Hausdorff and box dimensions.

90

3 A Middle Definition Between Hausdorff and Box Dimensions

Theorem 3.4.1 Let  be a fractal structure on a metric space (X, ρ) and F be a subset of X . Moreover, assume that diam (F, n ) → 0. The three following hold: 2

(1) dim3 (F) ≤ dim2 (F) ≤ dim (F). (2) If diam (A) = diam (F, n ) for all A ∈ An (F), then dim B (F) ≤ dim3 (F). (3) dimH (F) ≤ dim3 (F).

Proof (1) From Eq. (3.6), it holds that s Hn,3 (F) ≤ Nm (F) · diam (F, m )s for all m ≥ n.

Hence, if H3s (F) > 1 for some s ≥ 0, then there exists n ∈ N such that log Nm (F) ≥ −s · log diam (F, m ) for all m ≥ n. Thus, s ≤ limm→∞

log Nm (F) , − log diam (F, m ) 2

which leads to dim3 (F) ≤ dim2 (F) ≤ dim (F). (2) To calculate the box dimension of F, let Nδ (F) be the smallest number of sets of diameter at most δ that cover F (c.f. Theorem 2.3.3(i)). It is clear that dim B (F) = limδ→0

log Nδ (F) log Nδn (F) ≤ limn→∞ , − log δ − log δn

where we have denoted δn = diam (F, n ), for short. Assume that s < dim B (F). If ε > 0 satisfies log Nδn (F) , s + ε < limn→∞ − log δn then there exists n 0 ∈ N such that

log N δn (F) − log δn

> s + ε for all n ≥ n 0 . Hence,

Nδn (F) · δns > δn−ε for all n ≥ n 0 , which leads to Nδn (F) · δns → ∞. Since diam (A) = diam (F, n ) for all A ∈ An (F), then we have Nδn (F) · δns ≤ Hns (F) for all n ∈ N and s ≥ 0. Thus, for all R > 0, there exists n 0 ∈ N such that

3.4 Linking Fractal Dimension III to Some Fractal Dimensions

91

Hms (F) ≥ Nδm (F) · δms > R for all m ≥ n 0 . Since Hns0 ,3 (F) = inf{Hms (F) : m ≥ n 0 } (by Eq. (3.6)), then Hns0 ,3 (F) ≥ R. s s In addition, Hn,3 (F) ≥ Hns0 ,3 (F) ≥ R for all n ≥ n 0 , since Hn,3 is monotonic s increasing in n ∈ N (c.f. Remark 3.3.3). This leads to H3 (F) = ∞. Accordingly, since s ≤ dim3 (F) for all s < dim B (F), then dim B (F) ≤ dim3 (F). (3) Just notice that any covering in An,3 (F) is a δ−cover. The following corollary follows immediately from Theorem 3.4.1. Corollary 3.4.2 Let  be a fractal structure on a metric space (X, ρ) and F be a subset of X . In addition, assume that diam (F, n ) → 0. The two following hold: 2

(1) dimH (F) ≤ dim3 (F) ≤ dim2 (F) ≤ dim (F). (2) If diam (A) = diam (F, n ) for all A ∈ An (F), then 2

dimH (F) ≤ dim B (F) ≤ dim3 (F) ≤ dim2 (F) ≤ dim (F).

3.5 How to Calculate the Effective Fractal Dimension III It is worth mentioning that for a given subset F ⊆ X , the calculation of each term in the sequence Hns (F) (c.f. Eq. (3.1)) seems to be easier to be calculated than the s (F) (as described in Eq. (3.6)). In addition, as Remark 3.3.3 corresponding one in Hn,3 s is points out, fractal dimension III always exists provided that the set function Hn,3 selected to deal with its effective calculation. Following the above, the next theoretical result we provide allows the calculation of fractal dimension III from easier Eqs. (3.1) and (3.2). Theorem 3.5.1 Let  be a fractal structure on a metric space (X, ρ) and F be a subset of X . In addition, assume that there exists H s (F) with diam (F, n ) → 0. Then the fractal dimension III of F can be calculated as the critical point described as follows: dim3 (F) = sup{s ≥ 0 : H s (F) = ∞} = inf{s ≥ 0 : H s (F) = 0}.

Proof First of all, due to Eq. (3.6), we have s (F) = inf{Hms (F) : m ≥ n}. Hn,3

92

3 A Middle Definition Between Hausdorff and Box Dimensions

s Hence, it is clear that Hn,3 (F) ≤ Hns (F) for all subset F ⊆ X and each n ∈ N. Let ε > 0 be fixed but arbitrarily chosen. Since H s (F) = limn→∞ Hns (F) (c.f. Eq. (3.2)), then there exists a natural number n 0 such that

|H s (F) − Hns (F)| ≤ ε

(3.9)

for all n ≥ n 0 . Accordingly, H s (F) ≤ Hns (F) + ε, which implies that H s (F) ≤ s (F) + ε for all n ≥ n 0 . Letting n → ∞, Hn,3 H s (F) ≤ H3s (F) + ε.

(3.10)

Moreover, observe that Eq. (3.9) also leads to H s (F) ≥ Hns (F) − ε. Thus, similar arguments lead to (3.11) H s (F) ≥ H3s (F) − ε. Accordingly, the following chain of inequalities stands from both Eqs. (3.10) and (3.11): H3s (F) − ε ≤ H s (F) ≤ H3s (F) + ε for all ε > 0. Therefore, H s (F) = H3s (F). Finally, Eq. (3.7) leads to the result.

s 3.6 Measure Properties of Hn,3

As well as the Hausdorff dimension analytical construction is based on a measure, in this section we shall explore some measure properties regarding the set functions s , H3s , and H s that allow the calculation of the fractal dimension III of any Hn,3 subset F of X . To deal with, let P(X ) denote the class of all subsets of X . Recall that an outer measure is a set function μ : P(X ) → [0, ∞] satisfying the three following properties (c.f. , [25, Sect. 5.2], e.g.): (i) It assigns the value 0 to the empty set, i.e., μ(∅) = 0. (ii) It is monotonic increasing: if E, F ∈ P(X ) with E ⊆ F, then μ(E) ≤ μ(F). (iii) It is countably subadditive, namely, it satisfies that μ (∪n∈N An ) ≤

∞ 

μ(An )

n=1

for all sequence {An }n∈N of subsets of X . It is worth pointing out that Hns is an outer measure for all natural number n. Going s can be stated. beyond, the next result regarding the set function Hn,3

s 3.6 Measure Properties of Hn,3

93

Proposition 3.6.1 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , and assume that diam (F, n ) → 0. In addition, let s : P(X ) → [0, ∞] be the set function defined by Hn,3 s (F) = inf{Hms (F) : m ≥ n} Hn,3

(c.f. Eq. (3.6) or one of its equivalent expressions provided in Remark 3.3.2). s (F) is an outer measure for all n ∈ N. Then Hn,3

Proof To prove this result, we shall apply the so-called Method I of construction of outer measures (c.f. [25, Theorem 5.2.2]). In fact, let us define the family of coverings of F: A = ∪{Am (F) : m ≥ n} as well as the set function c : A → [0, ∞] which maps each A ∈ An,3 (F) to the s−power of its diameter, i.e., A → diam (A)s . Finally, for all n ∈ N, just identify s with the outer measure constructed in Method I. the set function Hn,3 The following counterexample points out that the limits as n → ∞ of both outer s measures Hns and Hn,3 are not, in general. Counterexample 3.6.2 Neither H s nor H3s are outer measures.

Proof Let  = {n }n∈N be the natural fractal structure (induced) on [0, 1] with levels  n =

 k k+1 n : k ∈ {0, 1, . . . , 2 − 1} , 2n 2n

and F = Q ∩ [0, 1]. Observe that each rational number qi ∈ F may belong at most to two elements in each level of the fractal structure . Hence, it becomes clear that Hns ({qi }) ≤

1 2ns−1

for all qi ∈ F and each natural number n. But going beyond, since each element in level n of  contains a rational number q ∈ F, then we have Hns (F) =

1 2n(s−1)

.

Therefore, for all s ∈ (0, 1), H s ({qi }) = 0 for all qi ∈ F. This leads to

94

3 A Middle Definition Between Hausdorff and Box Dimensions



H s ({qi }) = 0.

qi ∈F

On the other hand,

 H s (F) = H s ∪qi ∈F {qi } = ∞.

Accordingly, the set function H s (F) is not countably subadditive. The same counterexample remains valid to verify that H3s is not an outer measure.

3.7 Linking Fractal Dimension III to Fractal Dimensions I and II Another issue naturally arising consist of determining some reasonable conditions on the elements in each level of a fractal structure to guarantee the equality among fractal dimension III and fractal dimensions I and II previously explored in Chap. 2. In this way, the following result we provide allows the calculation of fractal dimension III from the fractal dimension I formula provided that fractal structures having and appropriate size (of 21n −order) are selected to deal with the calculations. First, recall that two sequences of positive real numbers f, g : N → R are said f (n) ∈ (0, ∞) (c.f. Definition 2.13.1). to be of the same order provided that limn→∞ g(n) Further, the next technical result follows immediately. Lemma 3.7.1 Let f, g, h : N → R be three sequences of positive real numbers. Assume that f and g are of the same order, namely, O( f ) = O(g) and lim

n→∞

h(n) ∈ (0, ∞). f (n)

Then there exists a constant κ ∈ (0, ∞) satisfying that lim

n→∞

h(n) h(n) = κ · lim . n→∞ g(n) f (n)

Proof The following chain of equalities stands:  h(n) f (n) · f (n) g(n) h(n) f (n) = lim · lim ∈ (0, ∞) n→∞ f (n) n→∞ g(n)

h(n) lim = lim n→∞ g(n) n→∞



(3.12)

3.7 Linking Fractal Dimension III to Fractal Dimensions I and II

95

since O( f ) = O(g), and also by the hypothesis in Eq. (3.12). Hence, lim

n→∞

h(n) h(n) = κ · lim , n→∞ g(n) f (n)

where we can choose κ = limn→∞

g(n) f (n)

∈ (0, ∞).

Following the above, next we shall prove a first result linking fractal dimension I to fractal dimension III. Theorem 3.7.2 Let  be a fractal structure on a metric space (X, ρ) and F be a subset of X . Additionally, assume that diam (F, n ) → 0 and O(diam (F, n )) = O( 21n ). If there exists the fractal dimension I of F, then dim1 (F) = dim3 (F).

Proof Let ε > 0 be fixed but arbitrarily chosen and write α = dim1 (F). Then there exists a natural number n 0 such that 2n(α−ε) ≤ Nn (F) ≤ 2n(α+ε)

(3.13)

for all n ≥ n 0 . Let us denote δn = diam (F, n ). Then we have s (F) ≤ Nm (F) · δms ≤ 2m(α+ε) · δms Hn,3

(3.14)

for all m ≥ n ≥ n 0 , due to Eq. (3.13). Letting n → ∞ in previous Eq. (3.14), the following expression holds: s (F) ≤ lim 2m(α+ε) · δms H3s (F) = lim Hn,3 n→∞ m→∞  ∞ if s < α + ε = κ s · lim 2m(α+ε−s) = 0 if s > α + ε, m→∞

where the second equality stands by Lemma 3.7.1. Hence, dim3 (F) ≤ dim1 (F) + ε.

(3.15)

On the other hand, let δ > 0 be another fixed but arbitrarily chosen real number. Thus, for all n ∈ N there exists m(n) ≥ n such that s s s (F) + δ ≥ Nm(n) (F) · δm(n) ≥ 2m(n)(α−ε) · δm(n) , Hn,3

where the second equality is due to Eq. (3.13). Letting n → ∞,

(3.16)

96

3 A Middle Definition Between Hausdorff and Box Dimensions

H3s (F) + δ ≥

s lim Nm(n) (F) · δm(n) ≥

m(n)→∞

= ρs ·

lim 2m(n)·((α−ε)−s)

m(n)→∞

s lim 2m(n)(α−ε) · δm(n)  ∞ if s < α − ε = 0 if s > α − ε, m(n)→∞

where Lemma 3.7.1 has been applied to deal with the first equality. Thus, if s < α − ε, then H3s (F) + δ = ∞ and the arbitrariness of δ throws H3s (F) = ∞. Accordingly, dim1 (F) − ε ≤ dim3 (F).

(3.17)

Finally, both Eqs. (2.9.3) and (3.17) and the arbitrariness of ε > 0 give the equality between fractal dimensions I and III for this kind of fractal structures. Regarding the existence of the fractal dimension I of F in previous theorem, it is noteworthy that if fractal dimension I does not exist, then Theorem 3.7.2 still throws the expected equality between (lower) fractal dimensions I and fractal dimension III. Next, we shall highlight that theoretical fact. Remark 3.7.3 Under the hypothesis of Theorem 3.7.2, suppose that fractal dimension I does not exist for a given subset F ⊆ X . Then we can affirm that dim1 (F) = dim3 (F).

Proof To deal with, let α = dim1 (F). Then there exists a subsequence 

log2 Nn k (F) nk





n k ∈N

such that α = lim

k→∞

⊆

log2 Nn (F) n

n∈N

log2 Nn k (F) . nk

Hence, continue by applying similar arguments to those carried out in the proof of Theorem 3.7.2. Next step is to find out appropriate conditions on the size of the elements in each level of a fractal structure to reach the equality between fractal dimensions II and III. The proof of the following result may be dealt with similarly to Theorem 3.7.2. Theorem 3.7.4 Let  be a fractal structure on a metric space (X, ρ) and F be a subset of X . Additionally, assume that diam (F, n ) → 0 and there exists a natural number n 0 such that diam (A) = diam (F, n ) for all A ∈ An (F)

3.7 Linking Fractal Dimension III to Fractal Dimensions I and II

97

and all n ≥ n 0 . If there exists the fractal dimension II of F, then dim2 (F) = dim3 (F). Under the hypothesis of Theorem 3.7.4, a weaker result than Theorem 3.7.4 can be stated in the case that fractal dimension II does not exist. This is similar to Remark 3.7.3 allowing dim1 (F) = dim3 (F) and can be proved just applying similar techniques to Remark 3.7.5. Remark 3.7.5 Under the hypothesis of Theorem 3.7.4, assume that fractal dimension II does not exist for a given subset F ⊆ X . Then we still have dim2 (F) = dim3 (F). From both Remarks 3.7.3 and 3.7.5, we can state that fractal dimension III generalizes both fractal dimensions I and II for fractal structures having 21n −order elements in each level n. Corollary 3.7.6 Let  be a fractal structure on a metric space (X, ρ) and F be a subset of X . Additionally, assume that diam (A) = diam (F, n ) for all A ∈ An (F) and O(diam (F, n )) = O( 21n ). Then dim1 (F) = dim2 (F) = dim3 (F). It is worth mentioning that Corollary 3.7.6 allows the calculation of the fractal dimension III of any subset with respect to a fractal structure  under the conditions provided therein via an easier box dimension type formula. Moreover, the following result establishes that all these fractal dimensions are equal in the context of Euclidean GF-spaces endowed with their natural fractal structures. In other words, fractal dimension III generalizes all the box dimension type models for fractal dimension including the classical one. Theorem 3.7.7 Let  be the natural fractal structure on Rd and F ⊆ Rd . Then the (lower/upper) box dimension of F equals the (lower/upper) fractal dimensions I, II, and III of F, namely, dim B (F) = dim1 (F) = dim2 (F) = dim3 (F).

98

3 A Middle Definition Between Hausdorff and Box Dimensions

Proof First, Theorem 2.9.3 gives dim B (F) = dim1 (F) = dim2 (F). Further, Remark 3.7.3 (or Theorem 3.7.2) leads to dim1 (F) = dim3 (F), since the diameters of all the elements in level n of the natural fractal structure on Rd are equal to √ d . 2n Previous Theorem 3.7.7 makes fractal dimension III to be understood as a hybrid approach to fractal dimension. In fact, though the analytical construction of fractal dimension III is based on a suitable discretization regarding the Hausdorff dimension, such a result states that fractal dimension III equals box dimension in the context of Euclidean subsets equipped with their natural fractal structures. It is also worth mentioning that Theorem 3.7.7 also allows the calculation of fractal dimension III for Euclidean subsets throughout easier box dimension type expressions such as those provided in Chap. 2.

3.8 Theoretical Properties of Fractal Dimension III In this section, we explore some theoretical properties for fractal dimension III similarly to Proposition 2.4.1 for fractal dimension I and Proposition 2.10.1 for fractal dimension II. Proposition 3.8.1 Let  be a fractal structure on a metric space (X, ρ) and assume that diam (F, n ) → 0. The following statements hold: (1) Fractal dimension III is monotonic. (2) Fractal dimension III is finitely stable. (3) There exist a countable subset F of X and a fractal structure  on X such that dim3 (F) = 0. (4) Fractal dimension III is not countably stable. (5) There exists a locally finite starbase fractal structure  defined on a certain subset F ⊆ X such that dim3 (F) = dim3 (F).

Proof (1) Let E, F be two subsets of X and assume that E ⊆ F. Hence, it becomes clear that Hns (E) ≤ Hns (F) for all n ∈ N. Letting n → ∞, we have H s (E) ≤ H s (F). Accordingly, dim3 (E) ≤ dim3 (F). (2) Let E, F be two subsets of X . First, note that Hns (E ∪ F) ≤ Hns (E) + Hns (F) for all n ∈ N. Letting n → ∞, it holds that H s (E ∪ F) ≤ H s (E) + H s (F). Further, let s ≥ 0 : H s (E ∪ F) = ∞ and assume, without loss of generality, that dim3 (E) ≤ dim3 (F). Thus, H s (F) = ∞, leading to dim3 (E ∪ F) ≤ dim3 (F). The opposite inequality becomes straightforward since fractal dimension III is monotonic (c.f. Proposition 3.8.1(1)). Accordingly,

3.8 Theoretical Properties of Fractal Dimension III

99

dim3 (E ∪ F) = max{dim3 (E), dim3 (F)}. (3) Since fractal dimension III generalizes fractal dimension I in the context of Euclidean subsets endowed with their natural fractal structures (see Theorem 3.7.7), the next counterexample, which was provided to verify Proposition 2.4.1(3), becomes also valid to deal with this fractal dimension III issue. More specifically, let X = [0, 1], F = Q ∩ X , and  be the natural fractal structure (induced) on X , whose levels are defined by  n =

 k k+1 n : k ∈ {0, 1, . . . , 2 − 1} . , 2n 2n

Hence, dim1 (F) = dim3 (F) = 1 > 0. (4) Notice that (3) gives (4). In fact, for all rational number qn ∈ F = Q ∩ [0, 1], let us denote Fn = {qn }. Thus, it becomes clear that dim3 (Fn ) = 0 for all n ∈ N, leading to sup{dim3 (Fn ) : n ∈ N} = 0. On the other hand, it holds that dim3 (F) = dim3

⎛

 Fn

= dim3 ⎝

n∈N

⎞ {qn }⎠ = 1,

qn ∈F

and hence, fractal dimension III is not countably stable. (5) Let  = {n : n ∈ N} be the fractal structure defined on the space X = ([0, 1] × {0})



 1 × [0, 1] : n ∈ N , 2n

whose levels are given as follows (see Fig. 2.1 for illustration purposes):  k k+1 n × {0} : k ∈ {0, 1, . . . , 2 − 1} n = , 2n 2n   

k k+1 1 n × n, n : k ∈ {0, 1, . . . , 2 − 1}, m ∈ N . 2m 2 2 

Additionally, define F = whereas



1 1 k∈N ( 2k+1 , 2k )

Hns (F) = Accordingly, dim3 (F) = ∞.

× {0} ⊆ X . Hence, dim1 (F) = 1,

∞ ∞   1 1 + = ∞. is 2 2ns i=1 i=1

100

3 A Middle Definition Between Hausdorff and Box Dimensions

3.9 An Additional Connection with Box Dimension Recall that in Theorem 2.12.3, some properties regarding the elements in each level of a fractal structure were provided to reach the equality between fractal dimension II and box dimension. We would like also to point out that box dimension may be also defined for metrizable spaces. The next result we provide has been carried out in the spirit of Theorem 2.12.3 and generalizes Theorem 3.7.7. To deal with, we shall assume that  is under the κ−condition (c.f. Definition 2.1). Theorem 3.9.1 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , assume that diam (F, n ) → 0, and suppose that  is under the κ−condition. If diam (A) = diam (F, n ) for all A ∈ An (F), and there exists dimB (F), then dimB (F) = dim3 (F).

Proof The result follows by applying both Theorems 2.12.3(2) and 3.7.4.

3.10 Fractal Dimension III for IFS-Attractors As it was stated previously in Sect. 2.14, the issue concerning the calculation of the fractal dimension for IFS-attractors via algebraic expressions involving only a finite number of known quantities arises naturally for each new definition of fractal dimension. It is worth mentioning that this kind of theoretical results is inspired on classical Moran’s Theorem 1.4.14 and usually assume that the similarities that give rise to the IFS-attractor are under the OSC hypothesis (see Definition 2.14.1). In fact, recall that this constitutes the main constraint required to an IFS to reach the equality between the Hausdorff and the box dimensions of its strict self-similar set (c.f. Theorem 1.4.14). The OSC is a strong hypothesis required to the pre-fractals of an IFS-attractor to control their overlapping. In this way, Theorem 2.14.3 stands under the OSC for fractal dimension II. Interestingly, the fractal dimension III model allows the calculation of the fractal dimension of strict self-similar sets via a Moran’s type equation (c.f. Eq. (4.19)) even if the similarities of the IFS are not under the OSC. This allows to generalize Moran’s Theorem 4.8.5 in the context of fractal structures. To prove such a theoretical result, both the natural fractal structure which any IFSattractor can be always endowed with (c.f. Definition 1.6.3 or Remark 1.6.4) and s do play a relevant role herein. Equivalent Definition (2) in Remark 3.3.2 for Hn,3

3.10 Fractal Dimension III for IFS-Attractors

101

Theorem 3.10.1 Let X be a complete metric space, F = { f 1 , . . . , f k } be an IFS whose IFS-attractor is K , ci be the similarity ratio associated with each similarity f i on X , and  be the natural fractal structure on K as a self-similar set. Then k  dim3 (K ) = s, where cis = 1. i=1

Additionally, for such a value of s, it holds that H3s (K ) ∈ (0, ∞).

Proof Recall that K is the unique non-empty compact subset of X satisfying the following Hutchinson’s equation: K =

k

f i (K ).

i=1

Since all the elements in each level of the fractal structure  intersect K , we have

{m : m ≥ n} for all n ∈ N. An,3 (K ) = k cis = 1 and define Jl = {(i 1 , . . . , il ) : i j = On the other hand, let s ≥ 0 : i=1 1, . . . , k, j = 1, . . . , l}. If we denote Ki1 ... il = f i1 ◦ · · · ◦ f il (K ), then we can write K = Jl Ki1 ... il . In addition, it is clear that ci1 · · · cil is the similarity ratio associated with the composition of similitudes f i1 ◦ · · · ◦ f il . Hence, s Hn,3 (K ) = inf



diam (A)s : A ∈ Am (K ), m ≥ n  diam (Ki1 ... im )s : m ≥ n = inf



Jm

= inf

 k i 1 =1

cis1 · . . . ·

k 

cism · diam (K )s : m ≥ n

i m =1

for all natural number n. Letting n → ∞, we have H3s (K ) = diam (K )s . / {0, ∞}, so s is the critical point where H3s (K ) “jumps” Accordingly, H3s (K ) ∈ from ∞ to 0, namely, s = dim3 (K ). Next, we verify that Theorem 3.10.1 cannot be improved in the sense that the similarities f i ∈ F , which give rise to the IFS-attractor K , cannot be weakened to merely contractions. To deal with, we provide an appropriate counterexample.

102

3 A Middle Definition Between Hausdorff and Box Dimensions

Counterexample 3.10.2 There exists a Euclidean IFS F = { f 1 , . . . , f k } whose (non-strict) IFS-attractor K , endowed with its natural fractal structure as a self-similar set, satisfies that dim3 (K ) = s :

k 

cis = 1.

i=1

Proof Let I = {1, . . . , 8} be a finite index set and (R2 , F = { f i : i ∈ I }) be a Euclidean IFS whose attractor is K = [0, 1] × [0, 1]. Define the self-maps f i : R2 → R2 as follows:  −y x ( 2 , 4 ) + ( 21 , i−1 ) if i = 1, . . . , 4; 4 f i (x, y) = −y x i−5 ( 2 , 4 ) + (1, 4 ) if i = 5, . . . , 8. In addition, let  be the natural fractal structure on K as a self-similar set. First of all, notice that K is not a strict self-similar set. Further, observe that the contractions f i are compositions of affine maps, including rotations, dilations (in the plane and with respect to one coordinate), and translations. Moreover, all the contractions f i have a common ratio, equal to 21 . It is also clear that s = 3 is the solution of the k equation i=1 cis = 1. On the other hand, we affirm that dim 3 (K ) = 2. To deal with, we shall calculate the fractal dimension III of K in the sense of Theorem 3.9.1. Consider all the even levels in . Thus, for all natural number n, each level 2n consists of squares with √ sides equal to 1/8n . Also, we have diam (A) = δ(K , 2n ) = 2/8n for all A ∈ 2n . Letting n → ∞, it holds that δ(K , 2n ) → 0. Next, we verify that  is under the κ−condition. We shall proceed by calculating the maximum number of elements in √ that the ratio 2n that are intersected by a subset B : diam (B) ≤ 2/8n . Observe √ between the diameter of each square in level 2n and its side is equal to 2 < 2, then it holds that the number of elements in A2n (B) is at most 3 in each direction for all subset B : diam (B) < δ(K , 2n ). Accordingly, κ1 = 9 provides a suitable constant for all the levels of even order in . Similarly, notice that all the levels of odd order in  consist of rectangles with dimensions 21 · 81n × 14 · 81n for all n ∈ N. It is worth noting√that all the elements in each odd level 2n + 1 have the same diameter, equal to 41 · 8n5 . Hence, the sequence of diameters δ(K , 2n+1 ) → 0. Finally, to check the κ−condition, observe that the following ratios between each diameter and the corresponding sides of each rectangle stand: √ 1 · 8n5 4 1 · 1 2 8n

√

5 0. By uniform continuity of α, there exists δ > 0 such that d(t, s) < δ implies d(α(t), α(s)) < ε. Moreover, since δ(F, n ) → 0, then there exists n 0 ∈ N such that δ(F, n ) < δ for each n ≥ n 0 . Let n ≥ n 0 and A ∈ n such that α(A) ∩ F  = ∅. Thus, A ∩ F = ∅, so diam (A) < δ. Therefore, diam (α(A)) ≤ ε. Accordingly, δ(F  , n ) ≤ ε for each n ∈ N, and hence, δ(F  , n ) → 0. Next, we define the fractal dimension of (the parametrization of) a curve via the induced fractal structure introduced in Definition 3.11.1. Definition 3.11.3 Let X be a metric space, α : I → X be a parametrization of a curve,  be a fractal structure on I , and  be the fractal structure induced by  on α(I ) ⊆ X . The fractal dimension of (the parametrization of) the curve α is defined throughout the next expression: dim (α) = dim (α(I )). If no additional information regarding the starting fractal structure  is provided, then we shall assume that  is the natural fractal structure on I ⊂ R as a Euclidean subset (c.f. Definition 2.2.1). In that case, the fractal dimension of a curve α, dim  (α), will be denoted merely as dim(α). It is noteworthy that the fractal dimension provided in Definition 3.11.3 actually takes into account all the overlappings among the elements in each level of . Notice that to explore the fractal behavior of curves throughout the classical models of fractal dimension, it is necessary to consider the graphindexgraphs of functions of the curve instead of its image set. In fact, it makes no sense to calculate either the box dimension or the Hausdorff dimension of the image set of a curve. Thus, we can highlight that this new definition of fractal dimension allows a deeper study regarding the complexity of curves since that dimension can be calculated with respect to different parametrizations of the same curve. Interestingly, Definition 3.11.3 can be applied for fractal dimension calculation purposes even if the curve α is not continuous. This allows α to be a time series, for instance. Regarding this, in upcoming Sect. 3.12, we shall apply the previous definitions to connect the fractal dimension of random processes with their selfsimilarity exponents.

