Teichmuller Theory Volume 2: Surface Homeomorphisms and Rational Functions [2, 1 ed.] 1943863008, 9781943863006

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8 The classification of homeomorphisms of surfaces In this chapter we present and prove the first of Thurston’s theorems involving Teichm¨ uller spaces: the classification of homeomorphisms of surfaces, Theorem 8.1.3. Understanding homeomorphisms and di↵eomorphisms of manifolds is a central problem of mathematics. Already understanding homeomorphisms and di↵eomorphisms of the circle is an immensely difficult problem with a huge literature. The 2-dimensional case is much harder yet; Thurston’s theorem is probably the main result in the field. The theorem concerns homeomorphisms up to homotopy, so it is in some sense crude, avoiding all delicate local study; in exchange, it provides vital global information. Moreover, the group of homotopy classes of homeomorphisms, also known as the mapping class group, is of central interest in geometric group theory. Here also Thurston’s theorem is of fundamental importance. We will present a proof due to Bers [12], which is more in keeping with the style of this book than Thurston’s proof. Thurston’s proof has been given in considerable detail by Fathi, Laudenbach, and Poenaru [40]; it is much longer and more elaborate.1 Thurston classifies homeomorphisms into three types: periodic, reducible, and pseudo-Anosov. This terminology is inspired by the classification of homeomorphisms of the torus.

Classification of homeomorphisms of the torus Let T denote the torus T := R2 /Z2 . A matrix A 2 SL2 Z defines an orientation-preserving homeomorphism fA : T ! T . Conversely, the parametrized closed curves ⇣ ⌘ ⇣ ⌘ t 7! 0t , 0  t  1 and t 7! 0t , 0  t  1

form a basis of the homology group H1 (T, Z) = Z2 , and any orientationpreserving homeomorphism f : T ! T gives a homomorphism f⇤ : Z2 ! Z2 that has a matrix A 2 SL2 Z. One can show that f is isotopic to fA . Thus 1 Apparently Jakob Nielsen has some claim to having proved the result long before Thurston. However, I have spoken with the people who know Nielsen’s work best, and they say that he never made any definition similar to “pseudoAnosov”. Without it, no classification theorem seems possible.

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Classification of homeomorphisms

the classification of homeomorphisms of T up to isotopy is the same as the classification of elements of SL2 Z up to conjugacy. This classification leads to three cases: a matrix A 2 SL2 Z may have complex nonreal eigenvalues, a double eigenvalue ±1, or real distinct eigenvalues. The eigenvalues of A are the roots of 2 (tr A) + 1 = 0. They can be nonreal only if tr A = 0 or tr A = ±1 or tr A = ±2 (remember that the trace tr A is an integer). If tr A = 0, the matrix has eigenvalues ±i, and A4 = I. If tr A = 1, then A3 = I, and if tr A = 1, then A6 = I. If tr A = ±2, then ±1 is an eigenvalue, and a corresponding eigenvector provides a simple closed curve on the surface that is mapped to itself (preserving or reversing the orientation). If |tr A| > 2, then A has two distinct real eigenvalues, necessarily irrational: the contracting eigenvalue | 1 | < 1 and the expanding eigenvalue | 2 | > 1. The directions of the eigenvectors provide invariant foliations on R2 /Z2 , which are contracted and expanded by fA . These homeomorphisms are called Anosov; in [8], Anosov studied them and in particular showed that they are structurally stable.

8.1 The classification theorem Thurston’s classification theorem is an analogue of the classification of homeomorphisms of tori; it applies to surfaces of any genus g 2. Anosov homeomorphisms are replaced by pseudo-Anosov homeomorphisms, which also have invariant foliations that are expanded and contracted; the leaves of these foliations are the horizontal and vertical trajectories of a quadratic di↵erential q. In this case, however, the foliations are singular. Some examples are shown in Figure 5.3.1. Definition in 8.1.1 (Classification of homeomorphisms of surfaces) Let S be a compact surface of genus g 2, and let f : S ! S be an orientation-preserving homeomorphism. The map f is 1. periodic if the iterate f

m

is the identity for some m

1

2. reducible if f is not homotopic to a periodic homeomorphism, and some nonempty multicurve is invariant under f (such a multicurve is called a reducing multicurve) 3. pseudo-Anosov if there exist an element ' : S ! X of Teichm¨ uller space TS , a holomorphic quadratic di↵erential q 2 Q(X), and K > 1 such that ' f ' 1 is a Teichm¨ uller mapping (X, q) ! (X, q/K); see Definition 5.3.6 June 25, 2015

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The classification theorem

3

Remark It is usually better to think of a pseudo-Anosov map as an area-preserving map f : (X, q) ! (X, q) rather than as a Teichm¨ uller map f : (X, q) ! (X, q/K). See Figure 8.1.1. 4 1 1

1

(X, q)

1

1

1/K

(X, q/K)

p K

p K p K

p 1/ K

(X, q)

Figure 8.1.1 At left, a piece of a Riemann surface X with quadratic di↵erential q, and a (blue) unit square in the natural coordinate for q. The two blue regions at right are the image of the unit square at left by the same map f (strictly speaking, by ' f ' 1 ). The two pictures on the right are identical, but with di↵erent metrics. Bottom right, f is a pseudo-Anosov homeomorphism from (X, q) to p (X, q); it stretches horizontal segments of curves by a factor of K, and shrinks vertical segments by the same factor, preserving area. Top right, the same map is seen as a Teichm¨ uller map f : (X, q) ! (X, q/K) that maps horizontal segments to horizontal segments of the same length, and shrinks segments by a factor of K. (Since the metric at upper right is smaller than that at left, the (yellow) unit square at right is of course larger.)

It is easy to find periodic homeomorphisms that are reducible, but reducible and pseudo-Anosov are mutually exclusive. Proposition in 8.1.2 A pseudo-Anosov homeomorphism and a reducible homeomorphism cannot be homotopic. Let q be a quadratic di↵erential on a Riemann surface. Recall from Section 5.3 (see equation 5.3.2) that in a neighborhood of any point where q 6= 0, there exists a natural local coordinate z such that q = dz 2 ; the element of length |dz| in such a coordinate is denoted |q|1/2 . June 25, 2015

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Proof Let f : S ! S be pseudo-Anosov, so there exists a Riemann surface X, a quadratic di↵erential q 2 Q(X), and a homeomorphism ' : S ! X such that if g := ' f ' 1 , then g : (X, q) ! (X, q/K) is a Teichm¨ uller map. We will show that g is not homotopic to a reducible homeomorphism, so that f isn’t either. With the metric |q|1/2 , the image of a geodesic by a Teichm¨ uller map is a geodesic. A closed geodesic is made up of finitely many segments, each with a slope. Suppose that a geodesic is mapped by g to a geodesic 0 homotopic to . Then either and 0 coincide, or together they bound a straight cylinder for the metric |q|1/2 . In either case, the slopes of the segments making up coincide with those making up 0 . However, a segment of slope a is mapped by g to a segment of slope a/K. Thus the only slopes that can appear for a segment of 0 are 0 and 1. Further, the horizontal and vertical parts of must be mapped to the horizontal and vertical parts of 0 , which must therefore have the same lengths. This contradicts the fact that g expands horizontal lines and contracts vertical ones. ⇤ The object of this chapter is to prove the following theorem, which will be proved in Section 8.4. Theorem in 8.1.3 (Classification of homeomorphisms of compact surfaces) Let S be a compact oriented surface of genus g, and let f : S ! S be an orientation-preserving homeomorphism. Then the map f is homotopic either to a periodic homeomorphism, or to a reducible homeomorphism, or to a pseudo-Anosov homeomorphism.

8.2 Periodic and reducible homeomorphisms It is reasonably easy to find examples of periodic and reducible homeomorphisms of surfaces.

Periodic homeomorphisms Let S be a compact oriented surface, with finite subset Z ⇢ S. Let ⇢ ⇡1 (S Z) be a normal subgroup of finite index, and let p : S ! S Z be the corresponding covering map. The surface S can be compactified by adding appropriate points above points of Z; let S be the resulting surface, and let p : S ! S be the extension of p; this extension p is now a ramified covering map. The deck transformations form a group isomorphic to ⇡1 (S Z)/ . All elements of this group are periodic homeomorphisms of S , and this construction yields all examples, for appropriate S, Z, . June 25, 2015

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Periodic and reducible homeomorphisms

5

It is not hard to find (lots of) normal subgroups of finite index. But understanding what one has found is a di↵erent matter: counting and classifying the conjugacy classes of normal subgroups of index n in the free group on m generators is an important and difficult problem in geometric group theory.