3.11 A New Fractal Dimension for Curves

109

3.11.3 Theoretical Properties of the New Fractal Dimension In this subsection, we shall explore some properties concerning the new fractal dimension for curves provided in Definition 3.11.3. First of all, let us prove the following technical lemma. Lemma 3.11.4 Let s ∈ (0, 1) and a, b, c ≥ 0 satisfying that a ≤ b + c. Then a s ≤ bs + cs .

Proof If b > a, then the result follows immediately. Indeed, then bs > a s leading to bs + cs ≥ a s + cs ≥ a s . Therefore, we shall focus on the case b < a. Thus, there exists r ∈ (0, 1) such that b = a · r . Hence, bs + cs ≥ a s · r s + a s · (1 − r )s = a s · (r s + (1 − r )s ) ≥ a s , where the first inequality stands since c ≥ a − b = a · (1 − r ). Further, the last inequality holds since x s ≥ x for all s, x ∈ (0, 1). In fact, this gives (1 − r )s ≥ 1 − r and r s ≥ r , as well, so (1 − r )s + r s ≥ 1 − r + r = 1. Next, we prove several properties regarding the new fractal dimension for curves from a theoretical viewpoint. Proposition 3.11.5 Let X be a distance space, α : I → X be a parametrization of a curve,  be the natural fractal structure on I ⊂ R, and  be the fractal structure induced by  on α(I ) ⊆ X . The following statements hold: (1) If α is a constant curve, then dim(α) = 0. (2) If α is a nonconstant curve, then dim(α) ≥ 1. (3) If α is a Lipschitz function, then dim(α) ≤ 1. In particular, this is satisfied by every differentiable map with bounded differential. (4) If α is a nonconstant Lipschitz function, then dim(α) = 1.

Proof (1) For all B ∈ n , there exists A ∈ n : B = α(A) = {α(t) : t ∈ A} = { p} since α is constant. Hence, diam (B) = diam (α(A)) = 0 for all B ∈ n . Thus,

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3 A Middle Definition Between Hausdorff and Box Dimensions

Hns (α(I )) =

 {diam (B)s : B ∈ n , B ∩ α(I ) = ∅}

= diam (B)s · Card (n ) = 0s · 2n = 0 = 0. Therefore, H s (α(I )) = 0 for all s ≥ 0. Accordingly, since dim(α) = inf{s ≥ 0 : H s (α(I )) = 0}, then we have dim(α) ≤ s for all s ≥ 0, namely, dim(α) = 0. (2) First, notice that for all A ∈ n , there exist B, C ∈ n+1 such that A = B ∪ C. Thus, diam (A)s ≤ diam (B)s + diam (C)s for all s ∈ (0, 1) by Lemma 3.11.4. This leads to s (α(I )) Hns (α(I )) ≤ Hn+1

for all n ∈ N and since α is not constant, then H1s (α(I )) > 0. Hence, H s (α(I )) > 0 for all s < 1, so dim(α) ≥ s for all s < 1. Accordingly, dim(α) ≥ 1. (3) Observe that each B ∈ n can be written as B = α(A), where A ∈ n . Even more, it can be stated that   k k+1 B=α , 2n 2n for some k ∈ {0, . . . , 2n − 1}. On the other hand, let L be the Lipschitz constant associated with α. Thus,   L k k+1 = n, , diam (B) ≤ L · diam n n 2 2 2 leading to diam (B)s ≤

L s

Hns (α(I )) =

2n

. Hence,

 {diam (B)s : B ∈ n , B ∩ α(I ) = ∅}

≤ L s · 2n(1−s) , since Card (n ) = 2n . Accordingly, H s (α(I )) = 0 for all s > 1. Therefore, dim(α) ≤ s for all s > 1, namely, dim(α) ≤ 1.

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111

(4) This stands by Proposition 3.11.5(2) and (3). Next, we summarize how the values provided by this fractal dimension can be understood. Remark 3.11.6 • Every nonconstant continuous curve has fractal dimension d ∈ [1, ∞). • A bigger fractal dimension means that the oscillations of the curve do increase at any range of scales. • Curves with smaller fractal dimensions are graphically described throughout smoother graphs. • If α is a smooth curve, then dim(α) = 1. • If α is a Brownian motion, then it holds that dim(α) = 2.

3.11.4 How to Construct Space-Filling Curves via Fractal Structures The main purpose in this section is to prove a theoretical result allowing to construct space-filling curves through fractal structures. Since Peano first described a plane-filling curve in 1890 [64], several curves of this kind have appeared in mathematical literature, being the Hilbert’s curve (first appeared in [43]), maybe, one of the most popular. The next result we provide is the key to construct continuous maps between two topological spaces. This will be helpful to illustrate how to deal with the construction of space-filling curves in upcoming sections. Theorem 3.11.7 Let  = {n : n ∈ N} be a starbase fractal structure on a metric space X and  = {n : n ∈ N} be a −Cantor-complete starbase fractal structure on a complete metric space Y . Moreover, for each natural number n, let f n : n → n be a family of maps. (1) Assume that { f n : n ∈ N} satisfies the two following conditions: • If A ∩ B = ∅ : A, B ∈ n for some n ∈ N, then f n (A) ∩ f n (B) = ∅. • If A ⊆ B : A ∈ n+1 , B ∈ n for some n ∈ N, then f n+1 (A) ⊆ f n (B). Then there exists a unique continuous map f : X → Y satisfying that f (A) ⊆ f n (A) for each A ∈ n and all n ∈ N. (2) Additionally, suppose that the fractal structure  is −Cantor-complete and also that f n fulfills the two following properties:

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3 A Middle Definition Between Hausdorff and Box Dimensions

(i) f n is onto. (ii) f n (A) = { f n+1 (B) : B ∈ n+1 , B ⊆ A} for all A ∈ n . Then f is onto and f (A) = f n (A) for all A ∈ n and all n ∈ N.

Proof (1) First of all, we take with the definition of the map f : X → Y . To deal with, observe that for each x ∈ X , there exists a sequence {An : n ∈ N} such that An ∈ n , An+1 ⊆ An , and x ∈ ∩n∈N An . Hence, { f n (An ) : n ∈ N} is a decreasing sequence with f n (An ) ∈ n . Moreover, since  is −Cantor-complete and starbase, then we have that ∩n∈N f n (An ) is a single point. In this way, let us define { f (x)} = ∩n∈N f n (An ). The next statements are fulfilled. • f is well-defined. Indeed, let x ∈ X and consider sequences {An : n ∈ N} and {An : n ∈ N} such that An , An ∈ n , An+1 ⊆ An , and An+1 ⊆ An for each n ∈ N, where x ∈ ∩n∈N An and x ∈ ∩n∈N An , as well. Further, let {y} = ∩n∈N f n (An ) and {z} = ∩n∈N f n (An ). Since x ∈ An ∩ An , then f n (An ) ∩ f n (An ) = ∅ by hypothesis on f n . Thus, if y = z, then f n (An ) ∩ f n (An ) = ∅ for some n ∈ N since  is starbase, a contradiction. Accordingly, y = z.  • f (A) ⊆ f n (A) for each A ∈ {n : n ∈ N}. This is clear by definition of f . • f is continuous. Let x ∈ X and n ∈ N. If y ∈ St (x, n ), then there exists A ∈ n : x, y ∈ A. Hence, f (x), f (y) ∈ f (A) ⊆ f n (A). Since f n (A) ∈ n , then f (y) ∈ St ( f (x), n ). • Uniqueness of f . Let g : X → Y be a map satisfying that g(A) ⊆ f n (A) for all  A ∈ {n : n ∈ N}. Given x ∈ X , there exists a sequence {An : n ∈ N} such that An ∈ n , An+1 ⊆ An , and x ∈ ∩n∈N An . Thus, g(x) ∈ ∩n∈N f n (An ) = { f (x)}, namely, f (x) = g(x) for all x ∈ X . (2) Assume that  is −Cantor-complete and also that f n is under the two additional properties  (2i)–(2ii) above. To deal with, let us verify that f n (A) ⊆ f (A) for all A ∈ {n : n ∈ N}. In fact, this yields that f is onto. Let n ∈ N, A ∈ n , and y ∈ f n (A). Moreover, let Bn = f n (A) and An = A. Then (by hypothesis) there exists An+1 ∈ n+1 such that An+1 ⊆ An and y ∈ f n+1 (An+1 ) ⊆ f n (An ). Denote Bn+1 = f n+1 (An+1 ). Hence, we can recursively construct the following sequences: {Bk : k ≥ n} and {Ak : k ≥ n} such that Ak ∈ k such that f k (Ak ) = Bk ∈ k , y ∈ Bk : k ≥ n, An = A, and Bn = f n (A). Then ∩k≥n Ak = {x} for some x ∈ X , since  is −Cantor-complete and starbase. Accordingly, by construction, we have that f (x) = y ∈ f (A) since x ∈ A. We would like to point out that Theorem 3.11.7 becomes the key to iteratively define functions or curves throughout fractal structures. Such a result could be understood as follows. Let us define the image of the first level of the starting fractal structure as the first approach to the definition of the function (curve). Then we refine that definition to the second level, and so on. Thus, if that refining process is under

3.11 A New Fractal Dimension for Curves

113

several natural conditions (just for the coherence of the definition), then there exists a map defined on the space which matches with the approaches that have been carried out in each level. It is also worth noting that, since the proof of Theorem 3.11.7 is constructive, then we can obtain different space-filling curves by choosing different fractal structures in a space or by choosing different chains in the construction, as well. This allows a great flexibility to deal with the construction of space-filling curves which will be illustrated in the upcoming sections.

3.11.5 The Fractal Dimension of the Hilbert’s Curve The main goal in the three forthcoming sections is to show, through some selected examples, how fractal structures allow to accurately describe several space-filling curves, including the classical Hilbert’s curve and a modified Hilbert’s curve, too. Moreover, for fractal dimension calculation purposes, we shall apply Definition 3.11.3 as well as classical models of fractal dimension. Thus, in upcoming Example 3.11.9 we prove that the new fractal dimension for curves throws a natural value for Hilbert’s space-filling curve, whereas in later Example 3.11.12, we shall verify that this model allows a deeper study concerning space-filling curves than by Hausdorff and box dimensions. Example 3.11.8 (The Hilbert’s curve, 1891) The smart construction regarding the classical Hilbert’s plane-filling curve can be iteratively carried out throughout fractal structures. To deal with, let  be a fractal structure on the closed unit interval I , whose levels are defined by n =

 k k + 1 2n : k ∈ {0, 1, . . . , 2 , − 1} . 22n 22n

On the other hand, let  be the natural fractal structure on I × I with levels     k2 k2 + 1 k1 k1 + 1 n × n, : k1 , k2 ∈ {0, 1, . . . , 2 − 1} . , n = 2n 2n 2 2n Next, we shall define the image of each level in  via a map α : I → X which iteratively approaches the Hilbert’s plane-filling curve. This is carried out by a sequence of maps {αn : n ∈ N}, where the definition of each map αn : n → n has been illustrated (for its first levels) in Fig. 3.3 and can be mathematically understood as follows:

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3 A Middle Definition Between Hausdorff and Box Dimensions

Fig. 3.3 First three levels in the iterative construction of the classical Hilbert’s curve

     1 1 1 = 0, × 0, . α 0, 4 2 2       1 1 1 1 , = , 1 × 0, . α 4 2 2 2       1 1 1 3 , = ,1 × ,1 . α 2 4 2 2       1 1 3 , 1 = 0, × ,1 . α 4 2 2 

This allows to define the whole covering α(1 ) = {α(A) : A ∈ 1 }. We can proceed similarly with the next levels of . The polygonal line in Fig. 3.3 displays how the plane is filled by α in each level of . Further, this recursive

3.11 A New Fractal Dimension for Curves

115

procedure allows to refine the definition of αn in each stage of that construction since additional information regarding the curve is provided as deeper levels are reached. Accordingly, if A ∈ n → B ∈ n via αn , then in the next level, A = ∪{C ∈ n+1 : C ⊆ A}, B = ∪{D ∈ n+1 : D ⊆ B}, and each C → D via αn+1 . Letting n → ∞, the Hilbert’s curve α stands as the limit of the sequence of maps {αn : n ∈ N}. Next step is to compare the fractal dimension of the Hilbert’s plane-filling curve to its Hausdorff and box dimensions. Remark 3.11.9 Let α : I → R2 be the parametrization of the Hilbert’s curve provided in Example 3.11.8, and ,  be the fractal structures considered previously on I and I × I , respectively. Then dim H (α(I )) = dim B (α(I )) = dim(α) = 2.

Proof To calculate the fractal dimension of the Hilbert’s curve according to Definition 3.11.3, we shall calculate the number of elements in each level of  that intersect the image set α(I ). In fact, we have Hns (α(I )) =

 {diam (B)s : B ∈ n , B ∩ α(I ) = ∅} = 2s/2 · 2n(2−s) ,

since there are 22n elements√in each level n that intersect α(I ), where the diameter of each of them is equal to 2/2n . Letting n → ∞,  H (α(I )) = s

∞ if s < 2; 0 if s > 2.

Accordingly, dim(α) = 2. It is noteworthy that the fractal dimension value of the Hilbert’s curve turns out to be quite natural since that curve fills the whole unit square and therefore, it must equal both its Hausdorff and box dimensions.

3.11.6 A Curve Filling the Sierpinski’s ´ Gasket In Sect. 3.11.5, we explored the fractal dimension of a plane-filling curve. Next, we explain how the technique applied therein can be further extended to deal with curves filling a whole self-similar set.

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3 A Middle Definition Between Hausdorff and Box Dimensions

Example 3.11.10 There exists a curve γ : I → X filling the whole Sierpi´nski’s gasket X .

Proof Let us apply Theorem 3.11.7 to deal with the construction of such a curve. In fact, let  be a fractal structure on the closed unit interval I with levels defined as follows:   k k+1 n : k ∈ {0, 1, . . . , 3 − 1} , , n = 3n 3n and X be the Sierpi´nski’s gasket (endowed with the Euclidean distance) contained in the equilateral triangle described through the set of points    1 (0, 0), , 1 , (1, 0) ⊂ R2 . 2 In addition, let  be the fractal structure on X as a self-similar set (c.f. Definition 1.6.3). It is worth noting that, in this case,  equals the fractal structure induced by  on γ (I ) ⊆ X (in the sense of Definition 3.11.1). Thus, the definition of the curve γ : I → X will be carried out throughout a sequence of maps {γn : n → n : n ∈ N} whose definition has been illustrated in Fig. 3.4 for its first two levels. Notice that the red line shows how the Sierpi´nski’s gasket is filled by γn in each level of that iterative construction. For instance, observe that 1 = γ (1 ) refers to the first level in the fractal structure induced by  on γ (I ) ⊆ X and consists of the three equilateral triangles described next:        1 1 1 1 is the triangle with vertices (0, 0), , , ,0 . γ 0,  3   4 2  2 1 2 1 3 1 γ , is the triangle with vertices ( , 0), , , (1, 0) . 4 2    3 3  2    2 1 1 1 3 1 γ , 1 is the triangle with vertices , , ,1 , , . 3 4 2 2 4 2 Note that since all of these triangles lie in the Sierpi´nski’s gasket, they are not fully filled. In this way, this procedure could be similarly applied to construct the following levels of . Next, we shall calculate the fractal dimension of the curve γ , whose construction was described in previous Example 3.11.10, and compare it to the classical dimensions of the image set γ (I ) ⊆ X . Remark 3.11.11 Let γ : I → X be the parametrization of the curve provided in Example 3.11.10, where X denotes the Sierpi´nski’s gasket. In addition, let ,  be the fractal structures considered previously on I and induced by  on

3.11 A New Fractal Dimension for Curves

117

Fig. 3.4 First two levels in the construction of a curve defined on the Sierpi´nski’s gasket which fills the whole self-similar set

γ (I ) ⊆ X , respectively. Then dimH (γ (I )) = dimB (γ (I )) = dim(γ ) = log2 3.

Proof First of all, we would like to point out that there are 3n equilateral triangles n with sides equal to 21 each, in level n of the fractal structure . Thus,  Hns (γ (I )) =

3 2s

n .

Letting n → ∞, we have ⎧ ⎨ ∞ if s < log2 3; H s (γ (I )) = 1 if s = log2 3; ⎩ 0 if s > log2 3. Accordingly, dim(γ ) = log2 3 since s = log2 3 is the critical point where s “jumps” from ∞ to 0. In this case, the fractal dimension of the curve γ equals both the box and the Hausdorff dimension of the whole Sierpi´nski’s gasket. This result also turns out to be quite natural since the image of this curve fills such a self-similar set (c.f. Example 3.11.10).

3.11.7 The Fractal Dimension of a Modified Hilbert’s Curve Nevertheless, though it may seem, at least at a first glance, that the fractal dimension for curves introduced in Definition 3.11.3 equals the classical fractal dimensions of space-filling curves, next we provide an appropriate counterexample to highlight

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3 A Middle Definition Between Hausdorff and Box Dimensions

some differences between these models and also to illustrate the actual behavior of this new fractal dimension. Example 3.11.12 (A modified Hilbert’s curve) There exists a Hilbert’s type curve β : I → I × I filling the whole unit square and crossing twice some elements in each level of its induced fractal structure  = β().

Proof Let  be a fractal structure on the closed unit interval I with levels given by  n =

 k k+1 n : k ∈ {0, 1, . . . , 5 , − 1} . 5n 5n

Moreover, let  be the fractal structure defined in Example 3.11.8 but adding one of the squares of level 1 twice, four of the squares of level 2 twice, and so on. Observe that such a fractal structure coincides with the fractal structure induced by  on β(I ) ⊆ X , where X = I × I ⊂ R2 is endowed with the Euclidean distance. The definition of the curve β : I → X has been made throughout a sequence of maps {βn : n → n : n ∈ N} by applying Theorem 3.11.7 and has been illustrated in Figure 3.5 for its first two levels. It is worth mentioning that the blue polygonal line shows how to fill out the whole unit square is via βn in level n of that iterative construction. For instance, 1 = β(1 ) contains all the elements in the first level of the fractal structure induced by  on β(I ) and can be described as follows:         1 2 1 1 1 =β , = 0, × 0, . β 0, 5 5 5 2 2       1 1 2 3 , = , 1 × 0, . β 5 5 2 2       1 1 3 4 , = ,1 × ,1 . β 5 5 2 2       1 1 4 , 1 = 0, × ,1 . β 5 2 2 That approach can be further applied to similarly deal with the construction of upcoming −levels. We also state that for such a square-filling curve, its fractal dimension does not equal neither its box nor its Hausdorff dimensions. This allows to highlight that Definition 3.11.3 becomes more accurate than the classical fractal dimensions since it also “quantifies” the complexity of the constructive process regarding each (filling) curve.

3.11 A New Fractal Dimension for Curves

119

Fig. 3.5 First two levels in the iterative construction of a modified Hilbert’s curve

Counterexample 3.11.13 Let β : I → I × I be the parametrization of the curve provided in Example 3.11.12. Moreover, let ,  be the fractal structures considered previously on I and I × I , respectively. Then dimH (β(I )) = dimB (β(I )) = 2 < dim(β) = log2 5.

Proof First, it is clear that dimH (β(I )) = dimB (β(I )) = 2 since the modified Hilbert’s curve described in Example 3.11.12 fills out the whole unit square I × I. √ 2 n On the other hand, observe that there are 5 subsquares (with diameters 2n , each) which intersect β(I ) in each level n of the induced fractal structure . Thus,  Hns (β(I )) = 2s/2 · Letting n → ∞,

5 2s

n .

⎧ ⎨∞ √ if s < log2 5 H s (β(I )) = 5 if s = log2 5 ⎩ 0 if s > log2 5,

and hence, dim(γ ) = log2 5. It is noteworthy that, though both the classical Hilbert’s curve and the modified Hilbert’s curve fill the whole unit square, their fractal dimensions (in the sense of Definition 3.11.3 with respect to an induced fractal structure) are not the same. This could be understood as this new fractal dimension for curves also takes into account the underlying structure of the curve as well as the constructive approach regarding each filling curve, whereas the classical fractal dimension models do not. Thus, we can state that our fractal dimension allows a deeper study regarding fractal patterns for (filling) curves than the classical models of fractal dimension since it can be calculated with respect to different parametrizations of a same curve.

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3 A Middle Definition Between Hausdorff and Box Dimensions

3.12 Applying Fractal Dimension to Deal with Random Processes In this section, the fractal dimension introduced in Definition 3.11.3 with respect to an induced fractal structure is applied to explore fractal patterns in random processes. In this way, we shall prove a strong connection between the fractal dimension and the self-similarity exponent (of a sample function) of a random process. It is noteworthy that a wide range of random processes, including Brownian motions (BMs), fractional Brownian motions (FBMs), and (fractional) Lévy stable motions (FLSMs) have been applied in scientific literature to deal with financial time series. Accordingly, the results provided in this section can be applied to study long-range dependence for these random processes. First of all, let s > 0. Recall that the absolute s−moment of a random variable X (provided that such a value exists) is defined through the expression m s (X ) = E [X s ], where E [·] denotes the mean of a given random variable X . Our first idea is as follows. Let s ≥ 0 and α : I → R be a parametrization of a real curve, where I denotes the closed unit interval [0, 1]. Moreover, let  be the natural fractal structure on I and  be the fractal structure induced by  on α(I ) ⊆ R. We shall apply Definition 3.11.3 for fractal dimension calculation purposes. Notice that each element A ∈ n can be written as A = A1 ∪ A2 , where A1 , A2 ∈ n+1 . If we suppose that the fractal structure  is regular enough, then we have diam (A1 )  diam (A2 ). Denote a = diam (A) and b = diam (A1 )  diam (A2 ). Then we can calculate the ratio between the diameter of elements in consecutive levels of , namely, rn = ab ∈ (0, 1). Let us suppose that there exists a common ratio r (which could be chosen to be the mean of the list of ratios {rn : n ∈ N}) for any two elements in consecutive levels of . Further, assume that there exists s ≥ 0 such that a s  2 · bs = 2 r s a s . Thus, we s (α(I )) for all n ∈ N, and hence, there exists H s (α(I )) have Hns (α(I )) = Hn+1 which is a finite quantity, so dim(α) = s. −1 On the other hand, if a s  2 · bs = 2 r s a s , then 2 · r s  1, namely, r  2 s . −1 throws a suitable estimation of the fractal dimension of the Accordingly, s  log 2r −1 . These ideas, which are corresponding random process. Therefore, dim(α)  log 2r formalized next, contribute a theoretical link between the fractal dimension and the self-similarity exponent of self-similar random processes. Theorem 3.12.1 Let α : I → R be a sample function of a random process X having stationary and self-affine increments with parameter H . Moreover, let  be the natural fractal structure on I and  be the fractal structure induced by  on α(I ) ⊆ R. The following statements hold:     1 1 H , ω ∼ 2 · M , ω . (i) M 2n 2n+1

3.12 Applying Fractal Dimension to Deal with Random Processes

(ii) If the 1/H −moment of the cumulative range M dim(α) =

121

1 2n

 , ω is finite, then

1 . H

Proof (i) Since X has stationary and self-affine increments with parameter H , then 

  H 1 1 ,ω ∼ · M(1, ω), n 2 2n

(3.18)

H   1 , ω ∼ · M(1, ω). 2n+1 2n+1

(3.19)

M which is equivalent to  M

1

The result follows immediately from both Eqs. (3.18) and (3.19). (ii) In Theorem 3.12.1(i), it is immediate that  M

1 ,ω 2n

 H1

 ∼2·M

1

2

,ω n+1

 H1

,

(3.20)

1

Moreover, observe that each collection {diam (A) H : A ∈ n } is a sample of 1 the random variable M( 21n , ω) H , since {diam (A) : A ∈ n } is a sample of the

1  random variable M 2n , ω . Since the 1/H −moment of the random variable 1 M( 21n , ω) is finite, then the mean of the random variable M( 21n , ω) H is also 1 finite. Thus, the mean of any sample of M( 21n , ω) H equals twice the mean of any 1 1 , ω) H by Eq. (3.20). Hence, sample of M( 2n+1 

1

{diam (A) H : A ∈ n } =2· 2n



1

{diam (B) H : B ∈ n+1 } , 2n+1

leading to 1

1

H (α(I )) Hn H (α(I )) = Hn+1 1

for all n ∈ N. Accordingly, there exists H H (α(I )) ∈ (0, ∞), and therefore, s = 1/H is the critical point where H s (α(I )) “jumps” from ∞ to 0, namely dim(α) =

1 . H

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3 A Middle Definition Between Hausdorff and Box Dimensions

It is worth noting the easiness concerning the proof of Theorem 3.12.1. In addition, we would like to highlight that such a result is quite general. For instance, FBMs and LSMs satisfy all the hypotheses therein as the following result points out. Corollary 3.12.2 Let α : I → R be a sample function of a random process X with parameter H , where X is either a FBM or a FLSM. In addition, let  be the natural fractal structure on I and  be the fractal structure induced by  on α(I ) ⊆ R. Then 1 dim(α) = . H

Proof Just notice that the increments of any FBM (resp., any FLSM) are stationary and self-affine with parameter H due to Remark 1.7.3 (c.f. [57, Theorem 3.3]) (resp. Remark 1.7.4). Hence, Theorem 3.12.1(ii) gives the result.