Dehn multi-twists To give examples of reducible and pseudo-Anosov homeomorphisms we first introduce Dehn multi-twists. Dehn twists were defined in Appendix A2, but here we will work with the piecewise linear definition of equation A2.2 rather than the smooth Dehn twist of Proposition and Definition A2.2. Let be a simple closed curve on an oriented surface X, and let C ⇢ X be a closed neighborhood of . Choose an orientation-preserving homeomorphism : R/Z ⇥ [0, 1] ! C. Define g : R/Z ⇥ [0, 1] ! R/Z ⇥ [0, 1] by g(s, t) := (s + t, t).

8.2.1

Recall that MCG denotes the mapping class group. Definition in 8.2.1 (Dehn twist) The Dehn twist D 2 MCG(X) is the isotopy class of the map X ! X given by ⇢ x if x 2 /C x 7! 8.2.2 1 g (x) if x 2 C. Figure 8.2.1 illustrates the construction. C

Figure 8.2.1 Left: A blue horizontal band crossing C (the tan); the curve is gold. Right: The image of the band under the metric Dehn twist D . On both boundary components of C, the band veers to the right under the twist.

Since g is the identity on @(R/Z ⇥ [0, 1]) = R/Z ⇥ {0, 1}, the map defined by equation 8.2.2 is continuous, in fact a homeomorphism. Of course, this map, called a metric Dehn twist, depends on the choice of C and . But the Dehn twist of Definition 8.2.1 is an isotopy class, and depends only on the homotopy class of the curve ; see Proposition and Definition A2.2. June 25, 2015

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Exercise 8.2.2 Show that if and are disjoint simple closed curves on X, then D D is homotopic to D D . Show that if you choose the neighborhoods of and disjoint, then D D = D D . } Remarks 1. One can easily imagine a twist map that twists in the opposite direction. You might think – I did for many years – that the direction depends on an orientation for the curve . Recall (Remark 7.6.2) that this is not the case! The construction of the Dehn twist depends only on an orientation for X. Our D has the property that as you approach the boundary of C from X C, you veer to the right. This is true for both components of @C. 2. Dehn twists are the easiest homotopy classes of homeomorphisms to imagine, though their compositions rapidly become inextricably complicated when the corresponding curves intersect. It is known [65] that on a surface of genus g one can choose 2g + 2 Dehn twists that generate the mapping class group; the relations between the generators are also known. 4 Because the (homotopy classes of) Dehn twists around disjoint curves commute, we can also define the Dehn multi-twist around a multicurve. In Definition 3.6.1 we defined a multicurve on a surface S as a family of simple closed curves on S whose elements are disjoint, with no two homotopic to each other, and none homotopic to a point. We now add the hypothesis that all of the curves 2 are nonperipheral, since a Dehn twist D with peripheral is the identity of the mapping class group, so D contributes nothing to the mapping class group. Definition in 8.2.3 (Nonperipheral curve) A curve on a surface X is nonperipheral if there exists a compact subset K ⇢ X with K 6= X such that any 0 homotopic to satisfies 0 \ K 6= ;. A nonperipheral curve cannot be “deformed” into the periphery so that it becomes disjoint from K. In particular, since K 6= X, the curve is not homotopic to a constant map. A typical example of a peripheral curve arises when X is a surface from which a point x has been removed. Then a small curve surrounding {x} is homotopically nontrivial but peripheral. On a compact surface, all nontrivial curves (i.e., curves not homotopic to a point) are nonperipheral. On a compact surface, a multicurve is always finite. More generally, on a compact surface with a finite number of points removed, a multicurve is always finite. June 25, 2015

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Exercise 8.2.4 Let X be a surface of genus g with n punctures, such that g 1 + 3n > 0. Show that a multicurve has at most 3g 3 + n elements. What happens when g = 1 and n = 0? What happens when g = 0 and n = 0, 1, 2, 3? } Definition in 8.2.5 (Dehn multi-twist) Let := { 1 , . . . , k } be a multicurve. The Dehn multi-twist around is the element of MCG(X) given by D := D

1

· · · D k.

8.2.3

Examples of reducible homeomorphisms Dehn multi-twists D are examples of reducible homeomorphisms: evidently the curves of are mapped to themselves. These are not the most general reducible homeomorphisms. We can compose Dehn twists with homeomorphisms of the components of the complement of a multicurve. Other reducible homeomorphisms permute the components of a multicurve before performing Dehn twists.

8.3 Pseudo-Anosov homeomorphisms Pseudo-Anosov homeomorphisms are by far the most common type of homeomorphisms2 , but they are much harder to imagine than periodic or reducible homeomorphisms (except perhaps in genus 1, where they are relatively straightforward). We give two classes of examples of pseudoAnosov homeomorphisms: those obtained from appropriate billiard tables, and those obtained from carefully chosen Dehn twists. The first class is discussed much more extensively by Gutkin and Judge in [46]. The second, due to Thurston, has been studied and generalized by, among others, Kariane Calta, Pascal Hubert, Curt McMullen, and T. A. Schmidt.

Square-tiled billiard tables Let X ⇢ C be a “square-tiled billiard table”: a simply connected region bounded by segments of horizontal and vertical lines, with the coordinates 2

Here is a way to make “most common” precise. There exist 2g + 2 simple closed curves on S such that every homeomorphism is homotopic to a composition of the Dehn twists around those curves. Consider elements of the mapping class group that can be represented by words of length at most n in these Dehn twists (some finite but huge family of homotopy classes of homeomorphisms): the proportion of these that is pseudo-Anosov tends to 1 as n tends to 1. June 25, 2015

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of all the vertices in Z2 ; see Figure 8.3.1. Double X along its boundary e ⇠, where X e is a second copy of X, to make the surface X := X [ X/ given the opposite of the orientation of X to make X oriented. Then X is a Riemann surface isomorphic to the Riemann sphere. There is a natural e map ⇡ : X ! X that is the identity on both X and X. If you denote by z the natural coordinate of C, then the quadratic di↵ere together define a ential dz 2 on X and the quadratic di↵erential d¯ z 2 on X quadratic di↵erential q on X, with simple poles at the angles ⇡/2 of X, and simple zeros at the angles 3⇡/2. The standard formula of plane geometry X (⇡ ↵i ) = 2⇡ 8.3.1 angles ↵i of X

gives

X

poles of q

⇡ 2

X

zeros of q

⇡ = 2⇡, 2

8.3.2

i.e., # (poles of q) # (zeros of q) = 4, as should be the case for a quadratic di↵erential on P1 . This is illustrated in Figure 8.3.1. becomes a zero in X

X become poles in X Figure 8.3.1 The polygon X can be doubled on its boundary to make a Riemann b The surface X isomorphic to the Riemann sphere and then a double cover X. b is a quadratic form on X has 10 poles and 6 zeros, and the Riemann surface X 1 double cover of P ramified at 16 points, hence has genus 7, by equation 8.3.5.

Note that geodesic flow on X corresponds exactly to billiards on X: e after each bounce; see Figure 8.3.2. motion on X flips to motion on X Billiards have provided a lot of the impetus for studying these sorts of homeomorphisms of surfaces. !

!

c b

c

b

Figure 8.3.2 Left: A trajectory on the double X of X, with the two sheets superposed. Right: If we unfold an edge, the trajectory becomes straight in the resulting flat surface. June 25, 2015

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Pseudo-Anosov homeomorphisms

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Cut X into horizontal bands, by cutting along horizontal lines through all the corners of X; two such bands are shaded tan in Figure 8.3.3. The inverse images under ⇡ of these bands in X are annuli with height hi and circumference 2li , where mi and li are integers. These horizontal annuli form a partition of X. Suppose there are k such annuli.

lj0

A0j hi

(

li Ai

h0j

Figure 8.3.3. Two horizontal bands are shaded tan; two vertical bands are outlined in blue. The inverse image in X of the horizontal band labeled Ai is an annulus with height hi = 2 and circumference 2li = 10, for a modulus 1/5. The inverse image of the vertical band labeled A0j has circumference 2h0j = 8 and height lj0 = 2.

Similarly, we can cut X along vertical lines through the corners of X, to obtain annuli A0j with circumferences 2h0j and heights lj0 (the heights are horizontal in the drawing). Two such bands are outlined in blue in Figure 8.3.3. Suppose there are n such bands. Now choose some positive integer M that is divisible by all the 2li , and set pi := M hi /2li ; similarly, choose M 0 divisible by all the 2lj0 and set p0j := M 0 h0j /2lj0 . Denote by Di : X ! X the metric Dehn twist in Ai , which exists because Ai is a metric cylinder. Denote by Dj0 the metric Dehn twist in A0j . Proposition in 8.3.1 The maps F := D1p1

· · · Dkpk

and G := D10

p01

· · · Dn0

p0n

8.3.3

are affine in the natural coordinates of q. Proof In a cylinder of modulus 1/m, with the circles horizontal, the metric Dehn twist has derivative  1 m . 8.3.4 0 1

We have chosen our numbers pi so that all the metric Dehn twists Dipi lift to the pi -fold cover of Ai ; these covers all have modulus 1/m. So along the common boundaries, all the Dipi fit to form affine maps. The situation is June 25, 2015

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similar for the vertical annuli A0j , but this time, the derivative of each of  1 0 p0 the Dj0 j is given by the matrix . ⇤ m0 1 Now consider F G, or more generally any element of the group generated by F and G; these maps are all affine, with derivatives in SL2 Z. But unless they are conjugates of powers of F and G, the eigenvalues 1 , 2 of the derivatives are integers in a real quadratic field that are not rational, and the directions of their eigenvectors give two foliations that are respectively expanded and contracted. Algebraically, these foliations are defined by the 1-densities |Re (↵ dz)| and |Re ( dz)|. Thus the map F G and all the elements of the group generated by F and G (unless they are conjugates of powers of F and G) are pseudo-Anosov for the quadratic di↵erential (Re (↵ dz) + iRe ( dz))2 . These maps are all defined on the sphere, and are associated to quadratic di↵erentials with poles. It isn’t difficult to use them to construct pseudoAnosov maps on surfaces of higher genus; these maps are associated to holomorphic quadratic di↵erentials (without poles). b of X on which q becomes orientable; by the Pass to the double cover X Riemann-Hurwitz formula (Theorem A3.4) it is a Riemann surface of genus

# (poles of q) + # (zeros of q) 1. 8.3.5 2 For instance, if we had chosen our polygon X in Figure 8.3.1 to be a recb would have genus 1. tangle, q would have four poles and no zeros, so X b puts us in the realm of compact surfaces and holomorphic Working on X quadratic di↵erentials (in fact, squares of complex 1-forms, also known as Abelian di↵erentials), although it is often more convenient to carry out the computations on X, or even in X directly. The meromorphic quadratic di↵erential q lifts as a holomorphic quadratic b This is easiest to see in terms of prongs (see Figure di↵erential q˜ on X. 5.3.1): at a simple pole q has one prong, and hence at the corresponding b the quadratic di↵erential q˜ has two prongs, so at that point q˜ point of X has neither a zero or a pole. Moreover, the inverse images of the annuli in X always consist of two annuli, each of which maps by a homeomorphism, since q is orientable in e as metric Dehn twists in each annulus. So we can lift F and G to Fe and G each of these annuli, hence as maps that are affine in the natural coordinates of q˜. Their compositions are pseudo-Anosov as above. b = g(X)

Pseudo-Anosov maps associated to two multicurves On a surface S of genus g 2, let := { 1 , . . . , k } and := { 1 , . . . , l } be two multicurves, in minimal position with respect to each other (see June 25, 2015

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Definition 3.3.10). We say that and form a web if every component of 0 1 k l [ [ A S @ 8.3.6 i [ j i=1

j=1

is simply connected. (Being a web is often referred to as filling the surface.) The Dehn twists D and D are reducible, as are all Dehn multi-twists; they preserve and respectively. If and do not form a web, then all elements of the group they generate are indeed, they fix any simple closed curve contained ⇣Salso reducible; ⌘ Sl k in S [ . i=1 i j=1 j

But if and form a web, the situation is completely di↵erent: Theorem 8.3.3 shows that compositions of D and D are usually pseudo-Anosov. Define M to be the matrix with k rows and l columns whose entry mi,j is the geometric intersection number of i and j , i.e., the minimal number of transverse intersections of curves in their homotopy classes. Figure 8.3.4 illustrates the construction. 2

2

H1

H2

00 2 0 2

00 2 0 2

1

3

1

H3

H4 1

1

Figure 8.3.4 Left: A surface S of genus 2 with two multicurves, := { 1 , 2 , 3 } and := { 1 , 2 }. The matrix 2 whose 3 entry mi,j is the geometric intersection 0 1 number of i and j is M = 4 2 2 5. The multicurves and cut S into four 0 1 hexagons, H1 , . . . , H4 . Right: Here we show that H1 is a hexagon; it has sides 0 00 1 , 1 (which is half of the 1 at left), 2 and 2 (which together make up half of 0 00 2 and bound the front and back side of the surface), and 2 and 2 (which make up one-fourth of 2 and bound the front and back side of the surface).

Exercise 8.3.2 Show that there exists N such that all entries of the matrix (M >M )N are strictly positive. Hint: Say that the curves 0 and 00 are “1-connected” if they both intersect some element of , and that they are “N -connected” if there exists a chain 0 , 1 , . . . , N with 0 = 0 and N = 00 , such that i , i+1 are 1-connected for all i = 0, . . . , N 1. Show that the i, jth entry of (M >M )N is the number of chains of length 2N connecting i to j (i.e., the number of ways in which i and j are N -connected). } June 25, 2015

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By Exercise 8.3.2 and the Perron-Frobenius theorem (Theorem C1.1 in Appendix C1), M > M has a dominant eigenvalue > 0. Theorem in 8.3.3 (Pseudo-Anosov maps as compositions of Dehn twists) Let two multicurves and form a web. If > 4, then each word in the group generated by D and D is isotopic to a pseudoAnosov homeomorphism, except for conjugates of the powers of the generators. We will illustrate the proof at each step with the special case shown in Figure 8.3.4, underlining the symbols relevant to the example of Figure 8.3.4 and leaving plain the symbols relevant to the general case. Proof By the Perron-Frobenius theorem, there is a vector v 2 Rl with strictly positive entries that is an eigenvector of M > M for the eigenvalue . Let µ be the positive square root of . Set w :=

Mv 2 Rk ; µ

8.3.7

then w is an eigenvector of M M > with eigenvalue3 . The vector w has strictly positive entries, and satisfies M > w = µv. In the case of Figure 8.3.4, we have 2 3  1 2 1 p 4 4 > > M M= , MM = 4 2 8 2 5 , = 5 + 17, 8.3.8 4 6 1 2 1 p 2 3  1 + 17 p 1 4p v= , w= p 8.3.9 p 4 10 + 2p 17 5 . 1 + 17 1 + 17 1 + 17 Now consider the graph A dual to [ . This is the graph that has a vertex in each component of 0 1 k l [ [ A, S @ 8.3.10 i [ j i=1

j=1

with an edge joining each vertex to the “middle of” each arc of i or j in its boundary, as shown in Figure 8.3.5. Let A be the union of the edges intersecting -curves (i.e., the i ), and let A be the union of the edges intersecting -curves. Our hypothesis that and together form a web says that each component of S A is a quadrilateral, with two sides intersecting a curve in and The eigenvalues for M > M and M M > are the same, except that if one matrix has larger dimensions, it will have the extra eigenvalue 0. 3

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two sides intersecting a curve in ; some sides may be identified. In the case of Figure 8.3.4, S A has six components, as shown in Figure 8.3.6. As shown in Figure 8.3.7, make each quadrilateral into a metric rectangle (an ordinary elementary school rectangle) using the entries of the eigenvectors v and w, assigning length wi to the (green) edge that intersects the (blue) edge i , and length vj to the (gold) edge that intersects the (red) edge j . The rectangle of Figure 8.3.7 becomes the left cylinder in Figure 8.3.8; the quadrilateral on the right side of Figure 8.3.6 becomes the right cylinder in Figure 8.3.8. The middle cylinder corresponds to the four remaining quadrilaterals of Figure 8.3.6, rotated by 90 degrees and glued together.