3.12.1 Introducing FD Algorithms to Calculate the Self-Similarity Exponent of Random Processes In this section, we shall introduce the so-called Fractal Dimension algorithms (FD algorithms, herein) to calculate the fractal dimension of a curve from the viewpoint of fractal structures. First, the next remark becomes especially appropriate to deal with empirical applications involving fractal structures. Remark 3.12.3 Though each fractal structure consists of a countable number of levels, in practical applications we shall only work with a finite number of them. That maximum number of levels will depend on the data number for each curve or time series to be explored. In fact, if l is the length of the data series, then the maximum level to be reached is n  log2 l. By Remark 2.7.1, {diam (A) : A ∈ n } is a sample of the random variable M( 21n , ω) for each natural number n. Let dn be the sample mean of {diam (A) : A ∈ n } which properly approaches the mean of M( 21n , ω). By Theorem 3.12.1(i), must be equal to a constant r = 21H . Thus, H = − log2 r and it holds that rn = ddn+1 n −1 . Theorem 3.12.1 yields dim(α)  log 2r The previous arguments lead to a first approach, named Algorithm FD1, valid to calculate both the fractal dimension and the self-similarity exponent of a random process X under the hypothesis of Theorem 3.12.1. Indeed, such a procedure can be described in the following terms.

3.12 Applying Fractal Dimension to Deal with Random Processes

123

Algorithm 3.12.4 (Algorithm FD1) 1. 2. 3. 4.

Calculate dn as the mean of {diam (A) : A ∈ n } for n = 1, . . . , log2 l. for all n = 1, . . . , log2 l − 1. Let rn = ddn+1 n Let r be the mean of {rn : n = 1, . . . , log2 l − 1}. −1 and H = − log2 r . Return dim(α) = log r 2

Recall that the collection {diam (A) : A ∈ n } is a sample of the random variable X n = M( 21n , ω) (c.f. Remark 1.7.1). Thus, Algorithm FD1 is valid to calculate the self-similarity index of any random process X under the condition E [X n ] = 2 H · E [X n+1 ]. In particular, if X n ∼ 2 H · X n+1 , which is the case of any random process with stationary and self-affine increments with parameter H , due to Theorem 3.12.1. It is noteworthy that GM2 approach is valid to calculate the self-similarity index of any random process satisfying the expression the E [X 1 ] = 2(n−1)H · E [X n ]. Since that equality is equivalent to E [X n ] = 2 H · E [X n+1 ], then the validity of algorithm GM2 is equivalent to the validity of algorithm FD1. Upcoming Fig. 3.6 graphically displays how to calculate the fractal dimension of a time series throughout the FD1 algorithm. In this case, they have been plotted the values of the coefficients rn as well as their mean value r (horizontal line) for a 2048−point time series (which makes a total amount of 11 levels of the induced fractal structure ) from a BM. Note that the quantity r leads to both the fractal dimension of the series and to its Hurst exponent, too (Fig. 3.6). Next, we provide two alternative approaches to calculate the fractal dimension of discretized curves which are based on statistical moments of certain random variables. To deal with, let {xk : k = 1, . . . , l} be a sample of length l from a random variable X . We shall calculate its sample s−moment throughout the following expression:

Fig. 3.6 Algorithm FD1 in action. Stars represent the values of the coefficients rn and the straight line corresponds to their mean (c.f. Algorithm 3.12.4). The graphical representation has been carried out a 2048−point BM

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3 A Middle Definition Between Hausdorff and Box Dimensions

m s (X ) =

l 1  s x . · l i=1 i

Next, we state another cornerstone to properly calculate the fractal dimension of a wide range of (sample) curves. Theorem 3.12.5 Let α : I → R be a sample function of a random process X ,  be the natural fractal structure on I , and  be the fractal structure induced by  on α(I ) ⊆ R. Moreover, let X n = M( 21n , ω) be a random variable providing the statistical distribution of the list of diameters {diam (A) : A ∈ n } and assume there exists s ≥ 0 satisfying the two following properties for each natural number n: (i) The s−moment of X n , m s (X n ), is finite. (ii) m s (X n ) = 2 · m s (X n+1 ). Then dim(α) = s.

Proof The sample s−moment of {diam (A) : A ∈ n } is approximately equal to m s (X n ), namely, 1  {diam (A)s : A ∈ n } m s (X n ) = n · 2 since each level of the induced fractal structure  consists of 2n elements. Hence,  {diam (A)s : A ∈ n } = 2n · m s (X n ). (3.21) Hns (α(I )) = Similarly, it holds that s (α(I )) = 2n+1 · m s (X n+1 ). Hn+1

(3.22)

From both Eqs. (3.21) and (3.22) and applying also hypothesis (ii) of Theorem 3.12.5, we can state that s (α(I )) Hns (α(I )) = Hn+1 for all n ∈ N. This leads to H s (α(I )) being positive and finite. Accordingly, dim(α) = s. The following result contains sufficient conditions to verify the hypothesis (ii) in Theorem 3.12.5. More specifically, it also allows to calculate the self-similarity exponent of a wide range of self-similar processes from their fractal dimensions.

3.12 Applying Fractal Dimension to Deal with Random Processes

125

Corollary 3.12.6 Let α : I → R be a sample function of a random process X with parameter H . Moreover, let X n = M( 21n , ω) and assume that the following expression is satisfied: X n ∼ TnH · X 0 , where Tn = namely,

(3.23)

1 . Then the hypothesis (ii) in Theorem 3.12.5 stands for s 2n

= 1/H ,

m H1 (X n ) = 2 · m H1 (X n+1 ). In particular, it holds that dim(α) =

1 . H

Proof Observe that Eq. (3.23) leads to X ns ∼ Tns H · X 0s for all s > 0. In particu1

1

lar, if s = H1 , then we have X nH ∼ Tn · X 0H for all natural number n. Since these random variables are identically distributed, their means must be equal, namely, 1

E [X nH ] = 1 H

1 2n

1

1

H · E [X 0H ] for each n ∈ N. Similarly, we can state that E [X n+1 ]= 1 H

1 H

1 2n+1

·

E [X 0 ]. Hence, E [X n+1 ] = · E [X n ]. Accordingly, m H1 (X n ) = 2 · m H1 (X n+1 ). Finally, Theorem 3.12.5 gives dim(α) = H1 . 1 2

It is worth mentioning that any random function X (t, ω) with self-affine increments with parameter H verifies Eq. (3.23) (c.f. Sect. 2.7.1). Following the above, the next result holds immediately from Corollary 3.12.6. Corollary 3.12.7 Let α : I → R be a sample function of a random process X having stationary and self-affine increments with parameter H . Then the hypothesis (ii) in Theorem 3.12.5 stands for s = 1/H . In particular, it is satisfied that dim(α) = H1 . The wide classes consisting of FBMs and FLSMs are under the hypothesis (ii) in Theorem 3.12.5 as the following result points out. Corollary 3.12.8 Let α : I → R be a sample function of a random process X with parameter H , where X is either a FBM or a FLSM. Then the hypothesis (ii) in Theorem 3.12.5 stands for s = 1/H . In particular, dim(α) = H1 .

Proof Just recall that the increments of any FBM (resp., any FLSM) are stationary and self-affine with parameter H by Remark 2.7.3 (c.f. [57, Theorem 3.3]) (resp. Remark 2.7.4). Finally, Corollary 3.12.7 gives the result.

126

3 A Middle Definition Between Hausdorff and Box Dimensions

Remark 3.12.9 The hypothesis (ii) in Theorem 3.12.5, namely, m s (X n ) = 2 · m s (X n+1 ),

(3.24)

where the random variable X n gives the distribution of each list of diameters {diam (A) : A ∈ n }, stands for many empirical applications. Key Theorem 3.12.5 throws two novel approaches to calculate both the fractal dimension and the self-similarity exponent of (sample) curves (resp., time series). To deal with, recall that if l is the length of the series, then log2 l is the deeper level of the induced fractal structure  that can be reached (c.f. Remark 3.12.3). It is also worth noting that the main condition the following algorithms are based on was provided in previous Eq. (3.24). Additionally, notice that such an expression is equivalent to m s (X k ) = 2 for k = 1, . . . , log2 l − 1. m s (X k+1 ) Algorithm 3.12.10 (Algorithm FD2) 1. For each s > 0, calculate the list ys = {yk,s : k = 1, . . . , log2 l − 1}, where yk,s =

m s (X k ) . m s (X k+1 )

2. Let ys be the mean of each list ys . 3. Find out s0 such that ys0 = 2. Observe that {(s, ys ) : s > 0} is s−increasing. 4. Return dim(α) = s0 (due to Theorem 3.12.5) and H = s10 (by Theorem 3.12.1). FD2 approach is valid to properly calculate the self-similarity index of any random 1/H 1/H process X under the condition E [X n ] = 2 · E [X n+1 ], namely, hypothesis (ii) in 1/H Theorem 3.12.5. In particular, it works for random functions such that X n ∼ 2 · 1/H 1/H 1/H H X n+1 . Going beyond, if X n ∼ 2 · X n+1 , then X n ∼ 2 · X n+1 and hence, both Theorems 3.12.1 and 3.12.5 allow to state that FD2 Algorithm is valid to estimate the self-similarity exponent of any random process X with stationary and self-affine increments with parameter H . Figure 3.7 displays a graphical approach regarding how to computationally deal with the calculations regarding FD2 Algorithm. This contains the graph of ys in terms of s. The s−increasing nature concerning the function ys makes easy to find out the value s0 of s for which ys0 = 2, namely, the fractal dimension of this sample function. In this occasion, a 2048−point point BM was considered for illustration purposes.

3.12 Applying Fractal Dimension to Deal with Random Processes

127

Fig. 3.7 Example of the graph representation of s versus ys for a 2048− point BM (c.f. Algorithm 3.12.10). In this case, the fractal dimension is close to 2 (a self-similarity exponent H = 0.5) and follows from the value of s making ys = 2

Next step is to describe the so-called FD3 Algorithm, an alternative to FD2 approach and also based on Theorem 3.12.5. To deal with, first we shall sketch some theoretical notes. In fact, it is clear that Eq. (3.24) is equivalent to m s (X n ) =

1 · m s (X 1 ). 2n−1

(3.25)

Taking 2−base logarithms in previous Eq. (3.25), it holds that log2 m s (X n ) = −n + γ ,

(3.26)

where γ = 1 + log2 m s (X 1 ) remains constant. Accordingly, Eq. (3.26) provides a linear relation between n and log2 m s (X n ) that we shall apply for fractal dimension calculation purposes. The algorithm based on the ideas described above is stated as follows. Algorithm 3.12.11 (Algorithm FD3) (1) For each s > 0, calculate the 2D−point cloud {(k, βk,s ) : k = 1, . . . , log2 l}, where βk,s = log2 m s (X k ). Moreover, let βs be the slope of the regression line for that cloud. (2) Consider the increasing function {(s, βs ) : s > 0} and determine the value of s, s1 , for which βs1 = −1. (3) Hence, dim(α) = s1 (due to both Theorem 3.12.5 and Eq. (3.26)) and H = s11 (by Theorem 3.12.1). FD3 Algorithm is valid to calculate the self-similarity exponent of any random 1/H 1/H 1 · E [X 1 ]. Since that expression is process X under the condition E [X n ] = 2n−1 1/H 1/H equivalent to E [X n ] = 2 · E [X n+1 ], then the validity of FD3 approach is equiva-

128

3 A Middle Definition Between Hausdorff and Box Dimensions

lent to the validity of FD2 Algorithm to calculate the self-similarity exponent of any random process X having stationary and self-affine increments with parameter H . Interestingly, FD3 Algorithm also allows to verify the condition m s (X n ) = 2 · m s (X n+1 ) (c.f. hypothesis (3.12.5) in Theorem 3.12.5). In fact, since Eq. (3.26) is equivalent to that constraint, we can check out that certain empirical data satisfy that condition provided that Eq. (3.26) holds, namely, if the regression coefficient in Eq. (3.26) is close to 1. In this way, a graphical representation similar to Fig. 3.7 could be carried out to illustrate that approach. Next theorem is the cornerstone to formally justify the validity of FD algorithms (and GM2 approach) to properly calculate the self-similarity exponent (and also the fractal dimension) of any random process nx having stationary and self-affine increments with parameter H . Theorem 3.12.12 Let X be a random process with stationary increments and assume that there exists a parameter H > 0 for which the following expression stands: M(T, ω) ∼ T H · M(1, ω). Then all FD algorithms (c.f. Algorithms 3.12.4, 3.12.10 and 3.12.11) and GM2 approach (c.f. [77, Sect. 4]) are valid to calculate that H .

Proof Let X n = M( 21n , ω) for all n ∈ N. Since the condition M(T, ω) ∼ T H · M(1, ω) holds by hypothesis, then we have the following: 1 1 (1) X n ∼ 2(n−1)H · X 1 . This implies that E [X n ] = 2(n−1)H · E [X 1 ], the condition that guarantees the validity of GM2 approach to calculate the self-similarity exponent H of the random process X . (2) Similarly, it is satisfied that X n+1 ∼ 21H · X n which leads to E [X n ] = 2 H · E [X n+1 ], the expression that justifies the validity of FD1 Algorithm to estimate H. 1/H 1/H 1/H (3) From the expression above, we have X n ∼ 2 · X n+1 . Thus, E [X n ] = 2 · 1/H E [X n+1 ], the condition which means that the FD2 estimator becomes a valid approach to calculate the self-similarity index H of X . 1/H 1/H 1 1 · X 1 , we obtain that X n ∼ 2(n−1) · X1 (4) Finally, from expression X n ∼ 2(n−1)H 1/H 1/H 1 for all natural number n. This implies that E [X n ] = 2(n−1) · E [X 1 ], leading to the validity of FD3 Algorithm to calculate H .

Theorem 3.12.12 gives the following two corollaries.

3.12 Applying Fractal Dimension to Deal with Random Processes

129

Corollary 3.12.13 Let X be a random process having stationary and selfaffine increments with parameter H . Then all the FD1, FD2, FD3, and GM2 approaches are valid to calculate that parameter H .

Proof Since the increments of the random process X are stationary and self-affine with parameter H , then the following expression holds: M(T, ω) ∼ T H · M(1, ω) (c.f. Eq. (1.26)). Hence, Theorem 3.12.12 can be applied leading to the result.

Corollary 3.12.14 Let X be a FBM or a FLSM with parameter H . Then all the FD1, FD2, FD3, and GM2 algorithms are valid to calculate that parameter H.

Proof Both Remarks 1.7.3 and 1.7.4 allow to affirm that FBMs and FLSMs are random processes having stationary and self-affine increments with parameter H . Thus, Corollary 3.12.13 concludes the proof.

3.12.2 Testing the Accuracy of FD Algorithms By previous Corollary 3.12.13, we proved theoretically that FD algorithms are valid to calculate the self-similarity index of random processes with stationary and selfaffine increments. That result holds, in particular, for the wide classes consisting of FBMs and FLSMs, as well (c.f. Corollary 3.12.14). In this section, though, we shall explore how accurate FD algorithms are for self-similarity exponent calculation purposes. It is worth mentioning that these novel approaches are especially appropriated to deal with short key advantage compared to other algorithms. For instance, notice that R/S analysis does not behave properly for this kind of series. Next, we explain how to check out the accuracy of FD algorithms to estimate the self-similarity exponent of random processes. To deal with, we have carried out Monte Carlo simulation as follows. First of all, we have generated 10000 BMs (recall that all of them have a Hurst exponent H = 0.5) with a length equal to 1024 and have calculated their Hurst exponents afterward considering that these processes (that could be understood as stock prices) do change about 128 times a day (actually, prices change much more usually, though 128 times is enough for simulation purposes). In this way, we can simulate both the maximum and the minimum values of each day.

130

3 A Middle Definition Between Hausdorff and Box Dimensions

It is noteworthy that the mean of the Hurst exponents was found to be equal to 0.50, whereas their standard deviation equals 0.04, quite close to 0. Thus, FD1 approach works fine as a self-similarity estimator since the Hurst exponent was found to be close to 0.5 (in mean) in the BM cases. Similar tests were carried out for selfsimilarity exponents H = 0.25 (a class of anti-persistent FBMs) and H = 0.75. In the first case, while the mean was found to be equal to 0.33, the standard deviation was equal to 0.04. On the other hand, for H = 0.75, the mean was found to be equal to 0.74 with a standard deviation equal to 0.07. Hence, it holds that for smaller values of H , FD1 algorithm lacks accuracy, whereas it remains quite accurate to deal with values of the self-similarity exponent of a process close to or greater than 0.5 (c.f. Fig. 3.9). We should mention here that FD1 procedure was theoretically proved to be asymptotically correct in Sect. 3.12.1. For all these values of H , namely, H ∈ {0.25, 0.5, 0.75}, similar tests were also carried out for 32− and 256−length time series. Table 3.1 contains all the results. On the other hand, both FD2 and FD3 approaches were similarly tested. Indeed, for both of them its accuracy was explored by Monte Carlo simulation throughout 10000 FBMs with different self-similarity exponents. The length of all the time series was chosen to be equal to 1024 and it was also considered that these processes changed about 128 times a day. Regarding the behavior of FD2 algorithm for BMs, it should be mentioned here that the estimated Hurst exponent was found to be equal to 0.50 (in mean) with a standard deviation equal to 0.05. Hence, FD2 procedure works properly to deal with random walks. Further, for H = 0.25, the mean of the Hurst exponents was found to be equal to 0.26 with a standard deviation equal to 0.04. In addition, for H = 0.75, the mean was found to be equal to 0.73 with a standard deviation equal to 0.07. Similarly, the behavior of FD3 Algorithm was also tested. The results were as follows. First, for H = 0.5, it was found out a mean equal to 0.49 with a standard

Table 3.1 Influence of the series length in the self-similarity exponent throughout different approaches. The results were obtained by Monte Carlo simulation carried out for 10000 FBMs with self-similarity exponents H ∈ {0.25, 0.5, 0.75} and time series of lengths 32, 256, and 1024. The data in bold corresponds to the best means in each case Algorithm

R/S

DFA

GM1

GM2

FD1

FD2

FD3

Index

Length

Mean Std

Mean Std

Mean Std

Mean Std

Mean Std

Mean Std

Mean Std

32

0.60 0.07

−0.00 0.37

0.30 0.31

0.31 0.06

0.37 0.09

0.25 0.08

0.24 0.09

0.25

256

0.41 0.06

0.11 0.09

0.28 0.11

0.30 0.03

0.34 0.05

0.26 0.05

0.25 0.05

1024

0.34 0.05

0.16 0.06

0.28 0.07

0.29 0.02

0.33 0.04

0.26 0.04

0.25 0.04

32

0.66 0.07

0.46 0.43

0.66 0.30

0.51 0.08

0.50 0.09

0.49 0.10

0.48 0.10

256

0.57 0.08

0.47 0.10

0.57 0.11

0.51 0.03

0.50 0.05

0.50 0.06

0.49 0.06

1024

0.54 0.08

0.48 0.06

0.54 0.07

0.51 0.04

0.50 0.04

0.50 0.05

0.49 0.04

32

0.69 0.07

0.68 0.42

0.98 0.27

0.73 0.11

0.72 0.13

0.71 0.15

0.69 0.15

256

0.71 0.10

0.71 0.12

0.86 0.11

0.74 0.07

0.73 0.08

0.73 0.09

0.72 0.10

1024

0.74 0.11

0.72 0.08

0.82 0.08

0.74 0.05

0.74 0.07

0.73 0.07

0.73 0.07

0.5

0.75

3.12 Applying Fractal Dimension to Deal with Random Processes

131

Fig. 3.8 Accuracy of FD2 approach for self-similarity exponent estimation purposes. The graph above provides a comparison between theoretical self-similarity exponents and empirical ones (calculated throughout FD2 algorithm). The experiments were carried out by Monte Carlo simulation consisting of 10000 FBMs with uniform and random self-similarity exponents lying in (0, 1), lengths equal to 1024, and 128 changes a day. Notice that each cloud point refers to one simulation

deviation equal to 0.04. The mean was quite close to 0.5 with a standard deviation quite small, so we concluded that FD3 procedure works also fine to deal with BMs. Moreover, Monte Carlo simulation was also carried out for anti-persistent FBMs (H = 0.25). Thus, the mean of the self-similarity exponent values was found to be equal to 0.25 with a slight standard deviation, equal to 0.04. In addition, the results for H = 0.75 were as follows: a mean equal to 0.73 and a standard deviation equal to 0.07. Table 3.1 displays all the results for each self-similarity value and each approach. In this way, from Table 3.1 and both Figs. 3.8 and 3.9, we argue that FD2 and FD3 procedures are accurate to deal with the whole range of self-similarity exponent values. In particular, Fig. 3.9 displays a graphical comparison among different algorithms to calculate the self-similarity exponent of a random process: FD algorithms (FD1, FD2, and FD3), geometric method-based procedures (GM1 and GM2), classical R/S analysis, and DFA. In this occasion, the graphs were plotted by a Monte Carlo simulation involving 1000 FBMs with self-similarity exponents lying in the unit interval (0, 1). The length of each FBM was equal to 256 and all of them were generated under the assumption of 128 changes a day. Hence, it is noteworthy that both FD2 and FD3 algorithms became particularly accurate to deal with persistent, anti-persistent, and random processes. However, for self-similarity exponents approximately 0 such that the following expression holds: M(T, ω) ∼ T H · M(1, ω).

(3.27)

Recall that such a power law is satisfied, in particular, by any random function X (t, ω) having self-affine increments with parameter H . Thus, if Eq. (3.27) is raised to the q−power, then we have M(T, ω)q ∼ T q H · M(1, ω)q for any q > 0. Let Tn = T . Thus,

1 2n

(3.28)

be an appropriate discretization regarding the time period M(Tn , ω)q ∼ Tnq H · M(1, ω)q

for all q > 0 and all n ∈ N. Hence, if X n denotes the Tn −period cumulative range of the random process X , namely, X n = M(Tn , ω) = M( 21n , ω) for all n ∈ N, then q qH q it holds that X n ∼ Tn · X 0 for all q > 0 and all natural number n. Accordingly, the following connection between the q−powers of consecutive period cumulative ranges stands: q X nq ∼ 2q H · X n+1 . Moreover, since the two previous random variables are identically distributed, namely, they have the same finite joint distribution functions, then their means must q q be equal, namely, E [X n ] = 2q H · E [X n+1 ] provided that such moments exist, leading to the following expression:

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3 A Middle Definition Between Hausdorff and Box Dimensions

m q (X n ) = 2q H · m q (X n+1 ),

(3.29)

regarding consecutive q−moments for all q > 0. It is worth noting that Eq. (3.29) is equivalent to the next one: m q (X n ) =

1 · m q (X 0 ). 2nq H

(3.30)

Taking 2−base logarithms on both sides of Eq. (3.29), we have  log2

m q (X n ) m q (X n+1 )

 = q H,

(3.31)

  m (X n ) which provides a linear relationship between q and log2 m q q(X n+1 provided that ) m q (X n ) exists for all n ∈ N. Hence, the self-similarity index of the random process X can be clearly calculated throughout Eq. (3.31) by the following: 1 H = · log2 q



 m q (X n ) . m q (X n+1 )

Along the sequel, FD will refer to this generic approach to estimate the self-similarity exponent of any random process X for any q > 0.

3.12.6 Generalizing GM2 and FD Algorithms Geometric method-based procedures were first introduced in [77] to efficiently deal with the estimation of the self-similarity exponent of time series. They consist of two algorithms, GM1 and GM2, both of them based on a well-known approach used in geometry to calculate the Hurst exponent. It is worth noting that they were revisited afterward from the viewpoint of fractal structures. This allowed to theoretically prove that GM algorithms are valid to estimate the self-similarity index of random processes with stationary and self-affine increments [82]. It is also worth mentioning that in [39], it was thrown some empirical evidence (by GM algorithms) that stocks from US Small Cap. and NASDAQ-100 cannot be modelized as BMs though their selfsimilarity exponents were usually close to 0.5. In addition, it was also justified by Monte Carlo simulation that GM algorithms are valid to properly calculate the self-similarity exponent of Lévy stable motions as well as they are especially accurate to deal with short series (which is usually the case of financial time series). On the other hand, a new range of procedures to test for long-memory in financial series was contributed in [76] throughout fractal techniques. In fact, FD algorithms, all of them based on the calculation of the fractal dimension of a curve, were proved to perform much better than classical algorithms, particularly for short-length series.

3.12 Applying Fractal Dimension to Deal with Random Processes

139

The accuracy of FD algorithms and geometric method-based procedures for selfsimilarity calculation purposes has been explored from a mathematical viewpoint in previous works (c.f. [76, 82]). Following the above, our next goal is to prove that the new FD approach introduced in Sect. 3.12.5 becomes a generalization of GM2 approach and FD algorithms, as well. Theorem 3.12.15 Let X be a random process with stationary increments and assume that there exists a parameter H > 0 for which the following expression stands: M(T, ω) ∼ T H · M(1, ω). Then all FD algorithms and GM2 approach remain as particular cases of the new approach to calculate the self-similarity exponent of random processes as it was introduced in Sect. 3.12.5.

Proof First, recall that Eq. (3.28) contains the power law that supports the new approach introduced in Sect. 3.12.5 for self-similarity exponent calculation purposes. Moreover, let Tn = 21n be an appropriate discretization regarding the time period T for all n ∈ N. Next, we shall deal with each of the following cases. • Let q = 1. Thus, (1) The power law in Eq. (3.28) becomes M(Tn , ω) ∼ TnH · M(1, ω) for all natural 1 · X 1 . Since these rannumber n, or equivalently, we can write X n ∼ 2(n−1)H dom variables are identically distributed, their means must be equal, namely, 1 · E [X 1 ]. This leads to the condition that guarantees the validE [X n ] = 2(n−1)H ity of GM2 procedure to calculate the self-similarity exponent of X (c.f. [39, Theorem 3.7 and Corollary 3.8]). 1 X 1 for all n ∈ N, then we have X n ∼ 2 H · X n+1 for all natural (2) Since X n ∼ 2(n−1)H number n. Hence, E [X n ] = 2 H · E [X n+1 ], which justifies the validity of FD1 algorithm to estimate the self-similar exponent of X (c.f. [75, Sect. 5]). Recall that the validity of FD1 is equivalent to the validity of GM2 approach. • Let q = 1/H . Hence, 1/H

1/H

1/H

(1) From X n ∼ 2 H · X n+1 , it holds that X n ∼ 2 · X n+1 . This leads to E [X n ] = 1/H 2 · E [X n+1 ], which is the expression that gives the validity of FD2 for selfsimilarity index estimation purposes (c.f. [75, Sect. 5]). 1/H 1/H 1 1 · X 1 leads to X n ∼ 2n−1 · X 1 for all (2) Observe that the expression X n ∼ 2(n−1)H 1/H 1/H 1 · E [X 1 ], namely, FD3 algorithm is n ∈ N. This implies that E [X n ] = 2n−1 valid to calculate the self-similarity index of the random process X . It is worth noting that such an expression also justifies the validity of FD2 approach to properly calculate the self-similarity exponent of X (c.f. [75, Sect. 5]).

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3 A Middle Definition Between Hausdorff and Box Dimensions

According to Theorem 3.12.15, the new approach we introduced and motivated in Sect. 3.12.5 to calculate the self-similarity exponent of random processes generalizes GM2 procedure and all the FD algorithms, as well. In this way, for q = 1, both algorithms GM2 and FD1 remain as particular cases of the new approach, whereas for q = 1/H , it holds that FD2 and FD3 are also generalized by this novel procedure.

3.12.7 An Open Gate to Multifractality Recall that Eq. (3.28) contains the power law that governs the working of our new approach. However, we shall modify it slightly throughout the next expression: M(T, ω)q ∼ T q H (q) · M(1, ω)q , leading to m q (X n ) = 2q H (q) · m q (X n+1 ) for all natural number n and all q > 0 (provided that there exist these moments). This allows us to distinguish between two kinds of processes: those for which H (q) = H is a constant independent of q, and those processes for which H (q) is not constant. In the first case, we deal with unifractal (also uniscaling) processes whose scaling behaviors are uniquely determined by the constant H , which equals their Hurst exponent, whereas in the second case, we tackle with multifractal (or multiscaling) processes whose scaling behaviors depend on each q−moment. Hence, their self-similarity exponents can be explored for each q−moment. Observe that for distinct q−values, the self-similarity exponent H (q) throws information concerning different features of the random process under study. Therefore, as it happens with the generalized Hurst exponent (GHE) [24] or the multifractal detrended fluctuation analysis (MF-DFA) [49], this approach provides a new procedure to properly deal with multifractal processes. In forthcoming subsections, though, we shall be focused only on unifractal processes.

3.12.8 Some Notes on the New FD4 Algorithm and Its Implementation First, to guarantee the existence of the q−moments m q (X n ) for unifractal processes, let q = 0.01. It is worth mentioning that any q = 0 may be chosen to deal with unifractal processes at a first glance. The only restriction consists of the existence of the sample q−moments. However, while this is not a problem in the FBM case, the sample moments may not exist if the process is a Lévy stable motion (LSM hereafter). In other words, it may happen that the sample moments m q (X n ) do not exist for some q0 and any q > q0 . Accordingly, the new FD4 algorithm will consist of the approach introduced in Sect. 3.12.5 for q = 0.01. On the other hand, from Eq. (3.30), we have

3.12 Applying Fractal Dimension to Deal with Random Processes

log2 m q (X n ) = −nq H + log2 m q (X 0 ).

141

(3.32)

Thus, the self-similarity exponent can be estimated throughout the slope of a linear regression comparing n versus log2 m q (X n ). A regression coefficient close to 1 will confirm that the condition in Eq. (3.28) holds. Another option is to apply Eq. (3.31), since it allows to calculate the self-similarity exponent of the random process X throughout the ratio between the q−moments of consecutive period cumulative ranges. Accordingly, Eq. (3.31) leads to H = 100 · log2

m q (X n ) . m q (X n+1 )

(3.33)

It is noteworthy that the calculation of m q (X n ) is based on a sample of the random variable X n . Since the length of that sample is equal to 2n , then the calculation of m q (X n ) will be more accurate if it is carried out by the greatest n. In this way, we can either take the two greatest n available and apply Eq. (3.33) to calculate H , or use all the values of n available and then calculate the ratios for each pair of consecutive moments. The calculation of the q−moments m q (X n ) can be dealt with as follows. For a given time series (of log prices), let us divide it into 2n nonoverlapping blocks of lengths k = length(series)/2n each. Then (1) Calculate the range of each block Bi = {B1 , . . . , Bk }, namely, Ri = max{B j : j = 1, . . . , k} − min{B j : j = 1, . . . , k} for i = 1, . . . , 2n . 2 n q Ri . (2) Calculate the q−moment of the block ranges: m q (X n ) = 21n · i=1 To calculate the range of each block Bi , we shall consider both the maximum and the minimum of each period (in fact, in financial series it becomes usual to know both the maximum and the minimum of each trading period). Thus, the greatest value of n gives length(series) and in that case, each block will contain one element only, though we can still calculate the range (the maximum minus the minimum) of that element. Another option to deal with the implementation of the algorithm is as follows. For a given n, we can start the block in any index 0, 1, . . . , k − 1, where k is the number of elements in each block, namely, k = length(series)/2n . This will throw k distinct estimations of m q (X n ), and hence, we can calculate their mean. Notice that such an idea is somehow equivalent to use overlapping blocks. In most of the financial data, the markets are not open 24 hours a day. When the market is closed, we cannot follow the movement of the stocks, so when we use daily series, both the maximum and the minimum of the data correspond to the observed (trading) period. One way to take this into consideration is to calculate the range as the so-called “true range,” where the maximum is defined as the maximum of the day and the opening of the next day and the minimum is defined as the minimum of the day and the opening of the next day.

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3 A Middle Definition Between Hausdorff and Box Dimensions

3.12.9 Testing the Accuracy of FD4 Algorithm Next, we shall carry out Monte Carlo simulation of unifractal processes to explore the performance of FD4. To test the accuracy of FD4 approach, we shall consider FBMs, which are connected with long-range dependence, and LSMs, which have been widely applied in finance (c.f. [56]). To deal with, we shall carry out an empirical analysis regarding the finite sample properties of the self-similarity exponent estimations being based on processes generated by standardized normal distributions such as FBMs. We would like to point out that in [39, Sect. 5] it was thrown some empirical evidence that stocks from NASDAQ-100 and US Small Cap. Indices do not follow BM patterns but they behave more like LSMs with self-similarity exponents close to 0.5. In this way, we shall involve also random processes generated by stable distributions with distinct parameters α. Recall that in the LSM case, α = 1/H . Both Tables 3.5 and 3.6 as well as Fig. 3.11 justify why FD4 algorithm is extremely accurate to deal with the self-similarity exponent of LSMs for any H ∈ [0.5, 1], even if the time series are too short (from 32 data). They display the results obtained for 1000 LSMs with self-similarity exponents in {0.5, 0.6, 0.7, 0.8, 0.9, 1}. The lengths of the time series that gave rise to Tables 3.5 and 3.6 were equal to 32, 64, 128, 256, 512, and 1024. On the other hand, the two graphics in Fig. 3.11 were generated by 128 and 1024−length LSMs, respectively. In [39, Sect. 4], it was proved that both GM2 and GHE (q = 1) approaches are quite accurate to calculate the self-similarity index of LSMs with H close to 0.5 though they lose accuracy as H increases. This fact may be due to the non existence of the corresponding q−moments (recall that GM2 procedure is a particular case of the new FD approach for q = 1, c.f. Theorem 3.12.15). Likewise, GHE algorithm (q = 0.01) is accurate for the whole range of self-similarity exponent values, as well. Both Tables 3.7 and 3.8 and also Fig. 3.12 provide empirical evidence that FD4 algorithm becomes quite accurate for FBMs with self-similarity exponents in the

Table 3.5 Influence of the time series length in the self-similarity exponent estimation (means in this case) through FD4 algorithm. The means were calculated by Monte Carlo simulation of 1000 LSMs with self-similarity exponents lying in {0.5, 0.6, 0.7, 0.8, 0.9, 1}. The lengths of the time series were chosen to be equal to 32, 64, 128, 256, 512, and 1024 Self-similarity exponent values (means) Length 32 64 128 256 512 1024

0.5 0.52 0.52 0.53 0.53 0.53 0.53

0.6 0.60 0.62 0.62 0.62 0.62 0.62

0.7 0.70 0.71 0.71 0.72 0.72 0.72

0.8 0.79 0.80 0.81 0.81 0.82 0.82

0.9 0.88 0.91 0.91 0.91 0.91 0.91

1.0 0.97 1.00 1.00 1.01 1.01 1.01

3.12 Applying Fractal Dimension to Deal with Random Processes

143

Table 3.6 Influence of the time series length in the self-similarity exponent calculation (standard deviations in this case) via FD4 approach. The standard deviations were obtained by Monte Carlo simulation of 1000 LSMs with self-similarity exponents in ∈ {0.5, 0.6, 0.7, 0.8, 0.9, 1}. Moreover, the lengths of the involved time series were equal to 32, 64, 128, 256, 512, and 1024 Self-similarity index (standard deviations) Length 32 64 128 256 512 1024

0.5 0.05 0.03 0.02 0.02 0.01 0.01

0.6 0.07 0.05 0.04 0.02 0.02 0.01

0.7 0.09 0.06 0.04 0.03 0.02 0.02

0.8 0.10 0.07 0.05 0.04 0.03 0.02

0.9 0.12 0.09 0.07 0.04 0.03 0.02

1.0 0.14 0.11 0.07 0.05 0.04 0.03

Fig. 3.11 Testing the accuracy of FD4 to estimate the self-similarity exponent of 128 and 1024−length LSMs. In this case, H ∈ [0.5, 1] since H = 1/α and α ∈ [1, 2]. Horizontal bars symbolize the confidence intervals of the empirical distribution thrown by FD4 approach at a confidence level of 90% Table 3.7 Influence of the length of the time series in the calculation of the selfsimilarity exponent (means in this case) using the FD4 algorithm. The means have been obtained by Monte Carlo simulation of 1000 FBMs with self-similarity exponents H ∈ {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1} and time series with lengths equal to 32, 64, 128, 256, 512, and 1024 Self-similarity exponent (means) Length 32 64 128 256 512 1024

0.0 0.21 0.21 0.21 0.21 0.21 0.21

0.1 0.25 0.25 0.25 0.25 0.25 0.25

0.2 0.29 0.30 0.30 0.30 0.30 0.30

0.3 0.35 0.35 0.36 0.36 0.36 0.36

0.4 0.41 0.42 0.42 0.42 0.43 0.43

0.5 0.52 0.52 0.53 0.53 0.53 0.53

0.6 0.57 0.58 0.58 0.59 0.58 0.59

0.7 0.66 0.67 0.68 0.68 0.68 0.68

0.8 0.77 0.77 0.78 0.78 0.78 0.78

0.9 0.88 0.88 0.90 0.90 0.90 0.90

1.0 0.97 0.99 1.00 1.00 1.00 1.00

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3 A Middle Definition Between Hausdorff and Box Dimensions

Table 3.8 Influence of the length of the time series in the calculation of the self-similarity exponent (standard deviations in this case) using the FD4 algorithm. The standard deviations in this table have been obtained by Monte Carlo simulation of 1000 FBMs with self-similarity exponents H ∈ {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1} and time series with lengths equal to 32, 64, 128, 256, 512, and 1024 Self-similarity exponent (means) Length 32 64 128 256 512 1024

0.0 0.02 0.02 0.01 0.01 0.01 0.00

0.1 0.03 0.02 0.01 0.01 0.01 0.00

0.2 0.03 0.02 0.02 0.01 0.01 0.01

0.3 0.04 0.02 0.02 0.01 0.01 0.01

0.4 0.04 0.03 0.02 0.01 0.01 0.01

0.5 0.05 0.03 0.02 0.02 0.01 0.01

0.6 0.06 0.04 0.03 0.02 0.02 0.01

0.7 0.08 0.05 0.04 0.03 0.02 0.02

0.8 0.09 0.07 0.06 0.04 0.03 0.02

0.9 0.11 0.09 0.07 0.06 0.05 0.05

1.0 0.00 0.00 0.00 0.00 0.00 0.00

Fig. 3.12 Testing the accuracy of FD4 approach to calculate the self-similarity exponent of 128 and 1024−length FBMs. Horizontal bars symbolize the confidence intervals of the empirical distribution provided by FD4 at a confidence level of 90%

range [0, 1]. Nevertheless, after a look at Fig. 3.12, it may seem at a first glance that FD4 approach is not accurate for self-similarity exponent values close to 0. Interestingly, it also holds for both FD1 and GM2 algorithms (c.f. [76, Fig. 5]). However, we can realize that this is due to the way we carry out our simulations. More specifically, notice that all of these algorithms (GM2, FD1, and FD4) use both the maximum and the minimum values of each period. Hence, to simulate a 1024−length process, let us generate a 1024 × 128−length series and then simulate the movement in each period via 128−length subseries, so we can get both the maximum and the minimum values of each of the 1024 periods. It turns out that processes with a small self-similarity exponent do oscillate too much, so subseries of greater length are needed to be able to capture their full movement in each period. Indeed, upcoming Figs. 3.13 and 3.14 display how FD4 approach becomes more accurate as the length of the subseries increases and once it is large enough, then we can verify that the mean of the estimated self-similarity exponent converges to the theoretical self-similarity index of the corresponding random process.

3.12 Applying Fractal Dimension to Deal with Random Processes

145

Fig. 3.13 Influence of the (2−base logarithm of the) length of the subseries considered to estimate both the maximum and the minimum values of one period in the calculation of the self-similarity exponent by FD4 algorithm. In this case, the series correspond to FBMs with H = 0.1

Fig. 3.14 Influence of the (2−base logarithm of the) length of the subseries considered to calculate both the maximum and the minimum values of one period in the calculation of the self-similarity exponent by FD4 algorithm. In this case, the series correspond to FBMs with H = 0.5

3.12.10 A Historic Study Regarding the Self-Similarity Exponent of S&P 500 Stocks In this section, we shall explore how the self-similarity exponent of stocks from the S&P500 index evolves in time. To deal with, we shall calculate the self-similarity exponent of all the stocks in the index for each year. Notice that a year contains approximately 256 periods (trading days), so these time series are relatively short. This makes an application of FD4 algorithm, especially appropriate to carry out this study. An alternative approach would consist of using intraday stock data if they are available. Figures 3.15 and 3.16 contain histograms regarding the self-similarity exponent distribution of S&P 500 stocks for each year (graph on the left) and the empirical distribution of the self-similarity exponent of S&P 500 stocks through years. FD4 algorithm was applied for self-similarity exponent calculation purposes. The horizontal bars symbolize the confidence intervals of the empirical distribution of the self-similarity exponent of S&P 500 stocks for each year at a confidence level of 90%. It is noteworthy that almost all the S&P 500 stocks are characterized by a self-

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3 A Middle Definition Between Hausdorff and Box Dimensions

Fig. 3.15 Histogram regarding the empirical distribution of the self-similarity exponent of S&P500 stocks (calculated via FD4 algorithm) for each year

Fig. 3.16 Histogram regarding the empirical distribution of the self-similarity exponent of S&P500 stocks through years. Horizontal bars symbolize the confidence intervals of the empirical distribution (provided by FD4 algorithm) for each year at a confidence level of 90%

similarity exponent significantly >0.5. Additionally, the mean of the self-similarity exponent of all the S&P 500 stocks has been moving from 0.6 to 0.55 in the last years. This is clearly different from a BM pattern for each year. It is worth mentioning that Fig. 3.16 is in line with the results provided in [15], where the evolution from 1983 to 2009 of the Hurst exponent of the S&P 500 index itself was explored throughout intraday (1−min) data.

3.13 Notes and References The definition of fractal dimension III for fractal structures was first contributed by Fernández-Martínez and Sánchez-Granero in [36] (2012). In that paper, they explored the hybrid behavior of this new fractal dimension and proved some of its theoretical properties. In particular, they connected it with the classical models of fractal dimension as well as with fractal dimensions I and II, studied in Chap. 2. That work also contains a proof of Theorem 3.10.1 which allows to easily calculate the fractal dimension III of IFS-attractors not required to satisfy the OSC. The natural fractal structure, which any self-similar set can be always endowed with, first

3.13 Notes and References

147

appeared in [11, Definition 4.4], where the concepts of self-similar symbolic space and self-similar set were characterized in terms of fractal structures. This work is due to Arenas and Sánchez-Granero (2012). On the other hand, 1990 Edgar’s book collects the so-called Method II of construction of measures (c.f. [25, Theorem s (F) is an outer measure for 5.2.2]) that allowed us to show that the set function Hn,3 all natural number n (see Proposition 3.6). Moreover, regarding the basics on outer measures, we refer the reader to [25, Sect. 5.2]. Though the word (and also the concept of) fractal was introduced by B. Mandelbrot in late seventies [55], several nonstandard objects had appeared in the decade of 1890 as mathematical monsters, due to their counterintuitive analytical properties. In this way, they had been provided by some recognized mathematicians as counterexamples or exceptional objects. They include both Peano (1890) and Hilbert (1891) space-filling curves [43, 64]. On the other hand, the concept of fractal structure induced by another one on the image set of a curve was first introduced in [76], where its authors also provided the definition of fractal dimension of a curve with respect to that induced fractal structure. In addition, Fernández-Martínez and Sánchez-Granero explored some theoretical properties regarding this novel concept of fractal dimension in [34]. In that work, they also proved Theorem 3.11.7 that allows to construct space-filling curves throughout fractal structures. In [76, Sect. 6], it was described in detail how both Theorems 3.12.1 and 3.12.5 can be applied in order to test for long-range dependence on actual S&P500 stocks. This was carried out by calculating the fractal dimension of each stock treating its evolution by a time series.

Chapter 4

Hausdorff Dimension Type Models for Fractal Structures

Abstract In this chapter, we shall study how to generalize the Hausdorff dimension for Euclidean sets throughout three new approaches of fractal dimension for a fractal structure. Thus, while two of such fractal dimensions will consist of appropriate discretizations regarding the classical Hausdorff dimension (the so-called fractal dimensions IV and V), the remaining will constitute a continuous approach from the viewpoint of fractal structures (fractal dimension VI). In this way, several results allowing to connect the new dimensions among them are provided. Also, we shall prove some theoretical results allowing to reach the equality among the new models with fractal dimensions I, II, III, and the classical fractal dimensions. Moreover, we shall explore how the analytic construction regarding fractal dimension VI is based on a measure as it is the case of Hausdorff dimension. The key result in this chapter consists of a generalization regarding the classical Hausdorff dimension in the context of Euclidean subsets endowed with their natural fractal structures in terms of both fractal dimensions V and VI. It is also worth mentioning that fractal dimension IV will equal the Hausdorff dimension of compact Euclidean subsets. Interestingly, as a consequence of the latter result, a novel algorithm to calculate the Hausdorff dimension of such a kind of sets will be developed in forthcoming Sect. 4.8. Finally, we shall contribute a Moran’s type theorem allowing to easily calculate these fractal dimensions for IFS-attractors lying under the OSC.

4.1 Improving the Accuracy of Fractal Dimension III In Chap. 3, the fractal dimension III of a subset F of a (metric) space X was defined with respect to a fractal structure (c.f. Definition 3.3.1). To deal with, it was explored s (F) (c.f. Eq. (3.6)), and hence, the critical value the asymptotic behavior of Hn,3 s where H3 (F) “jumps” from ∞ to 0 leads to the fractal dimension III of F (c.f. Eq. 3.7 and Fig. 3.1), which equals the box dimension of F in the context of Euclidean subsets. In this chapter, our aim is to generalize the classical Hausdorff dimension from the viewpoint of fractal structures. Next, we provide the basics for a new fractal dimension approach.

© Springer Nature Switzerland AG 2019 M. Fernández-Martínez et al., Fractal Dimension for Fractal Structures, SEMA SIMAI Springer Series 19, https://doi.org/10.1007/978-3-030-16645-8_4

149

150

4 Hausdorff Dimension Type Models for Fractal Structures

Fractal dimension approach 4.1.1 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , and assume that diam (F, n ) → 0. s (F) consists of minimizing the sum Recall that the calculation of Hn,3 of the s−powers of the diameters of all the elements in an appropriate diam (F, n )−cover of F, say {Ai }i∈I . Recall that all the elements Ai of the covering lie in a same level of  deeper than or equal to n. In fact, such a condition can be mathematically described in the following terms: there exists m ≥ n such that Ai ∈ m for all i ∈ I . A further consideration consists of allowing that given a level n of the fractal structure , each element Ai may lie in some level deeper than or equal to n, though not always being the same, necessarily. Formally, for all i ∈ I , there exists m(i) ≥ n such that Ai ∈ m(i) . Following the above, let us define the next collection of diam (F, n )−coverings of F:   Bn (F) = {Ai }i∈I : Ai ∈ ∪l≥n l , F ⊆ ∪i∈I Ai . Moreover, let s ≥ 0 and define    s s Dn (F) = inf diam (Ai ) : {Ai }i∈I ∈ Bn (F) and

(4.1)

(4.2)

i∈I

D s (F) = lim Dns (F), n→∞

then we affirm that the set function D s behaves similarly to the s−dimensional Hausdorff measure. In fact, let t ≥ 0 with t > s. Hence,   diam (Ai )s , diam (Ai )t ≤ diam (F, n )t−s · where Ai ∈ {Ai }i∈I ∈ Bn (F). Therefore, Dnt (F) ≤ diam (F, n )t−s · Dns (F). Letting n → ∞, we have D t (F) ≤ D s (F) · lim diam (F, n )t−s . n→∞

Thus, if D s (F) < ∞, then D s (F) = 0 since t > s and diam (F, n ) → 0. Accordingly, the equality sup{s ≥ 0 : D s (F) = ∞} = inf{s ≥ 0 : D s (F) = 0} throws a new Hausdorff type dimension for F.

4.1 Improving the Accuracy of Fractal Dimension III

151

Remark 4.1.2 Despite the topology induced by the fractal structure  usually equals the topology induced by the metric ρ on X , both topologies do not have to be the same, in general. Thus, we shall always refer to the topology induced by . It is worth mentioning that, similarly to Counterexample 3.2.1, the condition diam (F, n ) → 0 becomes necessary for upcoming fractal dimension calculation purposes. Counterexample 4.1.3 There exists a fractal structure  on a metric space (X, ρ) and a subset F of X with diam (F, n )  0, satisfying that inf{s ≥ 0 : D s (F) = 0} = sup{s ≥ 0 : D s (F) = ∞}.

Proof Indeed, let F = (0, 1] ⊂ R and  = {n }n∈N be a fractal structure on [0, 1] whose levels are given as follows: 

k k+1 n : k = 1, . . . , 2 , − 1 . (4.3) n = {[0, 1]} ∪ 2n 2n Since diam (F, n ) = 1 for all n ∈ N, then diam (F, n )  0. Additionally, the following identity stands:  1 if s ≤ 1 s D (F) = 0 if s > 1. In fact, (1) If s ≤ 1, then D s (F) = 1 since one of the two following cases may occur: (a) Assume that the covering of F we choose for fractal dimension calculation purposes contains the interval [0, 1]. Thus, (i) If that covering is {[0, 1]}, then Dns (F) = (diam ([0, 1]))s ≤ 1, and hence, D s (F) ≤ 1. (ii) Assume that the covering of F we choose, {Ai }i∈I , contains the interval [0, 1] together with some elements of the natural fractal structure on [0, 1] as a Euclidean subset. Then  (diam (Ai ))s ≥ 1. i∈I

Accordingly, D s (F) ≤ 1.

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4 Hausdorff Dimension Type Models for Fractal Structures

(b) On the other hand, assume that the covering of F we select, {Ai }i∈I , does not contain the closed unit interval [0, 1]. Accordingly, such a covering will consist of some elements of the natural fractal structure (induced) on [0, 1]. Since the fractal dimension of F with respect to such a natural fractal structure is equal to 1, then  (diam (Ai ))s ≥ 1. i∈I

Hence, D s (F) = 1. (2) If s > 1, then D s (F) = 0. In fact, let  = {n }n∈N be the natural fractal structure (induced) on [0, 1]. Hence, D s (F) = H s (F) = 0, due to Theorem 4.6.1. Let ε > 0 be a fixed but arbitrarily chosen real number. Thus, there exists a covering {Ai }i∈I ∈ Bn (F) such that for all i ∈ I , Ai ∈ k with k ≥ n, and  s i∈I diam (Ai ) < ε. One of the two following cases may occur: • Ai ∈  k : k ≥ n, or / k : k ≥ n. In this case, though, observe that • Ai = 0, 21k ∈   1 1 1 . 0, k = , 2 2k+α 2k+α−1 α≥1 Accordingly, a new covering B of F can be constructed

all the elements from in {Ai }i∈I but replacing those elements of the form 0, 21k by 

1

2

, k+α



1 2k+α−1

:α≥1 ,



instead. In fact, for each element of the form 0, 21k , we can write +∞  α=1

Thus,

+∞ 1  1 1 1 = ks · = ks · s s )α (2k+α )s 2 (2 2 2 −1 α=1 s   1 1 < ks = diam 0, k . 2 2

1

 B∈B

diam (B)s ≤



diam (Ai )s < ε,

i∈I

so Dks (F) < ε. Therefore, D s (F) = 0 for all s > 1. Another Hausdorff type dimension is sketched next in terms of finite coverings.

4.1 Improving the Accuracy of Fractal Dimension III

153

Fractal dimension approach 4.1.4 Similarly to both Eqs. (4.1) and (4.2), the following expressions lead to a discrete fractal dimension for finite coverings, which will become especially appropriate to deal with empirical applications involving Hausdorff dimension (c.f. Sect. 4.8). In fact, let   Ln (F) = {Ai }i∈I : Ai ∈ ∪l≥n l , F ⊆ ∪i∈I Ai , Card (I ) < ∞ , as well as  Kns (F)

= inf



 diam (Ai ) : {Ai }i∈I ∈ Ln (F) . s

(4.4)

i∈I

Thus, the asymptotic behavior of Eq. (4.4) as n → ∞ plays a similar role to Hausdorff measure. In fact, if K s (F) = lim Kns (F), n→∞

then it can be proved (similarly to Fractal dimension approach 4.1.1) that sup{s ≥ 0 : K s (F) = ∞} = inf{s ≥ 0 : K s (F) = 0}, namely, s is a (unique) critical point where K s (F) “jumps” from ∞ to 0 which leads to a Hausdorff type dimension for F.