6

1

2

5

4

3

3

2 1

4 6

5

Figure 8.3.5 We can represent each quadrant of S as a hexagon with six spokes going from the center to the middle of each edge; here we illustrate this for the upper left hexagon, marked H1 in Figure 8.3.4. Left: The spokes emanating from the purple circle (of which only the front is visible) are gold if they go to a red edge ( 1 or 2 ), green if they go to a blue edge (a i ). The spokes going around the back are fainter and are marked with italic numbers. Right: The corresponding graph. The graph A is the union of four such “pinwheels” (the graph at right minus the blue and red sides). Every spoke is half an edge, so we have twelve edges and four vertices; six edges belonging to A and six to A . Figure 8.3.6 The graph A cuts the surface S of Figures 8.3.4 and 8.3.5 into six quadrilaterals. There are two quadrilaterals in the center of the figure, one in front, one in back.

v2 2

2

w1 1

1

June 25, 2015

Figure 8.3.7 Turning the left-most quadrilateral of Figure 8.3.6 into an ordinary rectangle with width v2 and height w1 . It becomes the left cylinder in Figure 8.3.8.

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Chapter 8.

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Note that the circumference of the cylinder around i is given by the ith entry of M v. In Figure 8.3.8: 2 3 2 3 0 1  v2 v M v = 4 2 2 5 1 = 4 2v 1 + 2v 2 5 8.3.11 v2 0 1 v2 (The values for the entries of v are given in equation 8.3.9.) v 2 = (M v)1 #

2v 1 + 2v 2 = (M v)2 #

w1

w2

1

2

v 2 = (M v)3 #

2

1

2

w3 3

2

1

2

Figure 8.3.8 The three cylinders surrounding the core curves i 2 of Figure 8.3.4. The cylinder at left corresponds to the rectangle of Figure 8.3.7; that at right corresponds to the quadrilateral at the right of Figure 8.3.6. Both have circumference v 2 . The middle cylinder corresponds to the four middle quadrilaterals of Figure 8.3.6; imagine turning each into an ordinary rectangle and gluing them together. All have height w2 , two have width v 1 , and two have width v 2 , so this cylinder has circumference 2v 1 + 2v 2 .

Lemma 8.3.4 summarizes what we have accomplished so far: from two multicurves and forming a web on S we have found a complex structure on S and a quadratic di↵erential holomorphic for that complex structure. These are the ingredients required by the definition of a pseudo-Anosov map. Lemma 8.3.4 Let Ri,j ⇢ C be the rectangle n wj wj vi vi o Ri,j = z = x + iy 2 C x , y . 2 2 2 2 1. For each rectangle R of S A, crossed by i and j as in Figure 8.3.7, there exist orientation-preserving homeomorphisms R : Ri,j ! R such that if R1 , R2 are two adjacent rectangles of S A, then R21 R1 is an isometry on the identified sides. 2. There is a unique analytic structure on S such that R is analytic for each rectangle R of S A. We denote by X the Riemann surface S with this complex structure. June 25, 2015

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Pseudo-Anosov homeomorphisms

15

3. There is a unique quadratic holomorphic di↵erential q on X that 2 restricts to d R on each rectangle R of S A. The zeros of q are at the nodes of A, the union of the critical horizontal trajectories of q is A , and the union of the critical vertical trajectories of q is A . Proof It should all be clear from the discussion above. Adjacent rectangles are either crossed by the same i , in which case they have the same height vi , or crossed by the same j , in which case they have the same width. We leave to the reader to show that the complex structure and the quadratic di↵erential are holomorphic there. ⇤ Lemma 8.3.4 Proof of Theorem 8.3.3, conclusion If we consider the union of the rectangles that intersect i , we find a cylinder with height wi and circumference (M v)i . By equation 8.3.7, (M v)i = µvi , where µ is the positive square root of the dominant eigenvector of M > M , so for all the cylinders, the ratio of height to circumference is the same (in fact, 1/µ). So the metric Dehn twist D i in this cylinder is given by   x x + µy 7! . 8.3.12 y y

Since all the twists are the identity on A , they fit together to give a map D of the form of equation 8.3.12 in the natural coordinates for q. Similarly, if we cut the surface along H , we construct cylinders around the j , of height vj and circumference (M > w)j , and the same construction as above gives a representative of D , which in local coordinates is   x x 7! . 8.3.13 y y µx The composition D matrix

D

is affine in the natural coordinates for q, with 

1

µ2 µ

µ , 1

8.3.14

which has trace < 2 when µ2 = > 4 and is hence hyperbolic; see Exercise 8.3.5. So D D is pseudo-Anosov. Exercise 8.3.5

Show that an element of SL2 R induces

• an elliptic transformation if | tr A| < 2, • a parabolic transformation if | tr A| = 2, • a hyperbolic transformation if | tr A| > 2.

}

The same argument as above shows that a word in D , D is pseudoAnosov if and only if the corresponding product of the matrices   1 0 1 µ A := and B := 8.3.15 µ 1 0 1 June 25, 2015

16

Chapter 8.

Classification of homeomorphisms

is hyperbolic. Now we have to show that all words in D , D lead to products of A and B that are hyperbolic, except for the conjugates of the generators. In Proposition and Definition 3.9.8 we defined Schottky groups, and here we are in a position to apply that result. The region |Re z| < µ/2, z

1 1 1 1 > , z+ > µ µ µ µ

is a paired polygon (see Definition 3.9.2), and when looks like Figure 8.3.9.

8.3.16

> 4, i.e., µ > 2, it

µ A

B

0

1/µ

1/µ

1/µ

0 1/µ

1

Figure 8.3.9 Left: If µ > 2, the region described by equation 8.3.16 looks like the paired polygon shaded at left. The quotient of the paired polygon by the action of A and B is a trouser with two cusps (see Proposition 3.5.3). Right: A trouser with two cusps, the cusps corresponding to 0 and 1.

The group Gµ generated by A, B is free on the generators, and H/G is a trouser with two cusps. In particular, all elements of the fundamental group correspond to hyperbolic elements of PSL2 R except those represented by curves surrounding exactly one of the punctures; these are exactly the conjugates of the generators A and B. ⇤ Theorem 8.3.3 Exercise 8.3.6 Let DA and DB be Dehn twists for the multicurves A := ↵1 , ↵2 , ↵3 and B := 1 , 2 , 3 in Figure 8.3.10. Show that DA DB is pseudo-Anosov, and find the expanding and contracting eigenvalues. } Exercise 8.3.7 Let and be multicurves as in Theorem 8.3.3, and choose integers n1 , . . . , nk and m1 , . . . , ml , all strictly positive. Set A := Dn11 B := Dm1 1

· · · Dnkk

· · · Dml l .

June 25, 2015

8.3.17

8.4

Classification theorem

17

3

1

2

↵1

↵3

↵2

Figure 8.3.10 Figure for Exercise 8.3.6. Note that the pair A := ↵1 , ↵2 , ↵3 and B := 1 , 2 , 3 of multicurves defines di↵erent pseudo-Anosov homeomorphisms than those in Figure 8.3.4.

Show that all words in A and B are pseudo-Anosov except the powers of the generators. Hint: Show that if vectors v, w satisfy 2 3 2 3 n1 w1 m1 v1 . . M v = µ1 4 .. 5 and M > w = µ2 4 .. 5 , 8.3.18 nk wk ml vl

then the argument above (putting metric Dehn twists in appropriate cylinders) still works. } p Remark In the proof, we did not have to choose µ = (see the line before equation 8.3.7); we could have chosen any two numbers µ1 , µ2 > 0 such that µ1 µ2 = , and then set w :=

Mv µ1

so that v :=

Mw µ2

8.3.19

We could then construct rectangles with heights and widths given by the entries of w and v as above; with respect to these coordinates, the metric Dehn twists in the cylinders of X A and X A are still all affine with the same derivatives. This constructs a 1-parameter family of complex structures and quadratic di↵erentials, parametrized by µ1 2 (0, ). The same words in D and D are pseudo-Anosov as for the case µ1 = µ2 = µ. We will encounter this 1-parameter family in Section 8.4: it corresponds to the axis of f for a pseudo-Anosov homeomorphism f (see Lemma 8.4.5). 4

8.4 Proof of the classification theorem In this section we prove the classification theorem, Theorem 8.1.3, restated below. Note the similarities with the statement and proof of Proposition 2.1.14. June 25, 2015

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Chapter 8.