The following remark is analogous to Counterexample 4.1.3. Counterexample 4.1.5 There exists a fractal structure  on a metric space (X, ρ) and a subset F of X with diam (F, n )  0, satisfying that inf{s ≥ 0 : K s (F) = 0} = sup{s ≥ 0 : K s (F) = ∞}.

Proof Let F = (0, 1] ⊂ R and  be the fractal structure whose levels were defined previously in Eq. (4.3). Thus, any finite covering of F by elements of  must contain the closed unit interval [0, 1]. This implies that K s (F) = 1 for all s ≥ 0, leading to {s ≥ 0 : K s (F) = ∞} = {s ≥ 0 : K s (F) = 0} = ∅. It is worth pointing out that fractal structures also allow to provide continuous Hausdorff type dimensions. Next, we explain how to deal with.

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4 Hausdorff Dimension Type Models for Fractal Structures

Fractal dimension approach 4.1.6 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , and δ > 0. Moreover, let us define the following collection of coverings of F: Gδ (F) = {{Ai }i∈I : Ai ∈ ∪l∈N l , diam (Ai ) ≤ δ, F ⊆ ∪i∈I Ai } , as well as  Jδs (F)

= inf



 diam (Ai ) : {Ai }i∈I ∈ Gδ (F) . s

i∈I

Let us explore the asymptotic behavior of Jδs (F) by means of J s (F) = lim Jδs (F). δ→0

Let t ≥ 0. Thus,  i∈I

diam (Ai )t ≤ δ t−s ·



diam (Ai )s ,

(4.5)

i∈I

where the sums have been considered on Gδ (F). Taking infima in Eq. (4.5), we have Jδt (F) ≤ δ t−s · Jδs (F). Letting δ → 0,

J t (F) ≤ J s (F) · lim δ t−s . δ→0

Therefore, if J s (F) < ∞ and δ → 0 provided that t > s, then J s (F) = 0. Accordingly, the critical point s where J s (F) “jumps” from ∞ to zero leads to a Hausdorff type dimension for F, namely, sup{s ≥ 0 : J s (F) = ∞} = inf{s ≥ 0 : J s (F) = 0}. The previous models will be formalized in the upcoming section.

4.2 Hausdorff-Type Dimensions for Fractal Structures The fractal dimensions we shall explore in this chapter are provided in the next definition.

4.2 Hausdorff-Type Dimensions for Fractal Structures

155

Definition 4.2.1 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , and assume that diam (F, n ) → 0. In addition, consider the following expression:  s Hn,k (F) = inf



 diam (Ai )s : {Ai }i∈I ∈ An,k (F) , where

i∈I

  {Ai }i∈I : Ai ∈ ∪l≥n l , F ⊆ ∪i∈I Ai ,Card (I ) < ∞ if k = 4  An,k (F) = {Ai }i∈I : Ai ∈ ∪l≥n l , F ⊆ ∪i∈I Ai if k = 5, and define s (F) for k = 4, 5. Hks (F) = lim Hn,k n→∞

The fractal dimensions IV and V of F are defined, respectively, by the following (unique) critical points: dimk (F) = inf{s ≥ 0 : Hks (F) = 0} = sup{s ≥ 0 : Hks (F) = ∞} : k = 4, 5. The following link between fractal dimensions IV and V follows immediately from Definition 4.2.1. Proposition 4.2.2 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , and assume that diam (F, n ) → 0. Then dim5 (F) ≤ dim4 (F).

(4.6)

In Definition 4.2.1 as well as in the next one, we shall assume that inf ∅ = ∞. For instance, if An,4 (F) = ∅, then dim4 (F) = ∞. Definition 4.2.3 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , δ > 0, and assume that diam (F, n ) → 0. Moreover, consider the following expression:  s Hδ,6 (F)

= inf



 diam (Ai ) : {Ai }i∈I ∈ Aδ,6 (F) , where s

i∈I

Aδ,6 (F) = {{Ai }i∈I : Ai ∈ ∪l∈N l , diam (Ai ) ≤ δ, F ⊆ ∪i∈I Ai }

156

4 Hausdorff Dimension Type Models for Fractal Structures ∞

Fig. 4.1 Graph representation of s vs. Hks (F) for k = 4, 5, 6 s

Hk (F)

0 0

k

dimΓ (F)

and define s (F). H6s (F) = lim Hδ,6 δ→0

The fractal dimension VI of F is given by the following critical point: dim6 (F) = inf{s ≥ 0 : H6s (F) = 0} = sup{s ≥ 0 : H6s (F) = ∞}. Equivalently, from both Definitions 4.2.1 and 4.2.3, it holds that  Hks (F) =

∞ if s < dimk (F) 0 if s > dimk (F)

(4.7)

(for k = 4, 5, 6) provided that diam (F, n ) → 0. As a consequence of Eq. (4.7), we have that fractal dimensions IV-VI behave similarly to both Hausdorff measure and dimension (c.f. Eq. (1.10)) as well as fractal dimension III and its corresponding set function (c.f. Eq. (3.8)). Regarding this, Fig. 4.1 displays a graphical representation of s vs. Hks (F) for k = 4, 5, 6. The next remark states that it is not necessary to consider lower/upper limits (unlike it happens with box dimension) for Hks (F) calculation purposes (k = 4, 5, 6), which is also the case of both H3s (F) (c.f. Remark 3.3.3) and H Hs (F) (c.f. Sect. 1.2). Remark 4.1 s (1) Since Hn,k (F) is the general term of a monotonic nondecreasing sequence in n ∈ N for k = 4, 5, then the fractal dimensions IV and V of any subset F of X always exist.

4.2 Hausdorff-Type Dimensions for Fractal Structures

157

s (2) Since Hδ,6 (F) is nonincreasing for s ≥ 0, then H6s (F) also is (by definition), so the fractal dimension VI of any subset F of X always exists.

Recall that Hausdorff dimension becomes the main reference for new definitions of fractal dimension to be mirrored in. In fact, Hausdorff dimension satisfies several desirable properties as a dimension function (c.f. Theorem 1.2.8 and Propositions 1.2.9 and 1.2.10). Similarly to Proposition 2.4.1 for fractal dimension I, Proposition 2.10.1 for fractal dimension II, and Proposition 3.8.1 for fractal dimension III, next we explore the behavior of fractal dimensions IV-VI as dimension functions through a pair of theoretical results. The first of them contains some properties regarding fractal dimension IV, whereas the second result collects some properties for both fractal dimensions V and VI. Proposition 4.2.4 Let  be a fractal structure on a metric space (X, ρ) and assume that diam (n ) → 0. The following statements hold. (1) Fractal dimension IV is monotonic. (2) There exist a countable subset F of X and a fractal structure  on X such that dim4 (F) = 0. (3) Fractal dimension IV is not countably stable. (4) Fractal dimension IV is finitely stable. (5) dim4 (F) = dim4 (F) for all subset F of X .

Proof (1) Let E, F be two subsets of X and assume that E ⊆ F. In addition, let {Ai }i∈I be a covering of An,4 (F). Thus, Ai ∈ ∪l≥n l for all i ∈ I and E ⊆ F ⊆ ∪i∈I Ai , where Card (I ) < ∞. Hence, {Ai }i∈I ∈ An,4 (E), i.e., An,4 (F) ⊆ An,4 (E), so s s (E) ≤ Hn,4 (F) for all n ∈ N and all s > 0. Accordingly, H4s (E) ≤ H4s Hn,4 (F). Therefore, dim4 (E) ≤ dim4 (F). (2) Let  = {n }n∈N be the natural fractal structure (induced) on [0, 1], whose levels are given by 

k k+1 n : k ∈ {0, 1, . . . , 2 − 1} , n = 2n 2n for all n ∈ N and define F = Q ∩ [0, 1]. Thus, F is a countable set such that F = [0, 1]. Hence, Theorem 4.6.4 gives dim4 (F) = dimH (F) = 1. (3) Notice that (2) implies (3). In fact, let  be the natural fractal structure (induced) on [0, 1] and consider F = Q ∩ [0, 1]. By Proposition 4.2.4(2), we have that dim4 (F) = 1, though sup{dim4 ({qi }) : qi ∈ F} = 0. (4) The following inequality stands since fractal dimension IV is monotonic (c.f. Proposition 4.2.4(1)):

158

4 Hausdorff Dimension Type Models for Fractal Structures

max{dim4 (F1 ), dim4 (F2 )} ≤ dim4 (F1 ∪ F2 ), where F1 , F2 ⊆ X . Thus, we shall be focused on the opposite inequality. Indeed, let ε > 0 be fixed but arbitrarily chosen and s > 0 be such that s > max{dim4 (F1 ), dim4 (F2 )}. Since H4s (F1 ) = 0, then there exists a covering  1 l1  s 1 Ak k=1 of F1 such that lk=1 diam A1k < 2ε . Similarly, there exists a covering  s  2 l2 2 diam A2k < 2ε . Hence, Ak k=1 of F2 such that lk=1     1 2 F1 ∪ F2 ⊆ ∪lk=1 A1k ∪ ∪lk=1 A2k and l1 

l2  s   s ε ε diam A1k + diam A2k < + = ε. 2 2 k=1 k=1

Accordingly, H4s (F1 ∪ F2 ) = 0 for all s > max{dim4 (F1 ), dim4 (F2 )}, namely, dim4 (F1 ∪ F2 ) ≤ max{dim4 (F1 ), dim4 (F2 )}. (5) First of all, it is clear that dim4 (F) ≤ dim4 (F) since fractal dimension IV is monotonic (c.f. Proposition 4.2.4(1)). To tackle with the opposite inequality, let {Ai }i∈I ∈ An,4 (F). Then Ai ∈ ∪l≥n l for all i ∈ I , F ⊆ ∪i∈I Ai , and Card (I ) < ∞. Hence, F ⊆ ∪i∈I Ai = ∪i∈I Ai = ∪i∈I Ai sinceallthe elements in the cover  ing {Ai }i∈I are closed. Therefore, {Ai }i∈I ∈ An,4 F , so An,4 (F) ⊆ An,4 F   s s for all n ∈ N. Thus, Hn,4 F ≤ Hn,4 (F) for all s > 0. Letting n → ∞, we have   H4s F ≤ H4s (F) for all s > 0. Accordingly, dim4 (F) ≤ dim4 (F). Similarly to Proposition 4.2.4, the following result contains some properties regarding both fractal dimensions V and VI. Proposition 4.2.5 Let  be a fractal structure on a metric space (X, ρ) and assume that diam (n ) → 0. The following statements hold. (1) (2) (3) (4)

Both fractal dimensions V and VI are monotonic. Both fractal dimensions V and VI are countably stable. dim5 (F) = dim6 (F) = 0 for every countable subset F of X . There exists a locally finite starbase fractal structure  defined on a certain subset F ⊆ X such that dimk (F) = dimk (F) for k = 5, 6.

Proof (1) Let E, F be two subsets of X and assume that E ⊆ F. In addition, let {Ai }i∈I be a covering of An,5 (F). Thus, Ai ∈ ∪l≥n l for all i ∈ I and E ⊆ F ⊆ ∪i∈I Ai . Hence, {Ai }i∈I ∈ An,5 (E). Accordingly, An,5 (F) ⊆ An,5 (E). This leads to

4.2 Hausdorff-Type Dimensions for Fractal Structures

159

s s Hn,5 (E) ≤ Hn,5 (F) for all n ∈ N and all s > 0. Therefore, H5s (E) ≤ H5s (F), so dim5 (E) ≤ dim5 (F). On the other hand, let {Ai }i∈I ∈ Aδ,6 (F). Thus, Ai ∈ ∪l∈N l for all i ∈ I , diam (Ai ) ≤ δ, and E ⊆ F ⊆ ∪i∈I Ai . Hence, {Ai }i∈I ∈ Aδ,6 (E). This leads to s s (E) ≤ Hδ,6 (F) for all δ, s > Aδ,6 (F) ⊆ Aδ,6 (E) for all δ > 0. Therefore, Hδ,6 6 s s 0. Accordingly, H6 (E) ≤ H6 (F) for all s > 0, so dim (E) ≤ dim6 (F). (2) Since fractal dimension V is monotonic (c.f. Proposition 4.2.5(1)), then

sup{dim5 (Fi ) : i ∈ I } ≤ dim5 (∪i∈I Fi ) . To deal with the opposite inequality, let ε > 0 be fixed but arbitrarily chosen and s > 0 be such that s > sup{dim5 (Fi ) : i ∈ I }. Since H5s (Fi ) = 0 for all i ∈ I , then there exists a family of coverings of Fi , say {Aik }k∈N , such that  k

 s ε diam Aik < i 2

with Fi ⊆ ∪k Aik for all i ∈ I . Thus, ∪i∈I Fi ⊆ ∪k,i Aik and ∞  ∞  i=1

∞  s  ε diam Aik ≤ = ε. i 2 k=1 i=1

Hence, H5s (∪i∈I Fi ) = 0 for all s > sup{dim5 (Fi ) : i ∈ I }, so dim5 (∪i∈I Fi ) ≤ sup{dim5 (Fi ) : i ∈ I }. To show that H6s (∪i∈I Fi ) = 0 for all s > sup{dim6 (Fi ) : i ∈ I }, we shall apply that H6s is a (metric) outer measure (c.f. Theorem 4.5.3). In fact, H6s (∪i∈I Fi ) ≤



H6s (Fi ) = 0,

i∈I

since H6s (Fi ) = 0 for all i ∈ I . (3) Let F ⊆ X countable. Thus, we can write F = ∪i∈I {qi }. Since both fractal dimensions V and VI are countably stable (c.f. Proposition 4.2.5(2)), then it holds that dimk (F) = sup{dimk ({qi }) : i ∈ I } = 0 for k = 5, 6. (4) Let  = {n }n∈N be the natural fractal structure (induced) on [0, 1], whose levels ] : k ∈ {0, 1, . . . , 2n − 1}}. In addition, let F = Q ∩ [0, 1]. are n = {[ 2kn , k+1 2n Since F is countable and F = [0, 1], then Proposition 4.2.5(3) gives dim5 (F) = dim6 (F) = 0. On the other hand, Corollary 4.6.2 leads to     dim5 F = dimH (F) = dim6 F = 1.

160

4 Hausdorff Dimension Type Models for Fractal Structures

Table 4.1 The table summarizes all the theoretical properties that are satisfied by each definition of fractal dimension explored throughout this book Theoretical properties dimB dim1 dim2 dim3 dim4 dim5 dim6 dimH Monotonicity Finite stability Countable stability 0−countable cl− dim

✓ ✓

✓ ✓

✓

✓ ✓

✓

✓ ✓

✓ ✓ ✓ ✓

✓ ✓ ✓ ✓

✓ ✓ ✓ ✓

✓

Table 4.1 summarizes the behavior of all the fractal dimension models explored in this book (fractal dimensions I-VI) throughout the properties as dimension functions each of them satisfy. It is worth mentioning that fractal dimensions V and VI behave more similarly to Hausdorff dimension than the other models, whereas fractal dimension I is closest to box dimension.

4.3 Linking Fractal Dimensions V and VI Regarding the theoretical properties that fractal dimensions V and VI satisfy as dimension functions (c.f. Sect. 4.2), it holds that they behave similarly to classical Hausdorff dimension. In this section, we shall explore some conditions concerning the elements in each level of a fractal structure to theoretically connect them. With this aim, the first result we provide contains a first link between these fractal dimensions. It is worth pointing out that the only condition required therein concerns the sequence of diameters diam (F, n ). More specifically, we shall assume that the supremum of the diameters of all the elements in each level of the involved fractal structure is smaller as deeper levels are reached. Lemma 4.3.1 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , and assume that diam (F, n ) → 0. Then dim6 (F) ≤ dim5 (F).

(4.8)

Proof Let us consider the families of coverings of F, An,5 (F) and Aδ,6 (F) for each natural number n and all δ > 0, as they were described in both Definitions 4.2.1 and 4.2.3. In addition, let {Ai }i∈I ∈ An,5 (F) be a covering of F. Then for each i ∈ I , there exists a natural number l ≥ n such that Ai ∈ l . Thus, it holds that diam (Ai ) ≤ diam (F, l ) ≤ diam (F, n ),

4.3 Linking Fractal Dimensions V and VI

161

since diam (F, n ) → 0 as n → ∞ by hypothesis. Therefore, {Ai }i∈I ∈ Aδn ,6 (F), where we have denoted δn = diam (F, n ). Accordingly, An,5 (F) ⊆ Aδn ,6 (F), which s (F) for all natural number n. This leads to H6s (F) ≤ implies that Hδsn ,6 (F) ≤ Hn,5 s H5 (F) for all s > 0. Therefore, Eq. (4.8) follows. Next, a sufficient condition regarding the elements in each level of a fractal structure is provided to guarantee the equality between fractal dimensions V and VI. We shall refer to such a condition (regarding a fractal structure) as diameter-positive. Definition 4.3.2 Let  be a fractal structure on a metric space (X, ρ). We shall understand that  is diameter-positive provided that the following condition stands: inf{diam (A) : A ∈ n+ } > 0, where n+ = {A ∈ n : diam (A) > 0}. It is worth noting that there exist large families of diameter-positive fractal structures. Next, we collect some examples of fractal structures being diameter-positive. Remark 4.3.3 The following families of fractal structures are diameter-positive: (i) Any finite fractal structure. (ii) The natural fractal structure which any Euclidean subset can be always endowed with (c.f. Definition 2.2.1). (iii) The natural fractal structure which any IFS-attractor can be always endowed with. Interestingly, under the diameter-positive condition for a fractal structure, it can be proved that fractal dimensions V and VI coincide. Theorem 4.3.4 Let  be a diameter-positive fractal structure on a metric space (X, ρ), F be a subset of X , and assume that diam (F, n ) → 0. Then dim5 (F) = dim6 (F).

Proof Lemma 4.3.1 gives dim6 (F) ≤ dim5 (F). Thus, we shall be focused on the opposite inequality. To deal with, let {xn }n∈N be a sequence of real numbers whose general term is given by   xn = inf diam (A) : A ∈ n+ , A ∩ F = ∅ .

162

4 Hausdorff Dimension Type Models for Fractal Structures

Moreover, let {yn }n∈N be another sequence of real numbers whose general term is defined as yn = min{xk : k ≤ n}. The following statements follow clearly. • {yn }n∈N is a sequence of positive real numbers since  is diameter-positive. • yn ≤ xn for all n ∈ N. • {yn }n∈N is monotonic decreasing. In fact, yn+1 = min{xk : k ≤ n + 1} ≤ min{xk : k ≤ n} = yn . • limn→∞ yn = 0 since yn ≤ diam (F, n ) → 0. Let n be a fixed but arbitrarily chosen natural number and define ϕn = n1 . Further, assume that ϕn < y1 and let m(n) ∈ N be such that ym(n) > ϕn

ym(n)+1 ≤ ϕn .

Thus, m(n) = max{α ∈ N : yα > ϕn }. Equivalently, yk > ϕn for all k ≤ m(n) and yβ ≤ ϕn for all β > m(n). It is worth pointing out that such a natural number m(n) exists since {yn }n∈N is a decreasing sequence of positive real terms going to 0 and also due to the hypothesis ϕn < y1 . Let Ai ∈ {Ai }i∈I ∈ Aϕn ,6 (F). Then one of the two following cases may occur. (i) Let us suppose that diam (Ai ) > 0. If Ai ∈ k , then we can affirm that k ≥ m(n). In fact, assume the contrary. Thus, ϕn < yk ≤ xk ≤ diam (Ai ), a contradiction since {Ai }i∈I ∈ Aϕn ,6 (F) (and hence, diam (Ai ) ≤ ϕn for all i ∈ I ). (ii) Assume that diam (Ai ) = 0. Then Ai consists of a single point, namely, Ai = { p} ∈ l with l ∈ N. Thus, Ai ∈ k for all k ≥ l. In particular, Ai ∈ k for all k ≥ m(n). Accordingly, Ai ∈ {Ai }i∈I ∈ Am(n),5 (F), namely, Aϕn ,6 (F) ⊆ Am(n),5 (F) for all n ∈ s (F) ≤ Hϕsn ,6 (F) and letting n → ∞, we have H5s (F) ≤ N. This implies that Hm(n),5 s H6 (F), leading to the opposite inequality.

4.4 Additional Connections Among Fractal Dimensions III, IV, and V In this section, we prove that fractal dimension V (and hence, fractal dimension VI, by Lemma 4.3.1) can be linked to fractal dimension III. First, we state that fractal dimension V is always ≤ than fractal dimension III.

4.4 Additional Connections Among Fractal Dimensions III, IV, and V

163

Theorem 4.4.1 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , and assume that diam (F, n ) → 0. Then dim6 (F) ≤ dim5 (F) ≤ dim3 (F).

Proof To deal with, let us consider the families of coverings of F, An,3 (F) and An,5 (F) (c.f. Eq. (3.5) and Definition (4.2.1)). Let {Ai }i∈I ∈ An,3 (F) be a covering of F. Since each element of {Ai }i∈I belongs to a same level l of  with l ≥ n, then it is clear that {Ai }i∈I ∈ An,5 (F). Accordingly, An,3 (F) ⊆ An,5 (F) for all n ∈ N and letting n → ∞, we have H5s (F) ≤ H3s (F) for all s > 0. Therefore, dim5 (F) ≤ dim3 (F). Finally, Lemma 4.3.1 gives the result. The result provided below gathers several connections among Hausdorff dimension and fractal dimensions II (c.f. Definition 2.8.2), III (c.f. Definition 3.3.1), IV and V (both of them described in Definition 4.2.1), and VI (c.f. Definition 4.2.3). Corollary 4.4.2 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , and assume that diam (F, n ) → 0. The following two hold: (1) dimH (F) ≤ dim6 (F) ≤ dim5 (F) ≤ dim4 (F). 2 (2) dimH (F) ≤ dim6 (F) ≤ dim5 (F) ≤ dim3 (F) ≤ dim2 (F) ≤ dim (F).

Proof (1) First, it is clear that dimH (F) ≤ dim6 (F), since any covering of Aδ,6 (F) is also a δ−cover. In addition, Lemma 4.3.1 gives dim6 (F) ≤ dim5 (F) and Proposition 4.2.2 leads to dim5 (F) ≤ dim4 (F). (2) The chain of inequalities dimH (F) ≤ dim6 (F) ≤ dim5 (F) has been justified in Corollary 4.4.2(1). On the other hand, Theorem 4.4.1 gives dim 5 (F) ≤ dim3 (F) and finally, Theorem 3.4.1(1) gives the result. Interestingly, an additional link between fractal dimensions III and IV can be proved for finite fractal structures. Theorem 4.4.3 Let  be a finite fractal structure on a metric space (X, ρ), F be a subset of X , and assume that diam (F, n ) → 0. Then dim4 (F) ≤ dim3 (F).

(4.9)

164

4 Hausdorff Dimension Type Models for Fractal Structures

Proof Notice that by Remark 3.3.2(3), the fractal dimension III of F can be calculated throughout the following expression:  s Hn,3 (F)

= inf



 diam (A) : B ∈ An,3 (F) , s

A∈B

where An,3 (F) = ∪m≥n {Am (F)} (c.f. Eq. (3.5)). On the other hand, we shall calculate the fractal dimension IV of F by means of  s Hn,4 (F)

= inf



 diam (Ai ) : {Ai }i∈I ∈ An,4 (F) , where s

i∈I

  An,4 (F) = {Ai }i∈I : Ai ∈ ∪l≥n l , F ⊆ ∪i∈I Ai , Card (I ) < ∞ (c.f. Definition 4.2.1). Let {Ai }i∈I ∈ An,3 (F) be a covering of F. Then there exists a natural number l ≥ n such that Ai ∈ l and Ai ∩ F = ∅ for all i ∈ I . Clearly, F ⊆ ∪i∈I Ai . In addition, Card (I ) < ∞ since all the elements of the covering {Ai }i∈I belong to a same level l of  and such a fractal structure is finite. Moreover, for all i ∈ I there exists a natural number l ≥ n such that Ai ∈ l . This implies that s (F) ≤ {Ai }i∈I ∈ An,4 (F), i.e., An,3 (F) ⊆ An,4 (F) for all n ∈ N. Accordingly, Hn,4 s s s Hn,3 (F) for all natural number n and letting n → ∞, H4 (F) ≤ H3 (F). Hence, Eq. (4.9) stands. The last result we provide in this section involves all the fractal dimension models for a fractal structure from II to VI together with Hausdorff dimension for finite fractal structures. Corollary 4.4.4 Let  be a finite fractal structure on a metric space (X, ρ), F be a subset of X , and assume that diam (F, n ) → 0. The following chain of inequalities stands: dimH (F) ≤ dim6 (F) ≤ dim5 (F) ≤ dim4 (F) 2

≤ dim3 (F) ≤ dim2 (F) ≤ dim (F).

Proof The result follows as a consequence of Corollary 4.4.2(1) (which gives the first row above), Theorem 4.4.3 (which connects dim4 (F) with dim3 (F)), and Theorem 3.4.1(1).

4.5 Measure Properties of H6s

165

4.5 Measure Properties of H6s So far, some models to calculate the fractal dimension of any subset F of a GF-space (X, ) we have been introduced. The first of them, fractal dimension I (c.f. Definition 2.3.2), considers all the elements in each level n of a fractal structure as having the same size, equal to 1/2n . On the other hand, the fractal dimension II model (c.f. Definition 2.8.2) allows that elements with different diameters may appear in each level of , though for calculation purposes, such a definition actually uses the largest diameter of all the elements in each level n of  that intersect the given subset F. In addition, the number of elements in each level of the fractal structure that intersect F has to be counted to calculate both the fractal dimensions I and II of F. Anyway, fractal dimensions I and II are supported by discrete box dimension type formulae. An advantage regarding these new models of fractal dimension consists of their easy calculation in empirical applications (just as easy as the box dimension estimation), even in non-Euclidean contexts, where the classical box dimension may lack sense or cannot be calculated (c.f. Sect. 2.15). Moreover, a new methodology to deal with the calculation of the fractal dimension of a subset with respect to any fractal structure was provided in Definition 3.3.1. That approach is based on a suitable discretization regarding the classical Hausdorff model. In this case, the fractal dimension is calculated throughout the sum of the s−powers of the diameters of all the elements in a certain covering of F. It is worth mentioning that all the elements in such a covering lie in a same level of the fractal structure. In the present chapter, though, we have introduced new dimensions (c.f. both Definitions 4.2.1 and 4.2.3) where it is allowed that such a covering contains elements lying in different levels of the fractal structure (all of them deeper than a certain level). Interestingly, these Hausdorff dimension type models for a fractal structure inherit some measure properties from classical Hausdorff dimension as it was previously justified in the case of fractal dimension III. In fact, recall s are outer measures for all natural number n that the set functions Hns and Hn,3 (c.f. Proposition 3.6), though their limits as n → ∞, H s and H3s , respectively, are not (c.f. Counterexample 3.6.2). Following the above, our next goal is to show that the analytical construction regarding fractal dimension VI is also based on a measure. Indeed, unlike the set functions H s and H3s that allow the calculation of fractal dimension III, it holds that the set function H6s is an outer measure. To deal with, first, let us recall some concepts and results from probability and measure theories that will be useful for our purposes (c.f. [25, Sects. 5.2 and 5.4]). Definition 4.5.1 Let (X, ρ) be a metric space. We say that a pair of subsets A, B of X have positive separation, if and only if, ρ(A, B) > 0, i.e., if there exists r > 0 such that ρ(x, y) ≥ r for all x ∈ A and y ∈ B. Next, we recall a first approach to tackle with the construction of outer measures called as Method I. Let A be a collection of subsets of a set X that covers it. In

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addition, let c : A → [0, ∞] be a set function. The following result is the so-called Method I (of construction of outer measures). Theorem 4.5.2 (Method I Theorem) There exists a unique outer measure μ on X satisfying the two following conditions: (1) μ(A) ≤ c(A) for all A ∈ A . (2) If ν is any other outer measure on X such that ν(A) ≤ c(A) for all A ∈ A , then ν(B) ≤ μ(B) for all B ⊆ X . It is also worth pointing out that an outer measure μ : P(X ) → [0, ∞] is a metric outer measure if μ(A ∪ B) = μ(A) + μ(B) for any pair A, B of subsets of X having positive separation. The restriction of a metric outer measure to the class of its measurable sets is called a metric measure. Since the Method I provided in Theorem 4.5.2 may fail to provide a measure for which the open sets are measurable, the so-called Method II is applied to deal with this problem (c.f. [25, Sect. 5.4]). Next, we recall the basics on Method II. Method II (of construction of metric outer measures). Let A be a family of subsets of a metric space X and assume that for all x ∈ X and ε > 0, there exists A ∈ A such that x ∈ A with diam (A) ≤ ε. Let c : A → [0, ∞] be a set function. Then an outer measure can be constructed as follows. First, define Aε = {A ∈ A : diam (A) ≤ ε}. Moreover, let με be the outer measure provided by applying Method I to the set function c and the collection Aε . Notice that, for a given subset F of X , it holds that με (F) increases as ε decreases (c.f. [25, Proposition 5.2.3(a)]). Hence, it can be proved that the set function μ defined as follows: μ(F) = lim με (F) = sup με (F), ε→0

ε>0

is an outer measure. Let μ denote the restriction of μ to the class of measurable sets. Thus, such a construction of an outer measure μ from a set function c (and also, a measure μ from μ) is called as Method II of construction of metric outer measures. As Edgar states, though being more complicated than Method I, Method II guarantees that Borel sets are measurable (c.f. [25, Theorem 5.4.4]). Method II of construction of outer measures allows to justify that the restriction of the set function H6s to the class consisting of all the Borel sets of X is a measure. This has been carried out in the next pair of results.

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167

Theorem 4.5.3 Let  be a fractal structure on a metric space (X, ρ) and assume that diam (n ) → 0. The set function H6s : P(X ) → [0, ∞], given by s H6s (F) = lim Hδ,6 (F) δ→0

for all subset F of X , is a metric outer measure.

Proof Let A be the family of coverings of X given by A = ∪n∈N n . In addition, let c : A → [0, ∞] be the set function defined as c(A) = diam (A)s for all A ∈ A . The following two steps lead to the result:  (1) Let μ : P(X ) → [0, ∞] be the set function defined as μ(F) = inf A∈D c(A) for all F ∈ P(X ), where D = {{Ai }i∈I : Ai ∈ A , F ⊆ ∪i∈I Ai } . It is worth pointing out that μ is the outer measure provided by Method I (c.f. Theorem 4.5.2). (2) Let ε > 0 and x ∈ X be fixed but arbitrarily chosen. Since diam (n ) → 0, then there exists a natural number n ∈ N such that diam (n ) < ε. Thus, take A ∈ n such that x ∈ A. Hence, we have A ∈ A such that x ∈ A with diam (A) ≤ diam (n ) < ε. Therefore, we can apply Method II. To deal with, define the following collection of elements from A : Aδ = {A ∈ A : diam (A) ≤ δ} . P(X ) → [0, ∞] be the outer measure provided by Method In addition, let μδ :  I, i.e., μδ (F) = inf A∈D δ c(A), where Dδ = {{Ai }i∈I : Ai ∈ Aδ , F ⊆ ∪i∈I Ai } . It holds that μδ (F) increases as δ decreases. Hence, if we define the set function μ : P(X ) → [0, ∞] in the following terms: s μ(F) = lim μδ (F) = lim Hδ,6 (F) = H6s (F) δ→0

δ→0

(4.10)

for all F ∈ P(X ), then both Eq. (4.10) and [25, Theorem 5.4.4] allow to affirm that H6s is a metric outer measure. As an immediate consequence of Theorem 4.5.3, we can conclude that H6s is actually a measure provided that it is restricted to the class of all Borel sets of X , as the following result points out. It is worth mentioning that the only constraint therein is quite natural and regards the diameter of each level of the fractal structure (c.f. Definition 2.5.1(1)).

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Corollary 4.5.4 Let  be a fractal structure on a metric space (X, ρ) and assume that diam (n ) → 0. The restriction of H6s to the class of all Borel sets of X is a measure.

Proof Recall that H6s is a metric outer measure due to Theorem 4.5.3. Additionally, every Borel set of X is H6s −measurable (c.f. [25, Theorem 5.4.2]).

4.6 Generalizing Hausdorff Dimension In previous chapters, the box dimension for Euclidean subsets has been successively generalized throughout fractal dimensions I (c.f. Theorem 2.3.4), II (c.f. Theorem 2.9.3), and III (c.f. Theorem 3.7.7). It is worth pointing out that in all those cases, these fractal dimensions has been calculated with respect to the natural fractal structure that each Euclidean subset can be always endowed with (c.f. Definition 2.2.1). They also allow us to deal with the calculation of fractal dimension in a larger range of spaces and situations (c.f. Sects. 2.15–4.8). Recall that fractal dimensions I and II are both based on box dimension type definitions and coincide with the box dimension in the context of Euclidean subsets endowed with their natural fractal structures. Interestingly, the definition of fractal dimension III constitutes an attempt to discretize the classical Hausdorff model though it equals the box dimension in the context of Euclidean subsets endowed with their natural fractal structures (c.f. Theorem 3.7.7). Afterwards, fractal dimensions IV, V, and VI were introduced to provide better approaches to Hausdorff dimension from a discrete viewpoint (c.f. Definitions 4.2.1 and 4.2.3). In this section, we shall explore some connections of fractal dimensions IV, V, and VI with classical Hausdorff dimension. With this aim, next we contribute one of the main results in this book, where we shall prove that fractal dimension V generalizes Hausdorff dimension in the context of Euclidean subsets endowed with their natural fractal structures. Theorem 4.6.1 Let  be the natural fractal structure on Rd and F be a subset of Rd . Then the fractal dimension V of F equals the Hausdorff dimension of F, namely: dimH (F) = dim5 (F).

Proof First, the inequality dimH (F) ≤ dim5 (F) is clear by Corollary 4.4.2(1). Accordingly, we shall be focused on the opposite inequality. Let ε > 0 be fixed but arbitrarily chosen and s ≥ 0 be such that H Hs (F) = 0. Since

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169

H Hs (F) = lim Hδs (F) = sup Hδs (F), δ→0

δ>0

then there exists δ0 > 0 such that Hδs (F) < γ for all δ < δ0 , where we can choose γ =

ε . s d 2 · 3d

In addition, let n 0 ∈ N be such that 2−n 0 < δ0 . Thus, H2s−n0 (F) < γ . Then there exists {Bi }i∈I such that F ⊆ ∪i∈I Bi with diam (Bi ) < 2−n 0 for all i ∈ I . Moreover, it holds that  diam (Bi )s < γ . (4.11) i∈I

On the other hand, for each i ∈ I , let n i ∈ N be such that 1 1 ≤ diam (Bi ) < n −1 . n i 2 2i Since diam (Bi ) < of Bi :

1 2n 0

, then n i > n 0 for all i ∈ I . Define the following covering Ai = {A ∈ ni : A ∩ Bi = ∅}

and let A = ∪i∈I Ai . Notice that St (Bi , ni ) = ∪{A ∈ ni : A ∩ Bi = ∅} = ∪{A : A ∈ Ai } for all i ∈ I . Further, since diam (Bi ) < 21−ni , then Card (Ai ) ≤ 3d

(4.12)

for all i ∈ I . Hence, the following three stand. (1) A is a covering of F. In fact, F ⊆ ∪i∈I Bi ⊆ ∪i∈I ∪ A∈A i A = ∪ A∈∪i∈I A i A = ∪ A∈A A. (2) For all A ∈ A , there exists n i ∈ N with n i > n 0 such that A ∈ ni (and A ∩ Bi = ∅).  s (3) A∈A diam (A) < ε. Indeed,  A∈A

diam (A)s =

 i∈I A∈A i s

≤ d 2 · 3d ·

diam (A)s =  i∈I

 i∈I A∈A i

diam (Bi )s < ε,

 √ s d 2n i

(4.13) (4.14)

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4 Hausdorff Dimension Type Models for Fractal Structures

where the second equality in Eq. (4.13) stands due to the size of the elements in level n i of the natural fractal structure on Rd . Further, it is worth noting that Eq. (4.12) has been applied to deal with the first inequality in Eq. (4.14). It has been applied also that 2−ni ≤ diam (Bi ) for all i ∈ I . Finally, Eq. (4.11) gives the last inequality in Eq. (4.14). Therefore, it has been proved that for all ε > 0, there exists n 0 ∈ N such that s (F) < ε for all n ≥ n 0 . Hn,5

Accordingly, H5s (F) = 0 for all s > dimH (F). Thus, dim5 (F) ≤ dimH (F). As a consequence of Theorem 4.6.1, we can state that the fractal dimension VI and the Hausdorff dimension of Euclidean subsets endowed with their natural fractal structures are equal. Corollary 4.6.2 Let  be the natural fractal structure on Rd and F be a subset of Rd . Then both the fractal dimensions V and VI of F equal the Hausdorff dimension of F, namely, dimH (F) = dim6 (F) = dim5 (F).

Proof Corollary 4.4.2(1) leads to dimH (F) ≤ dim6 (F) ≤ dim5 (F) and Theorem 4.6.1 gives the proof. Interestingly, fractal dimension IV also generalizes Hausdorff dimension in the context of compact Euclidean subsets (endowed with their natural fractal structures) as the following result highlights. Theorem 4.6.3 Let  be the natural fractal structure on Rd and F be a compact subset of Rd . Then all the fractal dimensions IV, V, and VI of F equal the Hausdorff dimension of F, namely, dimH (F) = dim6 (F) = dim5 (F) = dim4 (F).

Proof First of all, Corollary 4.6.2 guarantees that dimH (F) = dim6 (F) = dim5 (F). Thus, we shall be focused on the proof of dimH (F) = dim4 (F), where F is a compact subset of Rd . Recall that Corollary 4.4.2(1) allows to affirm that dimH (F) ≤ dim4 (F). To tackle with the opposite inequality, we can proceed similarly to the proof of Theorem 4.6.1. In fact, let ε > 0 be fixed but arbitrarily chosen and s ≥ 0

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171

be such that H Hs (F) = 0. Since H Hs (F) = limδ→0 Hδs (F), then there exists δ0 > 0 s such that Hδs (F) < γ for all δ < δ0 . Define γ = ε/(d 2 · 3d ). Moreover, let n 0 ∈ N s −n 0 < δ0 . Thus, H2−n0 (F) < γ . Accordingly, there exists {Bi }i∈I satbe such that 2 isfying the following three: • F ⊆ ∪i∈I Bi . •  diam (Bi ) < 2−n 0 for all i ∈ I . • i∈I diam (Bi )s < γ . It is worth pointing out that the collection {Bi }i∈I can be chosen as a family of open balls on Rd (c.f. Proposition 1.2.12). Further, there exists a finite subset J ⊆ I such that F ⊆ ∪i∈J Bi since F is compact. For all i ∈ J , let n i ∈ N such that 2−ni ≤ diam (Bi ) ≤ 21−ni . Since diam (Bi ) < 2−n 0 , then n i > n 0 for all i ∈ J . Let Ai = {A ∈ ni : A ∩ Bi = ∅} and A = ∪i∈J Ai . Observe that St (Bi , ni ) = ∪{A ∈ ni : A ∩ Bi = ∅} = ∪{A : A ∈ Ai } for all i ∈ J . Additionally, Card (Ai ) ≤ 3d since diam (Bi ) < 21−ni . Similarly to Theorem 4.6.1, the following three stand: (1) (2) (3) (4)

J ⊆ I with Card (J ) < ∞. A is a covering of F. For exists n i > n 0 such that A ∈ ni .  all A ∈ A , there s diam (A) < ε. A∈A

s Accordingly, for all ε > 0, there exists n 0 ∈ N such that Hn,4 (F) < ε for all n ≥ 4 s n 0 . This leads to H4 (F) = 0 for all s > dimH (F), i.e., dim (F) ≤ s for all s > dimH (F). Hence, dim4 (F) ≤ dimH (F).

In summary, fractal dimension V generalizes Hausdorff dimension for Euclidean subsets endowed with their natural fractal structures (c.f. Theorem 4.6.1). In addition, fractal dimension IV throws an upper bound to Hausdorff dimension in the same context (c.f. Corollary 4.4.4). The equality between Hausdorff dimension and fractal dimension IV has been reached for compact Euclidean subsets (c.f. Theorem 4.6.3). Going beyond, it becomes also possible to weaken the hypothesis regarding the compactness of F to prove a further connection between the fractal dimension IV of F and its Hausdorff dimension. This issue will be discussed in the forthcoming result. Theorem 4.6.4 Let  be the natural fractal structure on Rd and F be a bounded subset of Rd . Then the fractal dimension IV of F equals the Hausdorff dimension of the closure of F, namely, dim4 (F) = dimH (F).

Proof Notice that F, the closure of F, is a compact subset of Rd since F is bounded. Thus, we have dimH (F) = dim4 (F) by Theorem 4.6.3. Finally, Proposition 4.2.4(5) gives dim4 (F) = dim4 (F).

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Interestingly, all the fractal dimension models explored in this book can be theoretically connected among them in the context of Euclidean subsets endowed with their natural fractal structures. Corollary 4.6.5 Let  be the natural fractal structure on Rd and F be a subset of Rd . In addition, assume that there exists one of the following dimensions: dimB (F), dim1 (F), or dim2 (F). The following two hold: (1) dimH (F) = dim6 (F) = dim5 (F) ≤ dim3 (F) = dim2 (F) = dim1 (F) = dimB (F). (2) If F is compact, then dimH (F) = dim6 (F) = dim5 (F) = dim4 (F) ≤ dim3 (F) = dim2 (F) = dim1 (F) = dimB (F).

Proof (1) The first row stands due to Corollary 4.6.2. In addition, Theorem 4.4.1 leads to dim5 (F) ≤ dim3 (F), whereas dim1 (F) = dim2 (F) = dim3 (F) = dimB (F) by Theorem 3.7.7. Assume that there exists dim1 (F) (otherwise, if it is assumed that there exists either dim2 (F) or dimB (F), the following arguments remain valid). Then 1 dim1 (F) = dim (F) = dim1 (F) and such a common value equals dimB (F) (c.f. Theorem 2.3.4). Hence, Theorem 2.9.3 allows to affirm that dim1 (F) = dim2 (F) = dimB (F). (2) This follows from both Corollary 4.6.5(1) and Theorem 4.6.3. It is worth mentioning that Corollary 4.6.5 provides a theoretical connection between all the definitions of fractal dimension for a fractal structure explored throughout this work: fractal dimensions I and II, that were described in terms of box-counting type formulae (c.f. Chap. 2), fractal dimension III, which was a first attempt to discretize the classical Hausdorff dimension from the viewpoint of fractal structures (c.f. Chap. 3), and finally, fractal dimensions IV, V, and VI which properly generalize the classical Hausdorff dimension. More specifically, it was proved that

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173

fractal dimensions I, II, and III coincide with box dimension for Euclidean subsets endowed with their natural fractal structures (c.f. Theorem 3.7.7), whereas fractal dimensions IV, V, and VI equal Hausdorff dimension in the same context (c.f. Theorem 4.6.3). In particular, the fractal dimension IV model introduced in Definition 4.2.1 results especially interesting from a theoretical viewpoint, since it becomes a middle dimension between box and Hausdorff dimensions, as we shall highlight in the next remark. Counterexample 4.6.6 (1) There exist a fractal structure  on a metric space (X, ρ) and a subset F of X such that dim4 (F) < dimB (F). (2) There exist a fractal structure  on a metric space (X, ρ) and a subset F of X such that dimH (F) < dim4 (F).

Proof (1) Let  be the natural fractal structure (induced) on [0, 1], and define F = {0, 1, 1/2, 1/3, . . .}. First, we have dimB (F) = 1/2 by [29, Example 3.5]. Moreover, Theorem 4.6.3 implies that dim4 (F) = dimH (F) since F is compact. Finally, observe that dimH (F) = 0 since F is countable. (2) Let  be the natural fractal structure (induced) on [0, 1] and consider F = Q ∩ [0, 1]. Hence, Theorem 4.6.4 gives dim4 (F) = dimH (F) = 1, whereas dimH (F) = 0 since F is countable. To conclude this section, we would like to point out that fractal dimension IV can be also applied for computational purposes. In other words, it becomes possible to computationally approach the fractal dimension IV of any compact Euclidean subset, which, by Theorem 4.6.3, equals its Hausdorff dimension. Therefore, such a fractal dimension will allow to computationally deal with the calculation of the Hausdorff dimension of compact Euclidean subsets. For illustration purposes, next we provide a preliminary example which shows how to computationally approach the Hausdorff dimension of the middle third Cantor set. Example 4.6.7 Let  be the natural fractal structure on the closed unit interval, [0, 1]. In this example, we shall consider a number of levels of  to approach the fractal dimension IV of the middle third Cantor set C (and hence, its Hausdorff dimension, due to Theorem 4.6.3). First, denote

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4 Hausdorff Dimension Type Models for Fractal Structures

 1 , A0 = 0, 2

 A1 =

1 ,1 , 2

 Ai j =

2i + j 2i + j + 1 , : i, j = 0, 1. 4 4

Thus, the first levels of  are 1 = {A0 , A1 } and 2 = {A00 , A01 , A10 , A11 }. Additionally, a discretization of C consisting of the 2048 extremes of the intervals involved in step 10 of the standard construction of C has been considered for computational purposes. The following approach to deal with the calculation of the fractal dimension IV of C has been applied. Given s ≥ 0, set H4s (C ) = ∞ provided that s (C ) (which is a minimum in this case) is obtained by a covering involving H1,4 at least one element in 1 . Otherwise, set H4s (C ) = 0. For s = 0.69, we have found that the minimum is obtained throughout the covering {A0 , A1 }, whereas for s = 0.7, the minimum is reached by means of the covering {A00 , A010 , A101 , A11 }. Hence, an approach of dim4 (C ) using only three levels of  lies between 0.69 and 0.7. Using five levels of  instead, it holds that for s = 0.63, the minimum is obtained by the covering {A0 , A1 } and for s = 0.64, the minimum follows throughout the covering {A000 , A00111 , A010 , A101 , A11000 , A111 }. Accordingly, the estimation of dim4 (C ) (using only five levels of ) lies between 0.63 and 0.64. Recall that the theoretical value of the Hausdorff dimension of C is log 2 ∼ 0.631. dimH (C ) = log 3 Example 4.6.7 has been provided to illustrate how fractal dimension IV allows to deal with the effective calculation of the Hausdorff dimension of compact Euclidean subsets. In forthcoming Sect. 4.8, we shall further develop such a procedure to computationally calculate the Hausdorff dimension of such a kind of sets.

4.7 Fractal Dimensions for IFS-Attractors Interestingly, the calculation of the fractal dimension of IFS-attractors can be also tackled with from the viewpoint of fractal structures. In fact, it was proved in classical Moran’s Theorem 1.4.14 that both the Hausdorff and the box dimensions of strict self-similar sets can be calculated as the unique solution of an expression which only involves the similarity ratios associated with each similarity of the IFS. It is also worth pointing out that the OSC (c.f. Definition 2.14.1) was required to be satisfied by the similarities of the corresponding IFS. Recall that the OSC is a hypothesis required to the pre-fractals of an IFS-attractor to someway control their overlapping.

4.7 Fractal Dimensions for IFS-Attractors

175

In this book, several attempts to reach Moran’s type theorems have been carried out in the context of fractal structures. Among them, we would like to highlight Theorem 3.10.1, where the fractal dimension III of any IFS-attractor was found to satisfy a Moran’s type equation without requiring the OSC to be satisfied by the pre-fractals of K . In addition, other theoretical results (under the OSC) have been proved for fractal dimensions I (c.f. Corollary 2.6.1) and II (c.f. Theorem 2.14.3), respectively. Next, we provide a generalization of classical Moran’s Theorem in the context of fractal structures. In fact, we shall prove that under the OSC, all the fractal dimension models for a fractal structure (from III to VI) coincide with both the Hausdorff and the box dimensions of IFS-attractors endowed with their natural fractal structures as self-similar sets. Moreover, they can be uniquely determined throughout a Moran’s type equation (c.f. Eq. (4.19)). In addition, if all the similarity factors are equal, then additional connections with fractal dimensions I and II follow. Theorem 4.7.1 Let F = { f 1 , . . . , f k } be a Euclidean IFS under the OSC whose IFS-attractor is K . Moreover, let ci be the similarity ratio associated with each similarity f i ∈ F and  be the natural fractal structure on K as a self-similar set. Then (1) dimH (K ) = dim6 (K ) = dim5 (K ) = dim4 (K ) = dim3 (K ) = dimB (K ) = s. (2) In addition, if all the similarities f i ∈ F have a common similarity ratio c ∈ (0, 1), then dimH (K ) = dim6 (K ) = dim5 (K ) = dim4 (K ) = dim3 (K ) = dim2 (K ) = dimB (K ) ≤ γc · dim1 (K ), where γc = − log 2/ log c. In both cases, s satisfies the following expression: k 

cis = 1.

(4.15)

i=1

Further, for such a value of s, HHs (K ), H js (K ) ∈ (0, ∞) : j = 3, 4, 5, 6.

Proof (1) First of all, notice that Corollary 4.4.4 can be applied since the natural fractal structure on K as a self-similar set is finite. Hence, dimH (K ) ≤ dim6 (K ) ≤ dim5 (K ) ≤ dim4 (K ) ≤ dim3 (K ).

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On the other hand, Theorem 3.10.1 guarantees that dim3 (K ) = s, where s is the solution of Eq. (4.15). Finally, Corollary 3.10.3 leads to dimH (K ) = dimB (K ) = dim3 (K ). (2) Similarly, the following chain of inequalities stands due to Corollary 4.4.4: dimH (K ) ≤ dim6 (K ) ≤ dim5 (K ) ≤ dim4 (K ) ≤ dim3 (K ) ≤ dim2 (K ), provided that dim2 (K ) exists. Additionally, dimB (K ) = dim2 (K ) by Theorem 2.14.3, since all the similarity ratios are equal. Further, recall that classical Moran’s Theorem 1.4.14 implies that dimH (K ) = dimB (K ). Moreover, Corollary 2.6.1 leads to dimB (K ) ≤ γc · dim1 (K ), where γc = −

log 2 . log c

Finally, it is worth pointing out that the following chain of inequalities stands: HHs (K ) ≤ H6s (K ) ≤ H5s (K ) ≤ H4s (K ) ≤ H3s (K ), where HHs (K ) > 0 by Moran’s Theorem 1.4.14 and H3s (K ) < ∞ due to Theorem 3.10.1.

4.8 How to Calculate the Hausdorff Dimension in Empirical Applications Most of the practical applications of fractal dimension have been carried out on Euclidean spaces through the box dimension since it can be easily estimated. Nevertheless, Hausdorff dimension throws the most accurate results since its technical definition is quite general and is based on a measure. This constitutes the actual reason for which it presents the best theoretical properties. Nevertheless, it can be hard or even impossible to calculate or to estimate in empirical applications. This justifies why only a few partial attempts to calculate it have been successfully carried out. In this way, there are not many results in scientific literature allowing to explicitly calculate the Hausdorff dimension for some kinds of sets under certain constraints. They include Julia sets (c.f. [48, 67]) and strict self-similar sets. The case of self-similar sets becomes especially interesting since only a few algorithms have been contributed to calculate their Hausdorff dimension. To deal with, the OSC is usually required to be satisfied by the pieces of the corresponding IFS-attractor. Under that assumption to control their overlapping, a well-known result due to P.A.P. Moran (1946, c.f. [59]) throws the Hausdorff dimension of IFS-attractors as the unique solution of an equation involving only the similarity ratios associated with that IFS. Following

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177

the above, next step consists of trying to weaken that constraint to deal with the case of IFS with overlapping. Yet in that case, some theoretical results and algorithms to calculate the Hausdorff dimension have already been contributed (c.f. [22, 61]). However, as far as the authors know, there are no generic procedures in literature to calculate the Hausdorff dimension. Accordingly, the main goal in upcoming subsections is to develop an overall procedure allowing to properly calculate the Hausdorff dimension of any compact Euclidean subset. With this aim, we shall be focused on the topological concept of fractal structure which was first sketched in [14] and formally defined later in [3] to characterize non-Archimedean quasi-metrizable spaces. This powerful tool links several topics in topology such as non-Archimedean quasi-metrization, transitive quasi-uniformities, metrization, topological and fractal dimensions, self-similar sets, and even space-filling curves (c.f. [74]). It is also worth mentioning that fractal structures provide a perfect context where new models of fractal dimension can be proposed. In this way, recall that in Chaps. 2 and 3, we explored in detail new fractal dimension definitions for fractal structures. Thus, some of them present good analytical properties, whereas other can be calculated with easiness. Moreover, several applications of these models to non-Euclidean contexts have been successfully carried out (c.f. Sects. 2.15 and 3.12, resp.). On the other hand, we should mention here that any algorithmic scheme to tackle with the calculation of the effective fractal dimension should be able to detect fractal patterns through a whole range of scales (from a theoretical viewpoint) or involving a discrete range of scales (especially in empirical applications). The new algorithm that will be introduced in forthcoming subsections is based on both Definition 4.2.1 (k = 4) of fractal dimension and Theorem 4.6.4. In this way, that definition throws a discrete model of fractal dimension for any subset with respect to a fractal structure throughout finite coverings (which becomes quite appropriate to deal with computational applications), whereas such a theoretical result guarantees that fractal dimension IV coincides with Hausdorff dimension for compact Euclidean subsets. In Example 4.6.7, we contributed a handy example about how to calculate that dimension based on finite coverings via fractal structures. Following the above, the main goal in upcoming subsections is to extend the theoretical work carried out in this chapter to provide the first overall algorithm to calculate the Hausdorff dimension in empirical applications. With this aim, we shall combine techniques from Fractal Geometry with methods from Machine Learning Theory. Following the above, the organization of the upcoming subsections is as follows. In Sect. 4.8.1, we shall describe a new algorithm to calculate the Hausdorff dimension of any compact Euclidean subset and we shall also explain there how to computationally deal with each stage of this procedure. Later, in Sect. 4.8.5, we shall tune up our algorithm to properly calculate the Hausdorff dimension. More specifically, in Sect. 4.8.6, we explain how to generate a wide collection of examples to train a support vector machine (SVM herein, see Sect. 4.8.7) which is used in the main algorithm. Later, in both Sects. 4.8.8 and 4.8.9, we develop both a cross-validation process and a final test to find out how our approach behaves to approach the theoretical Hausdorff dimension. The chapter ends by carrying out several external proofs that confirm the goodness of this novel approach (c.f. Sect. 4.8.10).