Classification of homeomorphisms

Theorem in 8.4.1 (Classification theorem, repeated from Section 8.1) Let S be a compact oriented surface of genus g, and let f : S ! S be an orientation-preserving homeomorphism. Then f is homotopic either to a periodic homeomorphism, or to a reducible homeomorphism, or to a pseudo-Anosov homeomorphism. Recall from Section 6.4 the action of the mapping class group MCG on Teichm¨ uller space TS ; recall that a point of TS is defined by a pair (X, '), where X is a Riemann surface and ' : S ! X is a homeomorphism. Proof The mapping f 2 MCG(S) induces the mapping given by f

f

: TS ! TS

(X, ') := X, ' f .

8.4.1

Let d be the Teichm¨ uller metric, and consider the number D(f ) := inf d ⌧, ⌧ 2TS

f (⌧ )

.

8.4.2

The following three propositions prove Theorem 8.4.1. Proposition in 8.4.2 If D(f ) is realized by some ⌧ 2 TS , and D(f ) = 0, then f is isotopic to a periodic mapping. Proposition in 8.4.3 If D(f ) is realized by some ⌧ 2 TS , and D(f ) > 0, then f is isotopic to a pseudo-Anosov homeomorphism. Proposition in 8.4.4 If D(f ) is not realized, then f is reducible. Proof of Proposition 8.4.2 If f (⌧ ) = ⌧ , where ⌧ is represented by a pair (X, '), then ' : S ! X and ' f : S ! X represent the same point of TS , i.e., they are Teichm¨ uller equivalent. So there exists an analytic isomorphism ↵ : X ! X such that the diagram '

S ! f# ' S !

X #↵ X

8.4.3

commutes up to homotopy. Since the group Aut(X) is finite, ↵ is of finite order. The mapping f is homotopic to ' 1 ↵ '. ⇤ Proposition 8.4.2 Remark We have proved along the way the following rather surprising statement, known as the Nielsen conjecture: if f k is homotopic to the identity for some k, then f is homotopic to a map of order k. We will see in Chapter 12 (Mostow’s theorem, Theorem 12.4.1) that the analogous statement is true for complete hyperbolic manifolds of any dimension. 4 June 25, 2015

8.4

Classification theorem

19

Proof of Proposition 8.4.3 Suppose that ⌧0 realizes the inf, i.e., that d(⌧0 , f (⌧0 )) = D(f ). Suppose that ⌧0 is represented by ' : S ! X, and let ↵ : X ! X be the Teichm¨ uller mapping homotopic to f (see Corollary 7.2.3). We require two lemmas. The geodesic ` of Lemma 8.4.5 is called the axis of f . The map f translates along that axis by D(f ). f (⌧0 )

Lemma 8.4.5 The Teichm¨ uller geodesic ` joining ⌧0 to ant under f .

is invari-

Proof It is enough to show that f 2 (⌧0 ) 2 `. Let ⌧1 be the midpoint of the segment [⌧0 , f (⌧0 )], so that ⌧2 := f (⌧1 ) is the midpoint of the segment [ f (⌧0 ), f 2 (⌧0 )], since f is an isometry. By the triangle inequality, d(⌧1 , ⌧2 )  d ⌧1 ,

f (⌧0 )

+d

f (⌧0 ), ⌧2

= d ⌧0 ,

f (⌧0 )

= D(f ),

8.4.4

so d(⌧1 , ⌧2 ) = D(f ), since D(f ) realizes the infimum. See Figure 8.4.1. Figure 8.4.1

d(⌧1 , ⌧2 ) # ⌧0 a

⌧1

b0 ⌧2 :=

f

2

(⌧0 )

f (⌧1 )

a0 f (⌧0 )

b

d(⌧0 ,

We have

f (⌧0 ))

= a + b.

Since f is an isometry, we have a = a0 , giving the equality in equation 8.4.4. Our argument shows that this picture is inaccurate; the two line segments are aligned.

Thus the triples ⌧0 , ⌧1 ,

f (⌧0 )

,

⌧1 ,

f (⌧0 ), ⌧2

,

f (⌧0 ), ⌧2 ,

2 f (⌧0 )

8.4.5

⇤ Lemma 8.4.5

are aligned.

Lemma 8.4.6 The initial and final quadratic di↵erentials of ↵ coincide. (Initial and final quadratic di↵erentials are defined in Definition 5.3.6.) Proof Lemma 8.4.5 says that ↵ ↵ is a Teichm¨ uller mapping. But the composition of two Teichm¨ uller mappings is never a Teichm¨ uller mapping if the final quadratic di↵erential of the first isn’t the initial one of the second. ⇤ Lemma 8.4.6 This completes the proof of Proposition 8.4.3. ⇤ June 25, 2015

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Chapter 8.

Classification of homeomorphisms

Proof of Proposition 8.4.4 This is an application of the gap principle, which we will encounter again in Chapters 10 and 16. Suppose that D(f ) is not realized, and that ⌧i is a sequence in TS such that d(⌧i , f (⌧i )) decreases to D(f ). Let ⌧i be represented by 'i : S ! Xi . Denote by S(S) the set of homotopy classes of simple closed curves on S. Let ' : S ! X represent a point ⌧ 2 TS . For each simple closed curve 2 S(S) denote by l (⌧ ) the hyperbolic length of the geodesic homotopic to '( ) on X, and further set w (⌧ ) := ln l (⌧ ); we call w (⌧ ) the log-length of . The logarithmic spectrum L⌧ ⇢ R is L⌧ := { w (⌧ ) |

simple closed curve on S } .

8.4.6

Note that the elements of L⌧i are labeled by the elements of S(S), of homotopy classes of simple closed curves on S. Lemma 8.4.7 The number inf 2S(S) l (⌧i ) tends to 0 as i tends to 1; equivalently, inf 2S(S) w (⌧i ) tends to 1 as i tends to 1.

Proof of Lemma 8.4.7 Suppose not; then the projection of the ⌧i into moduli space lies in a compact subset, by Mumford’s compactness theorem (Theorem 7.3.1). We can then extract a subsequence ⌧ij of the ⌧i and find elements ↵j 2 MCG(S) such that the sequence ↵j (⌧ij ) converges in TS , say to ⌧0 . Since the ↵j are isometries, we have ⇣ ⌘ ⇣ ⌘ d ⌧ij , f (⌧ij ) = d ↵j (⌧ij ), f (↵f (⌧ij )) . 8.4.7 Thus, extracting a further subsequence, we may assume that the sequence ↵j 1 f ↵j (⌧ij ) also converges. For any ✏ > 0, there exists N (✏) such that when j, k N (✏), ⇣ ⌘ d (↵j 1 ↵j ) 1 (↵k 1 ↵k )(⌧0 ) , ⌧0 < ✏. 8.4.8 f f

Since the orbits of the action of the Teichm¨ uller modular groups are discrete, this implies that for ✏ sufficiently small and j N (✏), all ↵j 1 ↵j coincide. This shows that D(f ) is then realized (and f is 0), a contradiction. ⇤ Lemma 8.4.7 By Lemma 8.4.7, the set L⌧i must have elements that are arbitrarily close to 1 for sufficiently large i. But only finitely many p (at most 3g 3, to be precise) may be smaller than W0 := ln ln(3 + 2 2); see Corollary 3.8.7. Thus for any C > 0 and all sufficiently large i we can write L⌧i = Ai t Bi with Ai 6= ;,

sup Ai < W0 ,

sup Ai + C < inf Bi .

8.4.9

The number C is the size of a gap between two elements of L⌧i ; see Figure 8.4.2. June 25, 2015

C

8.5

The reducible case

0

W0

Ai

21

Bi

Figure 8.4.2 The logarithmic spectrum L⌧i might look like this, for g = 3. There are then at most six “log lengths” smaller than W0 . Hence if any are very negative, there must be a large gap.

Choose C > d(⌧0 , f (⌧0 )), find an i such that Ai 6= ;, and consider the set of curves 2 S(S) with w (⌧i ) 2 Ai . The set is a multicurve since sup Ai < W0 : if ⌧i is represented by 'i : S ! Xi , all p the geodesics on Xi homotopic to 'i ( ) for 2 have length < ln(3 + 2 2), hence are disjoint. Observe two things. First, L⌧i and L⌧i+1 are the same sets of numbers (possibly with di↵erent labels), since they are the logarithmic spectra of the same Riemann surface. But if a log length has label in L⌧i , then it has label f 1 ( ) in L⌧i+1 . Second, by Theorem 7.6.4, the functions ⌧ 7! w (⌧ ) are all Lipschitz, with Lipschitz ratio 1. Moreover f is an isometry. So points with the same label in L⌧i and L⌧i+1 are at most C apart: |w (⌧i )

w (⌧i+1 )|  d(⌧i , ⌧i+1 ) = d(⌧i ,

f (⌧i ))

= d(⌧0 ,

f (⌧0 ))

< C.

It follows that for 2 , the geodesic in the homotopy class of f 1 ( ) has logarithmic length smaller than inf Bi , so f 1 ( ) 2 . Thus f is reducible: is invariant under f . ⇤ Proposition 8.4.4 This completes the proof of the classification theorem.

8.5 The structure in the reducible case When f is homotopic to a reducible homeomorphism h, Theorem 8.1.3 is unsatisfactory. If we reduce h, for instance by cutting along invariant curves, we simplify S, but the new surface Se we obtain is a surface with e boundary, and our theorem says nothing about the pieces that make up S. It is quite difficult to do Teichm¨ uller theory on a surface with boundary. We can solve this by collapsing the components of the boundary of Se to points; this gives a surface S without boundary but with marked points corresponding to the collapsed boundary components. (We have already encountered homeomorphisms of surfaces with marked points and the corresponding quadratic di↵erentials with poles; see Figure 8.3.1.) Teichm¨ uller theory for compact surfaces with marked points is very similar to the theory for compact surfaces, and with almost no modifications can be used to prove a theorem corresponding to Theorem 8.1.3. June 25, 2015

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Chapter 8.

Classification of homeomorphisms

The only change needed is a slight modification of the definition of pseudo-Anosov. Recall Definition 8.1.1: f : S ! S is pseudo-Anosov if there is an element ' : S ! X of TS , a holomorphic quadratic di↵erential q 2 Q(X), and K > 1 such that ' f ' 1 is a Teichm¨ uller mapping (X, q) ! (X, q/K). In the case of surfaces with marked points, we use the generalization of a Teichm¨ uller mapping given in the last subsection of Section 5.3, which we recall here. Let X be a compact Riemann surface, and Z ⇢ X a finite subset. Then Q1 (X Z) is equivalently (see Exercise 5.3.10) 1. the space of holomorphic quadratic di↵erentials on X R |q| < 1, or X Z

Z with

2. the space of meromorphic quadratic di↵erentials on X, holomorphic on X Z and with at most simple poles on Z. To deal with pseudo-Anosov homeomorphisms on surfaces with marked points, we need to allow such quadratic di↵erentials. Definition in 8.5.1 (Pseudo-Anosov homeomorphism on surface with marked points) Let S be a compact oriented surface, and P ⇢ S a finite subset. A homeomorphism f : (S, P ) ! (S, P ) is pseudo-Anosov if there exist • • • •

a Riemann surface X a finite subset Z ⇢ X an orientation-preserving homeomorphism ' : S q 2 Q1 (X Z)

such that ' f

'

1

P !X

Z

is a Teichm¨ uller map as in Definition 5.3.11.

Let S be a compact oriented surface of genus g, and let P := {p1 , . . . , pm } be m distinct points of S. Let f : (S, P ) ! (S, P ) be an orientationpreserving homeomorphism. Theorem in 8.5.2 (Classifying homeomorphisms of surfaces with marked points) The homeomorphism f is isotopic rel P either to a periodic homeomorphism, or to a reducible homeomorphism, or to a pseudo-Anosov homeomorphism (as defined in Definition 8.5.1). Theorem 8.5.2 generalizes Theorem 8.1.3 (it reduces to Theorem 8.1.3 when P = ;) and the proof requires no modification. The quadratic di↵erentials with poles appear for the following reason: in the line above Lemma 8.4.5, we invoked Corollary 7.2.3. This result is stated for Riemann surfaces June 25, 2015

8.5

The reducible case

23

of finite type, in particular for X Z if X is compact and Z ⇢ X is finite. However, the quadratic di↵erential involved in the Teichm¨ uller mapping belongs to Q1 (X Z), so we must allow simple poles on Z. ⇤ If f : S ! S is reducible, then f is isotopic to a homeomorphism g that maps a nonempty multicurve to itself. Denote by S the surface (usually not connected) obtained as follows: cut S along , so each element 2 becomes two boundary components 0 and 00 , and collapse each of these to a point. If | | = n, then the collapsed curves correspond to a set P of 2n points of S. Clearly g induces a homeomorphism g : (S , P ) ! (S , P ),

8.5.1

which permutes to components of S . Let T be such a component, and set PT := P \ T . There exists m such that g m maps (T, PT ) to itself. Thus we can apply Theorem 8.5.2 to g

m

: (T, PT ) ! (T, PT ),

8.5.2

which must be isotopic either to a homeomorphism of finite order, or to a pseudo-Anosov homeomorphism, or to a reducible homeomorphism. Note that if g m is reducible, then the reducing curves can be lifted to S, and these together with will form a larger reducing set 0 for f . Thus if is a maximal set of reducing curves, then g m : (T, PT ) ! (T, PT ) will never be reducible, hence will always be of finite order or pseudo-Anosov. We can finally explain the meaning of D(f ) when f is reducible. Proposition in 8.5.3 Let f : S ! S be a reducible homeomorphism, and let be a maximal reducing multicurve and g a homeomorphism isotopic to f mapping to itself. Then D(f ) =

sup T

component of S of period m

1 D(g Tm ). m

In particular, D(f ) = 0 if and only if the map g Tm : (T, PT ) ! (T, PT ) is isotopic to a map of finite order for every periodic component T of S . Exercise 8.5.4

Prove Proposition 8.5.3.

June 25, 2015

}

9 Dynamics of polynomials In this chapter we give an introduction to the dynamics of polynomials as endomorphisms of C. These results will be used in Section 10.5 in the context of quadratic polynomials, but here we avoid focusing on the quadratic case. Many of the results are true for rational functions, or even for entire functions, or functions meromorphic on C, but to keep the appendix of reasonable length we have avoided that generality.

9.1 Julia sets In complex dynamics in one variable the fundamental object of study is the Julia set. It is distinctly easier to study the Julia set of a polynomial than the Julia set of a more general mapping, because the Julia set of a polynomial is the boundary of the filled Julia set. Rational functions that are not polynomials have a Julia set but no filled Julia set. Definition in 9.1.1 (Julia set and filled Julia set) Let p be a polynomial of degree d 2. The sequence n 7! p n (z), i.e., the sequence z, p(z), p(p(z)), . . . is called the orbit of z under p. The Julia set Jp and the filled Julia set Kp are Kp := { z 2 C | the orbit of z under p is bounded } Jp := @Kp

Figure 9.1.1 represents six filled-in Julia sets Kp , for appropriate cubic polynomials. We chose d = 3 rather than d = 2 to avoid overlap with Chapter 10, and because these exhibit a larger variety of behavior. The set Kp is black, and the colors represent “rates of escape”: if the orbit n 7! p n (z) tends to infinity, there is a first n for which |p n (z)| > R, and if R is bigger than the number specified in Proposition 9.1.2, the picture will not depend in any essential way on R. If |f n (z)| > 10, then |f (n+1) (z)| ⇠ 1000 and |f (n+2) (z)| ⇠ 109 . When a point is well in the escape region, it really goes! A more crucial issue is when to color a pixel black: to choose an N such that if |f N (z)| < R, we decide that z 2 Kf . There di↵erent June 25, 2015

9.1

Julia sets

25

choices of N can produce quite di↵erent pictures; these pictures were made with N = 1000.

a = .5, b =

.485 + .4i

a = .5, b =

.67 + .44i

a = .5, b =

.065 + .605i

a = .5, b =

.105 + .595i

a = .5, b =

a = .5, b =

.655 + .575i

.145 + .57i

Figure 9.1.1 We see above two sets of 3 pictures of filled-in Julia sets for cubic polynomials, all written in the form z 3 3a2 z + b. The area represented is |Re z|  2, |Im z|  2 The sets Kp are in black; the colors represent “rate of escape to infinity”, i.e., the Green’s function of Kp (see Theorem 9.3.4). The three top pictures correspond to polynomials whose coefficients are quite close, as do the bottom three. The critical points are ±a = ±1/2. In the left pictures both critical points have bounded orbits, so the filled-in Julia sets Kp are connected. In both middle pictures one critical point has a bounded orbit and the other not; the sets Kp are not connected, but some components of Kp are not points (and are themselves fairly complicated in the botom center picture). For the polynomials pictured at right, both critical points escape, and Kp = Jp is a Cantor set, on which the polynomial is a homeomorphism conjugate to the one-sided shift on 3 symbols.