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4.8.1 Calculating the Hausdorff Dimension Next, we contribute the first-known algorithm to computationally calculate the Hausdorff dimension of any compact Euclidean subspace. In fact, that novel approach allows the calculation of the fractal dimension of any compact Euclidean subset with respect to its natural fractal structure (c.f. Definition 2.2.1). Moreover, we shall explain In Sects. 4.8.2–4.8.4 how to computationally deal with each step of that procedure. First, recall that though theoretically any fractal structure  has a countable number of levels {n }n∈N (c.f. Definition 1.6.1), in practical applications we shall always reach a maximum level n max of  which depends on the number of data available. s (F) Accordingly, the following expression will approach the theoretical quantity Hn,4 (c.f. Definition 4.2.1) to deal with the effective calculation of the fractal dimension of F:  k   s s k Hn,m (F) = min diam (Ai ) : Ai ∈ ∪n≤l≤m l , F ⊆ ∪i=1 Ai , (4.16) i=1

for all n ≤ m ≤ n max . Equivalently, following the formulation of Definition 4.2.1, the s (F) just provided in Eq. (4.16) can be also described in the following quantity Hn,m terms:  k   s s k diam (Ai ) : {Ai }i=1 ∈ An,m (F) , (4.17) Hn,m (F) = min i=1

where the family An,m (F) is given by   k k : Ai ∈ ∪n≤l≤m l , F ⊆ ∪i=1 Ai , An,m (F) = {Ai }i=1

(4.18)

for all n ≤ m ≤ n max . Thus, fixed a quantity s > 0, we have to tackle with the next three stages (recall both Definition 4.2.1 and Eq. (4.16)) to calculate the fractal dimension of any compact Euclidean subset. Algorithm 4.8.1 s (1) Calculate Hn,m (F) for all n ≤ m ≤ n max . (2) Decide the value of H s (F) from step (1). (3) Calculate dim4 (F).

A shorter version of Algorithm 4.8.1, which actually takes into account all the s (F), is provided next. values of Hn,m

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s (F) (resp., all the values of H s Table 4.2 The collection of all the values of Hn,m n,n max (F)) gives rise to a lower triangular matrix (resp., a row) as the input data for a SVM to decide the value of H s (F) (c.f. Sect. 4.8.3) s (F) H1,1 s (F) H1,2

s (F) H2,2

s (F) H1,3

s (F) H2,3

s (F) H3,3

.. .

..

s H1,n (F) max

s H2,n (F) max

s H3,n (F) max

···

.. .

.. .

. Hnsmax ,n max (F)

Remark 4.8.2 Let us fix m = n max to provide an alternative approach to Algorithm 4.8.1. This procedure only makes use of the last row in the lower triangular matrix displayed in Table 4.2 for Hausdorff dimension calculation purposes.

In upcoming subsections, we shall explain in detail how to deal with each stage of Algorithm 4.8.1 from a computational viewpoint. In this way, first we shall deal with steps (1) and (3), and we shall tackle with step (2) afterwards since that stage is more awkward.

s (F) for n ≤ m ≤ n 4.8.2 Calculation of Hn,m max

Let F ⊆ Rd , s > 0, and n, m ∈ N : n ≤ m ≤ n max . Next, we shall deal with the s (F), the first step of Algorithm 4.8.1. First of all, we prove a calculation of Hn,m theoretical result containing two properties that are always satisfied in the effective s (F) (see both Eqs. (4.17) and (4.18)). calculation of Hn,m Proposition 4.8.3 Let  be a fractal structure on a metric space (X, ρ), F be a subset of X , and assume that δ(F, n ) → 0. The two following inequalities hold for all n ≤ m ≤ n max − 1: s s (i) Hn+1,m+1 (F) ≤ Hn+1,m (F). s s (ii) Hn,m+1 (F) ≤ Hn+1,m+1 (F).

Proof k k (i) Let {Ai }i=1 ∈ An+1,m (F). Thus, Ai ∈ ∪n+1≤l≤m l with F ⊆ ∪i=1 Ai for all i = 1, . . . , k. Hence, it is clear that Ai ∈ ∪n+1≤l≤m+1 l for all i = 1, . . . , k, k ∈ An+1,m+1 (F). Accordingly, it holds that An+1,m (F) ⊆ leading to {Ai }i=1

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An+1,m+1 (F) for all n ≤ m ≤ n max − 1. The result follows by applying both Eqs. (4.17) and (4.18). k k ∈ An+1,m+1 (F). Then Ai ∈ ∪n+1≤l≤m+1 l and F ⊆ ∪i=1 Ai for all (ii) Let {Ai }i=1 i = 1, . . . , k. Thus, Ai ∈ ∪n≤l≤m+1 l for all i = 1, . . . , k, which implies that k ∈ An,m+1 (F). Therefore, An+1,m+1 (F) ⊆ An,m+1 (F) for all n ≤ m ≤ {Ai }i=1 s s (F) ≤ Hn+1,m+1 (F). n max − 1, and hence, Hn,m+1 It is worth mentioning that the scheme provided in Table 4.2 can be used to s (F) for n ≤ m ≤ n max . In this way, observe that if a deal with the calculation of Hn,m column in that matrix is fixed, then each value is ≥ than the quantity below. Similarly, fixed a row, then it holds each value is ≤ than the value on its right, according to Proposition 4.8.3. s (F) for Next, let us provide several comments on the pre-calculation of Hn,m d n ≤ m ≤ n max and any compact subset F of R . First, notice that for each level n of the fractal structure  with n ≤ n max , we have to know what elements in that level intersect F. Equivalently, we have to fully determine each collection An (F) = {A ∈ n : A ∩ F = ∅} for all n ≤ n max . In fact, the knowledge regarding these families An (F) will allow s (F) : n ≤ m ≤ n max . us to determine the minimal covering needed to calculate Hn,m k Formally, by a minimal covering, we shall understand the finite covering {Ai }i=1 that k s minimizes the quantity i=1 diam (Ai ) , where Ai ∈ ∪n≤l≤m l . Additionally, another question arising is as follows. Given a level of the fractal structure  and a point x ∈ Rd , how can we computationally determine what elements in that level contain x? Let us assume that the binary coordinates of that point are known. To deal with, first, we shall explain how to represent the elements in level n of the natural fractal structure  on Rd . Consider the next bijection d

ϕ : {0, 1}n × · · · × {0, 1}n → n , given by   n−1 ϕ(a11 a21 · · · an1 , . . . , a1d a2d · · · and ) = 0.a11 a21 · · · an1 , 0.a11 a21 · · · an1 + 0.0 · · · 01 d

× · · ·×   n−1 0.a1d a2d · · · and , 0.a1d a2d · · · and + 0.0 · · · 01 , j

j

j

where the numbers 0.a1 a2 . . . an : j = 1, . . . , d are written in binary. Next example is provided for illustration purposes regarding how this map works.

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Example 4.8.4 Let x be a point in R3 whose coordinates in binary are (0.1010101110, 0.00010010110, 0.0110101). Following the notation above, we know that x ∈ ϕ(1, 0, 0) in first level of , which will be denoted by (1, 0, 0) to simplify. In addition, x belongs to the element (10, 00, 01) ∈ 2 , to the element (101, 000, 011) ∈ 3 , . . ., and so on. Conversely, notice that the element (10, 00, 01) ∈ 2 corresponds to [0.10, 0.11] × [0.00, 0.01] × [0.01, 0.10] in binary, and also to [1/2, 3/4] × [0, 1/4] × [1/4, 1/2] in decimal. Therefore, the bijection ϕ allows to represent elements of n throughout points in d

{0, 1}n × · · · × {0, 1}n . It is also worth noting that for every compact subset F of Rd , we can always apply an appropriate similarity f to F so that f (F) ⊆ [0, 1]d . Since Hausdorff dimension is invariant under the action of similarities (c.f. [29, Corollary 2.4 (b)]), then we can assume that F ⊆ [0, 1]d . Moreover, recall that diam (A) = √ d/2n for each A ∈ n . Thus, a general procedure to deal with the calculation of s (F), for all F ⊆ [0, 1]d and all n ≤ m ≤ n max can be stated in the following Hn,m terms. Part (1) of Algorithm 4.8.1. Let F ⊆ [0, 1]d , s > 0, and n ∈ N. (i) For level m, the minimal covering is Mm = {A : A ∈ Am (F)}. (ii) For level m − 1, Initialize Mm−1 = ∅. For each A ∈ Am−1 (F), : B ⊆ A}. Find {B ∈ Mm  If diam (A)s < {diam (B)s : B ∈ Mm , B ⊆ A}, then Add A to Mm−1 . Else Add {B ∈ Mm : B ⊆ A} to Mm−1 . (iii) Repeat until reach level n of . (iv) Return covering Mn = {A1 , A2 , . . . , Ak } and the quankboth the minimal s diam (Ai )s (which gives Hn,m (F)). tity i=1 Next, we explain how to computationally tackle with some steps involved in Algorithm 4.8.2. With this aim, we shall use the binary notation introduced previously to denote each point of F ⊆ Rd . 1. How to calculate Ak (F) for all n ≤ k ≤ n max . (i) For level n max , Initialize An max (F) = ∅.

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For each point x ∈ F, Write it in binary: x = (0. p11 · · · pn1max , . . . , 0. p1d · · · pndmax ). Then An max (F) = An max (F) ∪ {( p11 · · · pn1max , . . . , p1d · · · pndmax )}. (ii) Given the family Ak+1 (F), how to determine the family Ak (F) for all n − 1 ≤ k ≤ n max − 1. Initialize Ak (F) = ∅. d 1 , . . . , p1d . . . pk+1 ) ∈ Ak+1 (F), For each element ( p11 · · · pk+1 d 1 1 Ak (F) = Ak (F) ∪ {( p1 · · · pk , . . . , p1 · · · pkd )}. 2. Given A ∈ Ak (F), then we can write A = ( p11 · · · pk1 , . . . , p1d · · · pkd ). Next, we shall explain how to calculate {B ∈ Mk+1 : B ⊆ A} for all n ≤ k ≤ n max − 1 (recall that Mk+1 is the minimal covering at stage k + 1). First, let us denote Ck+1 = {B ∈ Mk+1 : B ⊆ A}. Hence, Initialize Ck+1 = ∅. For each B ∈ Mk+1 , Let B = (q11 · · · qn1 , . . . , q1d · · · qnd ) where n ≥ k + 1. If B ⊆ A, namely, if q ij = pij for all i = 1, . . . , d, and all j = 1, . . . , k, then Add B to Ck+1 . 3. To calculate the diameter of any element of a given level of the fractal structure √ , just note that if B = ( p11 . . . , pk1 , . . . , p1d . . . pkd ) ∈ k , then diam (B) = d/2k . That consideration becomes useful to return the minimal covering Mn in Algorithm 4.8.2.

4.8.3 Calculation of H s (F) Given F ⊆ Rd and s > 0, in this subsection, we sketch how to deal with the calculation of H s (F). Indeed, by Definition 4.2.1, it holds that H s (F) only can reach the values 0 and ∞, except in the case s = dim4 (F), where H s (F) can equal 0, ∞, or s (F) : n ≤ m ≤ n max , any number in (0, ∞). Accordingly, from the values of Hn,m s we have to decide the value of H (F). Depending on how we carry out that task we shall obtain distinct approaches to the fractal dimension of F, and hence, to its Hausdorff dimension. It is worth mentioning that this becomes the most awkward step in the calculation of dim4 (F). s (F) for different values Both Figs. 4.2 and 4.3 illustrate the behavior of Hn,m of s. In fact, it seems that there is some kind of pattern inside the graphics. Thus, to determine if H s (F) is equal to ∞ or equal to 0, we have selected a support vector machine (SVM, c.f. [21]) with both Gaussian and linear kernels. The input s (F) for data for the SVM lie in the lower triangular matrix whose entries are Hn,m n ≤ m ≤ n max (c.f. Table 4.2).

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Fig. 4.2 Graphical s (F) representation of Hn,m for m = 2, . . . , 10 and s = 0.4 of a subset F ⊂ R such that dimH (F) = 0.5. In this case, Algorithm 4.8.1 should return H s (F) = ∞

Fig. 4.3 The figure above displays a graphical s (F) representation of Hn,m with m = 2, . . . , 10 and s = 0.6 of a subset F ⊂ R such that dimH (F) = 0.5. In this case, Algorithm 4.8.1 should return H s (F) = 0

4.8.4 Calculation of dim4 (F) Once we have determined the value of H s (F) for a given subset F ⊆ Rd , final step is to approach its fractal dimension throughout a bisection type algorithm. Thus, given ε > 0, in this subsection, we explain how to calculate dim4 (F) within an error ≤ ε. In this way, recall that H s (F) = ∞ for all 0 ≤ s < dim4 (F) and H s (F) = 0 for all s > dim4 (F). That procedure can be described as follows. Part (3) of Algorithm 4.8.1. Let F ⊆ Rd , s > 0, n ∈ N, and ε > 0. (i) Define s0 = 0, s1 = d, and s as the middle point of the interval (s0 , s1 ), 1 . namely, s = s0 +s 2 (ii) Hence, one of the following two cases occur. (1) If H s (F) = ∞, then take s0 = s. (2) If H s (F) = 0, then consider s1 = s.

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1 Anyway, s = s0 +s . 2 (iii) Repeat step (ii) until s1 − s0 ≤ ε.

4.8.5 Empirical Experimentation In upcoming subsections, we shall calibrate Algorithm 4.8.1 to accurately calculate the Hausdorff dimension of compact Euclidean subspaces. Accordingly, their organization will be as follows. In Sect. 4.8.6, we shall explain in detail how to computationally construct a wide collection of sets with a prearranged Hausdorff dimension. With this aim, we shall apply the classical Moran’s Theorem. Afterwards, in Sect. 4.8.7, we train a SVM to decide if H s (F) is equal to ∞ or equal to 0. A cross-validation process of the SVM with both Gaussian and linear kernels will be carried out In Sect. 4.8.8. This will allow to find out the optimal choice of parameters in each case. We shall also verify that there is no overfitting (see Sect. 4.8.9). Finally, the robustness of Algorithm 4.8.1 is tested in Sect. 4.8.10. To tackle with, first we will train our approach as in Sect. 4.8.7 and then we will apply it to calculate the Hausdorff dimension of two new classes of compact real subsets: nonoverlapping self-similar sets from a number of similarities >2 and overlapping self-similar sets.

4.8.6 Generating Collections of Sets with a Fixed Hausdorff Dimension First of all, we shall explain how to properly generate collections of compact real subsets to train and test Algorithm 4.8.1. To deal with, we shall be focused on nonoverlapping strict self-similar sets. In fact, the effective Hausdorff dimension of this kind of fractals can be theoretically calculated due to a well-known result contributed by Moran (c.f. [59]). We should mention here that our purpose consists of training Algorithm 4.8.1 by using the minimum possible information in a way that such a novel procedure can properly return the Hausdorff dimension of any other kind of compact Euclidean subsets. Next, we recall the standard construction of self-similar sets as provided by Hutchinson in [45]. Let (X, d) be a metric space. By the hyperspace of X , K0 (X ), we shall understand the collection of all non-empty compact subsets of X . The Hausdorff metric dH is defined as dH (A, B) < ε, if and only if, B ⊆ Bd (A, ε) and A ⊆ Bd (B, ε), where Bd (A, ε) = {x ∈ X : d(x, a) < ε for some a ∈ A} denotes the ball of radius ε > 0 and centered in A with respect to the metric d, as usual. Thus, the Zenor-Morita’s Theorem (c.f. [60, Theorem 1.5]) guarantees that (K0 (X ), dH ) is a complete metric space. Let { f i : i ∈ I } be a finite family of contractions from X into itself. The Hutchinson’s operator, given by F(A) = ∪i∈I f i (A), is a contraction in

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the hyperspace K0 (X ). Hence, there exists a unique non-empty compact subset K of X such that K = ∪i∈I f i (K ) due to the Banach’s fixed point theorem. The pair (X, { f i : i ∈ I }) is called an iterated function system (IFS) and K is called as its attractor. Moreover, if each mapping f i is not only a contraction but a similarity, then K is named a strict self-similar set. It is also worth noting that if E is any nonempty compact subset of X , then limn→∞ F n (E) = K with respect to the Hausdorff metric. Next, we recall the classical Moran’s Theorem (c.f. Theorem 4.8.5) which will allow to generate strict IFS-attractors with a prearranged Hausdorff dimension. Theorem 4.8.5 (Moran 1946) Let F = { f 1 , . . . , f m } be a Euclidean IFS under the OSC whose IFS-attractor is K and ci be the similarity ratio associated with each similarity f i ∈ F . Then dimH (K ) = dimB (K ) = s, where s is the unique solution of the following expression: m 

cis = 1.

(4.19)

i=1

Recall that the OSC is a hypothesis required to the similarities f i of an IFS (Rd , { f i : i ∈ I }) to control the overlapping of the pre-fractals f i (K ) of K . Technically, that condition stands, if and only if, there exists a non-empty bounded open subset V ⊂ Rd such that ∪i∈I f i (V ) ⊂ V with this union remaining disjoint (c.f. [29, Sect. 9.2]). Let d ∈ (0, 1) and m ≥ 2. The primary goal is to generate a strict self-similar set K under the three following conditions: (1) 0, 1 ∈ K and K ⊆ [0, 1]. (2) K is the attractor of an IFS (R, { f 1 , . . . , f m }) under hypothesis of Moran’s Theorem.. (3) dimH (K ) = d. We should mention here that our approach will be focused on IFSs on the real line to easily generate collections of training sets in upcoming subsections. Next, we explain in detail how to mathematically deal with the generation of that class of strict self-similar sets under conditions (1)–(3) above. Notice that the desired similarities will be of the form f i (x) = ai + ci · x for all i = 1, . . . , m. This way, ci ∈ (0, 1) will be the similarity factor associated with each f i ∈ F .

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(i) The first stage to define each similarity f i is to generate its similarity factor 1) : i ∈ I and define ci . To tackle with, let us randomly choose ci ∈ (0, ci = c · ci , where c ∈ (0, 1) has been chosen so that i∈I cid = 1. In other words, 1 . c=     d  d1 i∈I ci (ii) Next step is to define

ai . Take d1 = 0 and let us randomly choose m general, we shall randomly choose d j+1 ∈ ∈ 0, 1 − c d i=1 i . In  2 j m 0, 1 − i=1 ci − i=1 di , except for dm . In that case, we define dm = m m−1 1 − i=1 ci − i=1 di . Finally, we set a1 = 0, a2 = c1 + d2 , and in genj  j+1 eral, a j+1 = i=1 ci + i=1 di : j = 1, . . . , m − 1. The previous methodology allows to fully determine all the similarities f i ∈ F , so let K be the attractor of the IFS (R, { f 1 , . . . , f m }). In addition, we affirm that the following two hold: • f i (1) ≤ f i+1 (0) for all i = 1, . . . , m − 1. Indeed, it is clear that f i (0) = ai and f i (1) = ai + ci . Thus, since f i (1) = ai + ci , f i+1 (0) = ai+1 = ai + ci + di+1 = f i (1) + di+1 , and di+1 ≥ 0, then f i (1) ≤ f i+1 (0). f m (1) = am + • f m (1) = 1. In fact, m = am−1 m we have m−1 cm−1 +m cm−1 + dm + cm = 1 ci − i=1 di , am = i=1 ci + i=1 di , and am = f m−1 since dm = 1 − i=1 (1) + dm .

Next, we shall verify that the previous m−similarity IFS-attractor K is under conditions (1)–(3). (1) Observe that 0 = f 1 (0) < f 1 (1) ≤ · · · ≤ f i (0) < f i (1) ≤ f i+1 (0) < f i+1 (1) ≤ · · · ≤ f m (0) < f m (1) = 1. Hence, f i ([0, 1]) = [ai , ai + ci ] ⊆ m f i ([0, 1]) ⊆ [0, 1] [0, 1] for all i = 1, . . . , m leading to F([0, 1]) = ∪i=1 (here, F refers to the Hutchinson’s operator). Therefore, we can recursively calculate F n ([0, 1]) ⊆ [0, 1] for all n ∈ N. Accordingly, we have K ⊆ [0, 1] due to the Banach’s fixed point theorem (in fact, recall that this result throws that K = limn→∞ F n ([0, 1])). Further, we can affirm that 0, 1 ∈ K since 0 is a fixed point of f 1 and 1 is a fixed point of f m . (2) The similarities f i ∈ F are under the OSC. To deal with, let G = (0, 1). The arguments above allow to check out that f i (G) ⊆ G for all i = 1, . . . , m with the f i (G) being disjoint. m d ci = (3) Accordingly, the hypothesis of Moran’s Theorem stand and since i=1 1, then we have dimH (K ) = d.

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The Banach’s contraction mapping Theorem allows to affirm that the attractor K of the IFS { f i : i ∈ I } is the unique non-empty compact real subset that remains invariant under the action of the Hutchinson’s operator F, namely, K = F(K ) = ∪i∈I f i (K ). Hence, if E is any non-empty compact real subset, then F k (E) provides better and better approximations to the attractor K as k increases. This is the reason for which these iterative approaches F k (E) are sometimes called pre-fractals for K (c.f. [29, Chap. 9]). Our next goal is to theoretically prove that the asymptotic s s (K ) may be reached throughout Hn,m (Kl ). Roughly speaking, behavior of Hn,m this is somehow equivalent to consider an appropriate finite approach to the attractor K by means of a pre-fractal Kl for K for some l ∈ N. In other words, once the IFS (R, { f i : i ∈ I }) has been generated, we shall explore how many iterations l ∈ N we have to carry out so that Kl = F l ({0, 1}) properly approximates K provided that n max is prearranged. This fact will be stated in the next two results. Theorem 4.8.6 Let F = { f 1 , . . . , f m } be a real IFS whose associated attractor is K , ci be the similarity ratio associated with each similarity f i on R, and  be the natural. Moreover, let K0 = {0, 1} and Kl be the pre-fractal for K given by Kl = F(Kl−1 ) = F l (K0 ) : l ∈ N. The following three hold: (i) K0 ⊆ K1 ⊆ · · · ⊆ Kl ⊆ K for all l ∈ N. (ii) K = ∪n∈N Kn .   (iii) Given n max , let l =

1

−1

log2 maxi∈I {ci } n max

, where · denotes the ceiling func-

tion. Then Kl ∩ A = ∅ ⇐⇒ K ∩ A = ∅ for all A ∈ m with m ≤ n max .

Proof m (i) It is clear that K0 ⊆ ∪i=1 f i (K0 ) = F(K0 ) = K1 , since f 1 (0) = 0 and f m (1) = 1. Thus, let us assume that Kl−2 ⊆ Kl−1 and prove that Kl−1 ⊆ Kl . In fact, Kl−1 = F(Kl−2 ) ⊆ F(Kl−1 ) = Kl by induction hypothesis. On the other hand, recall that K0 ⊆ K . Hence, K1 = F(K0 ) ⊆ F(K ) = K , since the attractor K remains invariant under the action of the Hutchinson’s operator F. Accordingly, if we suppose that Kl−1 ⊆ K , then we have Kl = F(Kl−1 ) ⊆ F(K ) = K for all l ∈ N, which gives the result. (ii) First, since Kn+1 = F(Kn ) for all n ∈ N, then limn→∞ dH (Kn , K ) = 0. Let ε > 0 be fixed but arbitrarily chosen. Then there exists n 0 ∈ N such that dH (Kn , K ) ≤ ε for all n ≥ n 0 . Hence, the following two hold:

(a) K ⊆ B(Kn , ε) leading to K ⊆ B(∪n∈N Kn , ε).

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4 Hausdorff Dimension Type Models for Fractal Structures

(b) Kn ⊆ B(K , ε) for all n ≥ n 0 . Therefore, ∪n≥n 0 Kn ⊆ B(K , ε). Moreover, since Kn ⊆ Kn+1 due to Theorem 4.8.6 (i), then we have ∪n≥n 0 Kn = ∪n∈N Kn . Accordingly, ∪n∈N Kn ⊆ B(K , ε). Finally, observe that statements (iia) and (iib) lead to dH (K , ∪n∈N Kn ) < ε for all ε > 0. This implies that dH (K , ∪n∈N Kn ) = 0 and hence, K = ∪n∈N Kn , as desired.   1 . (iii) Given n max , let l = −1 log2 maxi∈I {ci } n max

(⇒) Assume that Kl ∩ A = ∅. Then Theorem 4.8.6(i) yields Kl ⊆ K . Hence, it becomes clear that K ∩ A = ∅. (⇐) On the other hand, suppose that K ∩ A = ∅, where A ∈ m and m ≤ k n max . Due to [29, Theorem 9.1], we can write K = ∩∞ k=0 F ([0, 1]), since [0, 1] is a compact real subset such that f i ([0, 1]) ⊂ [0, 1] for all i ∈ I . In addition, 1 1 since l ≥ by hypothesis. it is clear that (maxi∈I {ci })l < 2nmax −1 log2 maxi∈I {ci } n max

Let x ∈ K ∩ A. Thus, x ∈ F l ([0, 1]) ∩ A and in particular, x ∈ F l ([0, 1]). Accordingly, there exists (i 1 , . . . , il ) ∈ I l such that x ∈ f i1 ,...,il ([0, 1]) = f i1 ◦ · · · ◦ f il ([0, 1]) = [a, b], where a = f i1 ,...,il (0) and b = f i1 ,...,il (1). In addition, K j = { f i1 ,...,i j (0), f i1 ,...,i j (1) : (i 1 , . . . , i j ) ∈ I j } for all j ≤ n max , so a, b ∈ Kl . 1 Therefore, it is satisfied that x ∈ [a, b] ∩ A, where b − a < 2nmax , since a = f i1 ,...,il (0), b = f i1 ,...,il (1), and the contraction ratio associated with f i1 ,...,il 1 is ci1 · · · cil ≤ (maxi∈I {ci })l < 2nmax . Hence, we have that a ∈ A or b ∈ A 1 (observe that A ∈ m so A is an interval of length 21m ≥ 2nmax , x ∈ A, x ∈ [a, b], 1 and b − a < 2nmax ). Anyway, it holds that A ∩ Kl = ∅, as desired. The proof of the following result becomes straightforward from Theorem 4.8.6(iii). Corollary 4.8.7 Under the hypotheses of Theorem 4.8.6, it holds that s s (K ) = Hn,m (Kl ) Hn,m

 for each n ≤ m ≤ n max , where l =

1

−1

log2 maxi∈I {ci } n max

 .