Proposition in 9.1.2 The sets Kp and Jp are compact subsets of C. Proof Set p(z) = ad z d + ad If |z|

1z

d 1

A = max {(|ad o R := max 2+A |ad | , 1 then

|p(z)|

n

|ad ||z|d |z|d

1

|ad

(|ad ||z|

1 ||z

A)

+ · · · + a0 with ad 6= 0, and set

1|

d 1

+ · · · + |a0 |), 1}

| + · · · + |a0 |

2|z|.

June 25, 2015

|ad ||z|d

A|z|d

1

26

Chapter 9.

Dynamics of polynomials

Thus Kp ⇢ {z 2 C||z| < R} so Kp is bounded, and Kp =

1 \

p

n

(DR (0))

n=1

so it is an intersection of closed sets, hence closed. The boundary of a compact subset is always compact, so Jp is compact. ⇤ If f is a rational function or an entire function, the Julia set of f is defined as the set of non-normality of the sequence f, f 2 , f 3 , . . . . Proposition 9.1.3 shows that the definitions are consistent: for polynomials, the boundary of the filled Julia set is indeed the set of non-normality. Proposition in 9.1.3 Let p be a polynomial of degree d 2. Then z 2 / Jp if and only if z has a neighborhood on which the sequence p, p 2 , p 3 , . . . is normal. In this context, a sequence of functions that converges locally uniformly to infinity is normal, i.e., normal as a sequence of functions with values in P1 . Proof If z 2 / Jp , then either z is in the interior of Kp or is not in Kp . In

the first case, z has a neighborhood U ⇢ Kp , and the sequence n 7! p n is bounded on U , hence normal. In the second, there is some N such that |p N (z)| > R (R as in Proposition 9.1.2). There is then a neighborhood U of z such that |p N (w)| > R for all w 2 U . Since for all n N we then have |p n+1 (w)| > 2|p n (w)|, n the sequence n 7! p converges uniformly to 1 on U . Conversely, if z 2 Jp , then every neighborhood of z contains points with bounded orbits (including z itself), and open subsets on which the sequence n 7! p n converges to infinity, so the sequence has no subsequence converging uniformly on compact subsets. ⇤ Proposition in 9.1.4 a. For any polynomial p of degree d and Jp are totally invariant: p(Jp ) = Jp ,

p

1

(Jp ) = Jp ,

p(Kp ) = Kp ,

p

1

2. both Kp

(Kp ) = Kp .

Conversely, if X ⇢ C is a closed set satisfying p(X) = p then either

1

(X) = X,

1. X = {a} is a single point, p is conjugate by an affine map to z 7! z d and a corresponds to 0 under the conjugacy, or 2. Jp ⇢ X. June 25, 2015

9.1

Julia sets

27

Proof The first part is obvious: a point z has a bounded orbit under p if and only if the orbit of p(z) is boun ded if and only if for each z1 2 p 1 (z) the orbit of z is bounded. The second part requires Montel’s theorem. If X contains 2 or more points, the sequence n 7! p n restricted to C X omits X, hence n 7! p n is normal on C X by Montel’s theorem, so Jp is a subset of X. If X = {a} is a singleton, then a is a fixed point of, and is the only solution of p(z) = a. Thus p(z) = b(z a)d +a for some b, i.e., p is conjugate to Z 7! bZ d . A further conjugacy, setting Z = W ⌘ with ⌘ d 1 = 1/b makes p conjugate to W 7! W d . ⇤ Proposition in 9.1.5 1. If V is a component of the interior K p , then the boundary of V is contained in Jp . 2. Every component of the filled Julia set Kp is full, i.e, its complement in C is connected. 3. The Julia set Jp is perfect: it has no isolated points. Proof 1. Let z be a point of the boundary of some component V of Kp . If z 2 / Jp , then z has a connected neighborhood U on which the sequence n 7! p n is normal. Since U \ V 6= ;, no subsequence can converge to 1. So the sequence is bounded, and U ⇢ Kp , so z is not a boundary point of any component.

2. If some component V of K p disconnects C, at least one component U of the complement C V must be bounded. The sequence n 7! p n is bounded on the boundary of U, hence bounded on U by the maximum modulus principle, so U ⇢ Kp . 3. This is more subtle. Let Jp0 be the derived set of Jp , i.e., the set of accumulation points of sequences of distinct points in Jp . Since Jp is closed, Jp0 ⇢ Jp . It is easy to see that Jp0 is forward and backwards invariant, so by Proposition 9.1.4, Jp ⇢ Jp0 . Thus Jp = Jp0 , so Jp can have no isolated points. ⇤ Theorem 9.1.6 is illustrated by Figure 9.1.6. This result is due to Fatou, and shows that the first thing to study when iterating a polynomial is the orbits of the critical points; to a large extent they control the topology of Kp . Let ⌃d = {0, . . . , d 1}N and d : ⌃d ! ⌃d be the one-sided shift ( d (a))i = ai+1 if a = (a0 , a1 , . . . ) 2 ⌃d . Note that ⌃d with the product topology is a Cantor set, and with this topology d is continuous and d to 1. June 25, 2015

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Chapter 9.

Dynamics of polynomials

Theorem in 9.1.6 Let p be a polynomial of degree d 2, and denote by Critp the set of finite critical points of p: CritP := { z 2 C | p0 (z) = 0 }. Then 1. Critp ⇢ Kp if and only if Kp is connected.

2. If Critp \ Kp = ;, then Kp is a Cantor set, and p|Kp is conjugate to the one-sided Bernoulli shift on d symbols. Remark The conjugacy in part 2 means that there exists a homeomorphism ' : Kp ! ⌃d such that the diagram '

!

Kp p# Kp

'

!

⌃d # d ⌃d

commutes. Since any two such homeomorphisms ' di↵er by postcomposition with an automorphism of (⌃d , d ), the homeomorphism ' is unique up to an element of Aut(⌃d , d ). The group Aut(⌃d , d ) is quite well understood, unlike the group of automorphisms of the full shift on d symbols, which is very mysterious even when d = 2. In particular, when d = 2, the only automorphism of the 1-sided shift is the exchange of symbols; when d > 2 it is an infinitely generated group (with known generators), and this fact will complicate the proof of Theorem 9.1.6. 4 Proof 1. Note first that p extends to an analytic map P1 ! P1 , still denoted p, with {1} fixed and its only inverse image. Thus deg1 p = d. Let DR ⇢ C be the closed disc of radius R centered at the origin with R sufficiently large so that p 1 (DR ) is contained in DR as a compact subset. Set U0 := P1 DR and Un := p 1 (Un 1 ) for n 1. Then we have U0 ⇢ U1 ⇢ U2 ⇢ . . .

1 [

with

Un = C

Kp .

9.1.1

n=0

Since any component of Un must contain an inverse image of 1, and 1 has only itself as inverse image, all the Un are connected. If Critp ⇢ Kp , then for every n > 0 the mapp : Un ! Un 1 is a covering map of degree d ramified only at 1 (with local degree d as we saw). Since U0 is a disc, we have (U0 ) = 1. Suppose by induction that (Un 1 = 1, then by the Riemann-Hurwitz formula, (Un ) = d (Un

1)

(d

1) = 1.

The only connected orientable surface with Euler characteristic 1 is the disc, so all Un are homeomorphic to discs. Their complements p n are nested Jordan domains, so their intersection is connected. June 25, 2015

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29

Conversely, suppose Critp 6⇢ Kp . Then some first Un 1 contains at least one critical value, and by the Riemann-Hurwitz formula, (Un )  0. Then Un is a connected but not simply connected subset of the sphere, and its complement must have at least two components. Each must contain points of Kp , so Kp is not connected. 2. Choose R large as above. If Critp \ Kp = ;, then for m large enough, Critp \ (p m ) 1 (DR ) = ;. Write V := (p

m

)

1

(DR ) and W := (p

(m 1)

)

1

(DR ).

9.1.2

Then V ⇢ W , and p : V ! W is a covering map. Let (z, ⇠) be a vector tangent to V at z; denote by |(z, ⇠)|V its hyperbolic length in V , and by |(z, ⇠)|W its hyperbolic length in W . Since V is relatively compact in W , there exists C > 1 such that |(z, ⇠)|V

C|(z, ⇠)|W .