4.8.7 Training the Algorithm Recall that step (2) of Algorithm 4.8.1 consists of a classification problem since H s (F) may reach values in {0, ∞}. Next, we computationally deal with this stage of the approach throughout a SVM. It is worth mentioning that for computational tasks involving a SVM we have used the Scikit-learn package for Python (c.f. [65]).

4.8 How to Calculate the Hausdorff Dimension in Empirical Applications

189

Thus, the SVM will work with either a Gaussian kernel (determined by a pair of parameters (γ , c)) or a linear kernel given by a single parameter c. The aim of the s (F) : n ≤ m ≤ SVM is to decide the value of H s (F) with regard to the inputs Hn,m n max for a subset F ⊆ [0, 1]. In this way, we shall train the SVM by nonoverlapping 2−similarity self-similar sets. Recall that we tackled with their computational construction in Sect. 4.8.6. To deal with, for each pair (K , d), where K is a 2−similarity strict selfsimilar set under conditions (1)–(3) and d = dimH (K ) is randomly chosen in (0, 1), we shall randomly choose 6 values of s in (0, d) and another 6 values of s in (d, 1). Furthermore, to improve the accuracy of the approach, let us also consider some values of s being close to each theoretical Hausdorff dimension d. Accordingly, we shall select values of s of the form s = d − ε and s = d + ε, where ε ∈ {0.005, 0.01, 0.015, 0.01, 0.02, 0.03, 0.05, 0.1, 0.15}. This makes a total amount of 30 values of s for each IFS-attractor K . It is worth noting that the procedure should return H s (K ) = ∞ for all s < d and H s (K ) = 0 for all s > d.

4.8.8 Cross-Validation of the SVM Let us explain how to properly choose the parameters of the SVM. This will be carried out by a cross-validation procedure. First of all, we have generated 200 pairs (K , d), where K is a 2−similarity strict self-similar set under conditions (1)–(3) and d = dimH (K ) has been randomly chosen in the interval (0, 1), according to details provided in Sect. 4.8.6. It is worth mentioning that 30 values of s has been considered for each IFS-attractor K which makes a total amount of 6000 training examples. Then we have applied a SVM to the training set with both Gaussian and linear kernels to decide the value of H s (K ) as advanced previously in Sect. 4.8.7. For experimental purposes, the range of parameters has been selected as follows: γ , c ∈ {0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 1, 3, 10, 30, 100, 300, 1000}. Accordingly, the main goal of the cross-validation we shall develop next is to find out the optimal choice of parameters for each kind of kernel. In this way, we shall understand that combination of parameters is optimal provided that it throws the lowest mean error 1 after comparing the theoretical Hausdorff dimension versus the empirical fractal dimension IV (returned by Algorithm 4.8.1) of each IFS-attractor in the training set. We should mention here that the cross-validation we shall carry out is slightly distinct from a standard cross-validation process since we do not try to optimize the prediction of H s (K ) but the accuracy of the approaches to dimH (K ).

1 By

mean error we shall understand hereafter the mean absolute deviation between a theoretical value and its corresponding approach.

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4 Hausdorff Dimension Type Models for Fractal Structures

Table 4.3 The table above contains the mean errors (for n max = 7) obtained by a SVM with both Gaussian and linear kernels to approach dimH (K ) (c.f. step 2 of Algorithm 4.8.1). These values have been calculated from a training set consisting of 6000 examples as described in Sect. 4.8.7. The data in bold correspond to the lowest mean errors for each kind of kernel. They lead to the optimal choice of parameters in each case Mean errors for Gaussian kernel SVM (n max = 7) γ \c

0.001 0.003 0.01

0.03

0.1

0.3

1

3

10

30

100

300

1000

0.001 0.235 0.211 0.110 0.051 0.033 0.031 0.027 0.022 0.017 0.014 0.012 0.011 0.010 0.003 0.142 0.102 0.049 0.033 0.031 0.028 0.022 0.017 0.015 0.012 0.011 0.010 0.010 0.01

0.067 0.044 0.033 0.031 0.028 0.022 0.018 0.015 0.012 0.010 0.010 0.010 0.010

0.03

0.036 0.033 0.031 0.028 0.023 0.019 0.015 0.012 0.010 0.010 0.009 0.010 0.010

0.1

0.033 0.032 0.029 0.025 0.020 0.015 0.011 0.010 0.010 0.009 0.010 0.010 0.011

0.3

0.033 0.031 0.026 0.021 0.015 0.011 0.009 0.009 0.009 0.010 0.010 0.011 0.011

1

0.031 0.031 0.024 0.018 0.013 0.011 0.010 0.009 0.010 0.010 0.011 0.011 0.011

3

0.286 0.286 0.116 0.037 0.014 0.013 0.011 0.013 0.013 0.012 0.012 0.013 0.015

10

0.385 0.385 0.385 0.300 0.146 0.063 0.024 0.024 0.024 0.023 0.024 0.024 0.025

30

0.398 0.398 0.398 0.390 0.348 0.221 0.134 0.128 0.128 0.128 0.128 0.128 0.128

100

0.401 0.401 0.401 0.401 0.392 0.359 0.262 0.250 0.250 0.250 0.250 0.250 0.250

300

0.403 0.403 0.403 0.403 0.403 0.389 0.338 0.334 0.334 0.335 0.335 0.335 0.335

1000 0.404 0.404 0.404 0.404 0.404 0.400 0.366 0.365 0.365 0.365 0.365 0.365 0.365 mean errors for linear kernel SVM (n max = 7) c

0.030 0.025 0.020 0.015 0.013 0.012 0.011 0.011 0.011 0.012 0.012 0.012 0.012

Table 4.3 contains the mean errors provided by Algorithm 4.8.1 for both kinds of kernels in the case of n max = 7. In this way, the cross-validation displays that the optimal choice of parameters for the Gaussian kernel SVM is (γ , c) = (0.3, 10) leading to a lowest mean error equal to 0.009. In addition, the optimal parameter for the linear kernel SVM is c = 3 which gives a lowest mean error equal to 0.011. It is worth noting that the returned mean errors after calculating dimH (K ) through Algorithm 4.8.1 are quite small for both kernels. Moreover, Fig. 4.4 highlights that the mean errors obtained are quite stable. In fact, that plot, which contains the surface of mean errors from the data contained in Table 4.3, points out that it becomes possible to find out a wide area close to the lowest mean error where the mean error remains small ( 3 and overlapping self-similar sets. It is also worth mentioning that we shall train the SVM throughout the minimum possible information, as in Sect. 4.8.7. The first external proof we shall carry out is to test Algorithm 4.8.1 by nonoverlapping strict self-similar sets generated by k > 3 similarities. Indeed, for experimental purposes, we shall selected k ∈ S = {3, 4, 5, 10}. It is worth mentioning that an overall method to construct a collection of such sets was provided in Sect. 4.8.6. Thus, for each k ∈ S , we shall generate 200 pairs (K , d), where K is a nonoverlapping k−similarity strict self-similar set such that d = dimH (K ), as in Sect. 4.8.8. Observe that we have developed the cross-validation of the SVM and its training via nonoverlapping 2−similarity strict self-similar sets, though now we shall test

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4 Hausdorff Dimension Type Models for Fractal Structures

Algorithm 4.8.1 by nonoverlapping k−similarity strict self-similar sets, where k ∈ S. It is worth noting that our simulations were performed by the whole matrix version of Algorithm 4.8.1 through a Gaussian kernel SVM with optimal parameters (γ , c) = (0.3, 10) and n max = 7, which is the most accurate version of our procedure for dimH (K ) calculation purposes. The mean errors returned were as follows. For k = 3, the mean error was found to be 0.015; for k = 4, the mean error was equal to 0.012; for 5 similarities, the mean error was equal to 0.011, and finally, for k = 10, the mean error was found to be equal to 0.008. Accordingly, Algorithm 4.8.1 results quite accurate to tackle with the calculation of the Hausdorff dimension of nonoverlapping strict self-similar sets, if we recall that the SVM was trained only by self-similar sets generated from 2−similarities. On the other hand, let us show how accurate Algorithm 4.8.1 is in the case of overlapping self-similar sets. It is noteworthy that they are IFS-attractors generated by similarities that do not lie under the OSC, so they provide a class of compact real subsets quite different in nature from the sets of the training collection. To test our approach for self-similar sets with overlaps, we shall apply some interesting results from the scientific literature. They allow to construct overlapping self-similar sets with a prearranged dimension.  Hausdorff √  3− 5 First, let ρ ∈ 0, 2 . Moreover, let K be the attractor associated with the ρ, and IFS whose similarities are given as follows: f 1 (x) = ρ · x, f 2 (x) = ρ · x + √ log(3+ 5)−log 2 f 3 (x) = ρ · x + 1. By [61, Example 5.4], we know that dimH (K ) = . − log ρ Thus, we have generated 200 overlapping self-similar sets in this way, randomly √ choosing ρ ∈ (0, 3−2 5 ). In this case, the mean error was equal to 0.031. Additionally, we shall also test Algorithm 4.8.1 by means of another family of selfsimilar sets with overlaps. To deal with, let p, q ∈ (0, 1) with 2 p + q − pq < 1 and K be the attractor of the IFS defined by the similarities f 1 (x) = p · x + 1, f 2 (x) = q · x, and f 3 (x) = p · x + q1 . Hence, [22, Example 9] guarantees that dimH (K ) = s, where s is the unique solution of the equation 2 p s + q s − p s q s = 1. To check the performance of Algorithm 4.8.1 through this kind of IFS-attractors, we have generated 200 of these self-similar sets as follows. First, we randomly choose p, s ∈ (0, 1) and then we find out the value of q from the equation 2 p s + q s − p s q s = 1, checking that both constraints 2 p + q − pq < 1 and 0 < q < 1 are satisfied. In this way, for this class of overlapping self-similar sets, the mean error was found to be 0.019. These two external proofs throw some evidence that our approach becomes quite robust to deal with the approximation of the Hausdorff dimension of compact real subsets, even if they are IFS-attractors with overlapping. It is worth mentioning that Algorithm 4.8.1 has been trained only by 2−similarity strict self-similar sets under the OSC. Finally, let us carry out a further external proof. With the same training set, namely, by a collection of 2−similarity strict self-similar sets under the OSC on R, next we shall verify how Algorithm 4.8.1 performs in the case of self-similar sets on the Euclidean plane R2 . Of course, in this case, it is not expected that our approach performs as accurate as in the previous cases, since this time, the SVM will not be

4.8 How to Calculate the Hausdorff Dimension in Empirical Applications

195

trained through a family of sets on the plane. With this aim, let us generate 200 selfsimilar sets under the OSC which are slight modifications of the Sierpi´nski triangle, all of them with a theoretical Hausdorff dimension equal to log(3)/ log(2)  1.585 (just applying Theorem 4.8.5). In this way, the mean error was found to be equal to 0.19. This result allows to conclude that the previous training set is not wide enough to properly apply our procedure in this novel context. Moreover, it holds that the easiest version of Algorithm 4.8.1, namely, the one that uses both a linear kernel SVM and the last row (c.f. Remark 4.8.2) yields a mean error equal to 0.11. In addition, if instead of the usual training set, we use a broader training set including self-similar sets constructed from 3, 4, and 5 similarities (under the OSC) and overlapping selfsimilar sets such as those described previously in Sect. 4.8.10 (all of them lying in the real line), the variant of Algorithm 4.8.1 using both a linear kernel SVM and the last row yields a mean error equal to 0.08. These results suggest that for examples that are quite different in nature from the training set, the version of Algorithm 4.8.1 with both a linear kernel SVM and the last row (which is the easiest of the four versions of our approach) becomes the most stable. In short, the wider the training set, the better the results we can reach.

4.9 Notes and References It is worth pointing out that the Hausdorff dimension type models described in this chapter (namely, fractal dimensions IV, V, and VI) were first explored by M. Fernández-Martínez and M.A. Sánchez-Granero in [37]. The results from measure and probability theories we applied to prove that the set function H6s is a measure can be found out in Sect. 5.4 of Edgar’s book. More specifically, both Methods I and II of construction of outer measures appear in [25, Theorem 5.2.2. and Sect. 5.4]. In addition, [25, Theorem 5.4.2] states that the restriction of a metric outer measure to the class of Borel sets is a measure.

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Index

Symbols δ−cube, 18 δ−parallel body, 2, 24 σ −algebra, 6 s−dimensional Hausdorff measure, 8 s−moment, 120, 121, 123, 124, 137, 138, 140–142 s−set, 12, 14, 34 (δ−)cover, 7–11, 16–19, 21, 22, 36

A Affine transformation, 5, 102 Affinity, 5, 13 Area, 9, 10, 190, 191 Artificial intelligence, 80 A. S. Besicovitch (1891-1970), 7 Attractor, 22, 23, 31–34, 38, 39, 50, 57, 64– 66, 68, 71–73, 83, 100–104, 146, 149, 161, 174–176, 185–187, 189, 194 Average, 81

B Ball, 2, 3, 9, 16–19, 21, 33–35, 73, 171, 184 Banach’s contraction mapping principle, 30, 31 Banach’s contraction mapping theorem, 30, 187 Banach’s fixed point theorem, 30, 31, 185, 186 Bijection, 2, 4, 6, 180, 181 Bi-Lipschitz function, 5, 13, 15, 66 Borel set, 4, 6–9, 12, 168 Boundary, 3 Box(-counting) dimension, 1, 7, 16–22, 33, 36, 149, 156, 160, 165, 168, 172–176

Brownian motion, 106, 107, 111, 120 fractional Brownian motion, 120

C Cardinal number, 9 C. Carathéodory (1873-1950), 7 Closed ball, 2, 18, 21 Closed set, 2–4, 7, 13, 14, 16, 18, 21, 25, 27, 31, 51, 106, 113, 116, 118, 120, 152, 153, 158 Codomain, 4 Compact set, 3, 17, 24, 26–28, 31–33, 101, 170–174, 177, 178, 180, 181, 184, 185, 187, 188, 193, 194 Complement of a set, 2, 3, 15 Composition of functions, 4, 5, 34 Confidence interval, 132, 135–137, 143–146 Confidence level, 132, 135–137, 143–146 Congruence, 4 Connected component, 3 Connected set, 3 Continuous function, 6, 10, 23 Contraction, 23, 24, 28–31, 58, 66, 68, 73, 101–103, 184, 185, 187, 188 Contraction mapping theorem, 30 Contraction ratio, 23, 24, 29–31 Convergence, 3 Covering, 3, 7, 16–19, 21, 22, 34, 36, 38, 39, 51, 59, 75, 88, 91, 104, 106, 114, 150– 154, 157–160, 163–165, 167, 169, 171, 174, 177, 180–182 Cube, 2, 18, 19 Cumulative range, 121, 137, 141 Curve, 9, 10

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202 D Dense set, 3, 13–15 Derivative, 6 Diameter, 2, 8, 9, 17–19, 21, 22, 32, 34, 36, 57, 59, 64, 65, 67–69, 74, 85, 86, 90, 98, 102, 115, 119, 120, 124, 126, 150, 160, 165, 167, 182 of a level of a fractal structure, 55, 57, 60, 167 of a subset in a level of a fractal, 102 of a subset in a level of a fractal structure, 55–59, 68, 86, 160 Difference of sets, 2 Differentiability continuous, 6 Dimension, 1, 5, 7, 8, 11–22, 33, 36, 49–53, 55–69, 71–83, 85–92, 94–100, 102– 106, 108, 109, 111, 113, 115–120, 122–124, 126–128, 133, 138, 146, 147, 149–165, 168, 170–179, 181– 185, 189, 192–195 box(-counting), 1, 7, 16–22, 33, 36, 49– 52, 55–58, 62–67, 71–74, 76, 79, 85, 89, 90, 97, 98, 100, 103, 105–108, 113, 115, 117, 118, 149, 156, 160, 165, 168, 172–176 calculation of, 17, 18, 33, 49, 52, 57, 60, 64, 72, 76, 82, 89, 91, 92, 94, 97, 98, 100, 106, 108, 113, 120, 127, 138, 151, 165, 168, 173, 174, 177–179, 182, 191, 193, 194 capacity, 17 entropy, 17 estimation of, 49, 52, 58, 120, 165, 174 for a fractal structure, 13, 51, 105, 165, 172 fractal dimension I, 49, 50, 52, 53, 55–64, 69, 74, 75, 77–83, 85, 94–99, 104, 146, 149, 157, 160, 165, 168, 172, 173, 175 fractal dimension II, 50, 59–69, 71– 73, 83, 85, 89, 94, 96–98, 100, 104, 146, 149, 157, 163–165, 168, 172, 173, 175 fractal dimension III, 88, 89, 91, 92, 94–100, 102–106, 108, 146, 149, 156, 157, 162–165, 168, 172, 173, 175 fractal dimension IV, 149, 155–158, 163, 164, 168, 170–174, 177, 189 fractal dimension V, 149, 155, 156, 158–163, 168, 170–173 fractal dimension VI, 149, 156–165, 168, 170, 172, 173, 175

Index Hausdorff, 1, 7, 11–17, 20–22, 33, 53, 62, 65, 72, 85, 89, 92, 98, 100, 103– 106, 108, 113, 115, 117, 118, 165, 168, 170–179, 181, 182, 184, 185, 189, 192– 195 Hausdorff dimension, 7, 12 Hausdorff–Besicovitch, 12 information, 17 lower box(-counting), 18–22, 52, 62, 63, 97 metric dimension, 17 of self-similar sets, 22, 33, 36 upper box(-counting), 18–22, 52, 62, 63, 97 Dimension function, 7, 12, 13, 17, 20, 62, 63, 157, 160 Distribution, 124, 126, 137, 142 empirical, 132, 143–146 normal, 142 Domain, 4, 6 E Euclidean set, 66, 68, 71–75, 79, 83, 85, 87, 97–99, 102, 103, 106, 108, 116, 118, 149, 161, 165, 168, 170–178, 184, 185, 194 F Fixed point, 30, 31, 185, 186 Fractal, 17, 23, 49 Fractal dimension properties Closure dimension property, 13, 20 Countable sets, 12, 14, 20 Countable stability, 12, 14, 20 Finite stability, 12, 20 Geometric invariance, 13 Lipschitz invariance, 13 Monotonicity, 12, 14, 20 Open sets, 13 Smooth manifolds, 13 Fractal structure, 1, 13, 23, 38–40, 49–69, 71–79, 82, 83, 85–91, 93–109, 111– 113, 115–120, 122–124, 126, 138, 146, 147, 153, 155, 157–161, 163– 165, 167, 168, 170–175, 177–180, 182, 187 diameter-positive, 161, 162 finite, 39, 59, 60, 83, 161, 163, 164, 175 level of a, 38–40, 50–56, 58–61, 63, 64, 67–69, 71, 74–79, 82, 85–88, 93, 94, 96–104, 106–108, 112–120, 122–124, 126, 150, 161, 165, 167, 180–182

Index locally finite, 50, 53, 62, 98, 158 natural fractal structure as a Euclidean set, 49–52, 54, 55, 58, 59, 62, 64, 66, 83, 85, 87, 93, 97–99, 107–109, 113, 120, 122, 124, 149, 151, 152, 157, 159, 161, 168, 170–173, 178, 180 as a Euclidean set fractal structure on K as a real subset, 187 as a self-similar set, 23, 38–40, 57, 58, 63, 64, 66, 68, 71–73, 100–104, 146, 175 starbase, 39, 50, 53, 56, 59–62, 83, 86, 98, 111, 112, 158 tiling, 50, 51 topology induced by a, 61, 151 Function, 4–6 ceiling, 187 dimension, 7, 12, 13, 17, 20, 62, 63, 157, 160 set function, 6, 8, 14, 88, 91–94, 147, 150, 156, 165–167

G GF-space, 55, 56, 97, 165 Golden ratio, 77

H Hausdorff, 7 Hausdorff dimension, 1, 7, 11–17, 20–22, 33, 53, 62, 72, 85, 92, 98, 104, 108, 117, 118, 149, 150, 157, 160, 163, 165, 168, 170–174, 176–179, 181, 182, 184, 185, 189, 192–195 Hausdorff measure, 7–11, 14, 22, 34, 150, 153, 156 Hausdorff metric, 24–26, 28, 184, 185 Homeomorphism, 6 Hutchinson’s equation, 34

I IFS-attractor, 22, 31–34, 38, 39, 50, 57, 64– 66, 68, 71–73, 83, 100–104, 146, 149, 161, 174–176, 185–187, 189, 194 Image, 4, 6, 105–108, 112, 113, 115–117, 147 pre-image, 4 Infimum, 2, 8–12, 16, 23, 29, 154 Injection, 4, 71 Interior, 3, 33 Intersection, 2–4, 32, 105

203 Interval, 2, 13, 14, 51, 104, 106, 113, 116, 118, 120, 131, 151–153, 174, 183, 188, 189 Invariance, 14 geometric, 13, 15 Lipschitz, 13 Invariant set, 24, 31, 187 Inverse function, 4, 6, 15 Inverse image, 4 Isometry, 4, 5, 11 Iterated function scheme, 1, 23, 24, 29, 31– 33, 36, 38, 39, 57, 58, 64, 66, 68, 71– 73, 100–104, 174–177, 185–187, 194 Iterated function system, 1, 23, 24 Iteration, 187 J Julia set, 176 K K. Falconer, 12, 13, 17, 33, 36 Kolmogorov entropy, 17 L Lebesgue measure, 9 Length, 9, 10, 13–15, 58, 71, 72, 75, 77, 79, 122, 123, 126, 129–131, 133–136, 138, 141–145, 188 length, 130, 133 Lévy stable process, 120, 138, 140, 142 Limit, 3, 5, 6, 8, 18, 26, 30, 52, 59, 61, 67, 70, 88, 89, 93, 115, 156, 165 lower, 5, 18, 52, 59, 61, 67, 70, 88, 89, 156 upper, 5, 18, 59, 88, 89, 156 Linear transformation, 4, 5 Lipschitz function, 5, 10, 11, 13, 15, 23, 109, 110 Lipschitz invariance, 13 Logarithmic density, 17 Logarithms, 6, 127, 138, 145, 191 M Map isometry reflection, 4 rotation, 4 similarity, 4 translation, 4 Lipschitz, 10 Mapping, 4, 5, 10, 23, 30, 31, 69, 185, 187

204 Mass distribution, 7, 22, 23, 34 Mass distribution principle, 22, 23, 34, 35 Mean, 82, 120–123, 125, 126, 130, 131, 133, 137, 139, 141–144, 146, 189–195 mean, 130 Mean value theorem, 10 Measure, 6–9, 14, 16, 22, 88, 92–94, 106, 147, 149, 150, 153, 156, 159, 165– 168, 176 Hausdorff, 7–11, 14, 22, 34, 85, 88, 92, 150, 153, 156 Lebesgue, 9 on a set, 6, 7 outer, 7 restriction of, 166, 168 Middle third Cantor set, 7, 58, 63, 64, 83, 173 Monotonicity, 12, 14, 17, 20, 63, 160 Monte Carlo simulation, 129–131, 133, 134, 136, 138, 142–144 Moran’s theorem, 22

N Neighborhood, 3, 56, 61 Neighborhood base, 56, 61 Normal distribution, 142

O Onto function, 4, 112 Open ball, 2, 16, 171 Open set, 3, 4, 13, 16, 34, 35, 104, 166 Open set condition, 33, 34, 36, 71–74, 83, 100, 103, 104, 146, 149, 174–176, 185, 186, 194, 195 Othello, 52, 75, 80–82

P Power law, 5 Pre-fractal, 100, 174, 175, 185, 187 Pre-image, 4 Product, Cartesian, 2, 60

R Random function, 125, 126, 137 Random process, 105, 108, 120, 122–129, 131–133, 137–142, 144 Random variable, 120–126, 137, 139, 141 Random walk, 130 Range of a block, 141

Index Range of scales, 17, 49, 79, 80, 83, 86, 111, 177 Rigid motion, 4, 5 Rotation, 4, 11, 13, 102

S Sample function, 120, 122, 124–126 Scaling property, 9, 137, 140 Self-affine increments, 120–123, 125, 126, 128, 129, 137, 138 Self-similarity exponent, 105, 108, 120, 122–124, 126–133, 135–146 Self-similar set, 23, 24, 31, 32, 38–40, 57, 58, 63, 64, 66, 68, 71–73, 83, 100– 104, 115, 117, 146, 147, 174–177, 184, 185, 189, 193–195 Sierpi´nski gasket, 32, 33, 39, 40, 106, 116, 117 Similarity, 4, 5, 13, 23, 33, 58, 64, 72, 73, 101, 103 Simulation, 129, 131, 144, 194 Stability, 13 countable, 12–14, 17, 20, 53, 62, 63, 98, 99, 157–160 finite, 12, 17, 20, 53, 62, 63, 98, 157, 160 Stationary increments, 120–123, 125, 126, 128, 129, 137–139 Strong open set condition, 71, 83 Support of a measure, 7, 22 Supremum, 2, 29, 66, 160 Surjection, 4

T Tends to, 5 Totally disconnected, 3, 4, 15, 16 Transformation, 4, 5, 14, 15 Translation, 4, 11, 102

U Uncountable set, 2 Union, 2–4, 39, 185

V Volume, 9, 10, 33

W W. Sierpi´nski, 32