9.1.3

Since the restriction p : V ! W is a covering map, it is an infinitesimal isometry, so it is strongly expanding, in the sense that |(p(z), p0 (z)⇠)|W = |(z, ⇠)|V

C|(z, ⇠)|W .

9.1.4

Let M be the maximum of the diameters of the components of V , as measured in the hyperbolic metric of W . Then the maximum diameter of T a component of (p n ) 1 (V ) is M C n , and since Kp = (p n ) 1 (V ), it follows that the components of Kp are points. Since Kp is compact and perfect, it is a Cantor set. The proof that p|Kp is conjugate to the one-sided Bernoulli shift on d symbols is a refinement of the above, but we must be more careful about our choice of domain. Choose a closed topological disc D ⇢ C that contains Kp in its interior and contains no critical value of p. Then p : p 1 (D) ! D is proper of degree d, and it is a local homeomorphism, since the domain p 1 (D) contains no critical points. Thus it is a covering map, necessarily trivial since D is simply connected. So the inverse image p 1 (D) consists of d topological discs, which we label D0 , . . . , D(d) ; see Figure 9.1.2. It would be nice if we had p 1 D ⇢ D, but that is a bit delicate to arrange. As a substitute, we use the following lemma. Lemma 9.1.7 Let D ⇢ C be a closed topological disc that contains Kp in its interior and contains no critical value of p. Then there exists n such that (p n ) 1 (D) is a compact subset of the interior of D. June 25, 2015

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D0

Dynamics of polynomials

.... ........ ... ..... .. . .

D00

.. ....... . .... ... . .. .

Figure 9.1.2 The (green) set D is a topological disc that contains Kp and omits all critical values; this may force it to be convoluted. The heavy black dots represent the omitted critical values; the sprinkling of light dots represents Kp .

.

D .. . ..... ....... .... ......... . . . .. . . .... . .. . .. ..

D000

Proof of Lemma 9.1.7 Choose a number R sufficiently large so that D ⇢ DR and p 1 (DR ) ⇢ DR . Define the numbers M := sup

inf { m | |p

z2@D m2N

Mn :=

z2(p

inf n )

1 (D)

m

(z)| > R }

inf { m | |p

m2N

m

(z)| > R } .

9.1.5

If we think of the points of C DR as “escaped” (to infinity), then M measures how fast the slowest point of @D escapes: M is large if the slowest escaping point of @D escapes slowly. The number Mn measures how fast the fastest point of (p n ) 1 (D) escapes. The number M is finite, since all the points of @D do escape. Clearly Mn+1 = Mn + 1, so that there exists n such that Mn > M ; we leave it to the reader to check that for this value of n the lemma is true. ⇤ Lemma 9.1.7 Let Kp,i := Kp \ D(i) . By Proposition 9.1.4, Kp = [di=1 Kp,i and each p : Kp,i ! Kp is injective. Define the mapping : Kp ! {1, . . . , d}N by (z) := s0 (z), s1 (z), s2 (z), . . . ,

where p n (z) 2 D(sn (z)) ,

9.1.6

i.e., the nth term of the sequence (z) remembers which of the discs D0 , . . . , D(d) contains the point p n (z). The sequence is well defined since p n (z) 2 Kp for all n, and clearly (p(z)) = S( (z)), where S(s0 , s1 , s2 , . . . ) = s1 , s2 , s3 , . . .

9.1.7

is the one-sided shift on d symbols. We need to show that is injective, surjective, and continuous. The continuity should be clear: if two points z1 , z2 2 Kp are close, their orbits will be close for a long time, so long initial segments of (z1 ) and (z2 ) will coincide. The surjectivity follows easily from the fact that p : Kp,i ! Kp is bijective. June 25, 2015

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The injectivity requires Lemma 9.1.7: by the same argument that showed that Kp is totally disconnected, it follows from Lemma 9.1.7 that the components of (p kn ) 1 (D) shrink to points as k ! 1; hence, since n is fixed, the components of (p k ) 1 (D) shrink to points as k ! 1. But (z1 ) = (z2 ) means that z1 and z2 are in the same component of (p k ) 1 (D) for all k, hence z1 = z2 . ⇤ Theorem 9.1.6 Remark 9.1.8 The isomorphism Kp ! {0, . . . , d 1}N is a lot easier to understand and a lot more natural when deg p = 2, as illustrated in Figure 9.1.3, left.

0

1

0

1

Figure 9.1.3 Left. For the polynomial p : z 7! z 2 + 2, the orbit of the critical point 0 is 0, 2, 6, 36, . . . tends to infinity, so the Julia set is a Cantor set, visible as a thin dust. We have drawn the circle |z| = 2 (containing the critical value). Its inverse image is the “vertical” lemniscate |z 2 + 2| = 2, whose two lobes have been labeled 0 and 1. The orbit of a point z 2 Kp visits these lobes in some order, producing a homeomorphism Kp ! {0, 1}N , called the kneading sequence of z. Right. This represents the (highly atypical) polynomial z 7! z 2 2. Here the unique critical orbit is 0, 2, 2, 2, . . . , so Kp is connected; in fact it is the segment [ 2, 2]. The inverse image of the circle |z| = 2 is the horizontal lemniscate, but this time the double point belongs to Kp , and corresponds to two elements of {0, 1}N , namely 010 and 110. This is very similar to the identification .9 = 1.0 of the decimal system: allowing it maps the Cantor set 0, . . . , 9}N to the continuum [0, 1].

We can follow the same plan as in the general case. Choose R large enough so that p 1 (DR ) has compact closure in DR . If the (unique) critical point !p escapes to 1, there exists a smallest m such that |p m (!p )| R. Then the restriction p : (p m ) 1 (DR ) ! (p (m 1) ) 1 (DR ) is an unramified double cover of (p (m 1) ) 1 (DR ), which is homeomorphic to a disc. Thus (p m ) 1 (DR ) is a union of two components U0 , U1 . We define the map June 25, 2015

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Chapter 9.

Dynamics of polynomials

: Kp ! {0, 1}N by ( (z))n :=

⇢

0 1

if p n (z) 2 U0 if p n (z) 2 U1

9.1.8

Then as above (but more easily: the inverse image (p m ) 1 (DR ) has compact closure in (p (m 1) ) 1 (DR ), so we can bypass Lemma 9.1.7) we see that is a conjugacy. The only ambiguity in choosing this conjugacy is in labeling the components U0 and U1 . There is no such natural conjugacy when d > 2; the situation is explored in [14], [17], [18]. 4

9.2 Fixed points Let U ⇢ C be a neighborhood of a point z0 and let f : U ! C be an analytic map with f (z0 ) = z0 . Then for |z z0 | sufficiently small, f (z z0 ) is approximately f 0 (z0 )(z z0 ), so if |f 0 (z0 )| < 1, then |f (z z0 )| < |z z0 |, and if |f 0 (z z0 )| > 1, then |f (z z0 )| > |z z0 |. This justifies the following terminology. Definition in 9.2.1 (Classifying fixed points) The fixed point z0 is • superattracting if f 0 (z0 ) = 0 • linearly attracting if 0 < |f 0 (z0 )| < 1 • attracting if it is superattracting or linearly attracting (i.e., if |f 0 (z0 )| < 1) • indi↵erent if |f 0 (z0 )| = 1 • repelling if |f 0 (z0 )| > 1 The number f 0 (z0 ) is called the multiplier of f at the fixed point z0 . The indi↵erent case is divided into the rationally indi↵erent case, also called the parabolic case, where f 0 (z0 ) is a root of unity, i.e., f 0 (z0 ) = e2⇡i✓ , ✓ 2 Q/Z; and the irrationally indi↵erent case, where f 0 (z0 ) = e2⇡i✓ , with ✓ 2 (R Q)/Z. The irrationally indi↵erent case is a big subject, not discussed in this book. It is itself subdivided into the linearizable case (also called Siegel) and nonlinearizable case (also called Liouville). Remark A point z such that f k (z) = z is called a periodic point of f with period k. The orbit of a periodic point is called a cycle; the period of z is the length of the cycle. All points of a cycle of length k are fixed points of f k , and by the chain rule the multiplier of f k is the same at each point of the cycle; this number is called the multiplier of the cycle. We label a cycle of length k as attracting, repelling, superattracting, or indi↵erent if the points of the cycle are fixed points of the same sort for f k . 4 June 25, 2015

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Fixed points

33

In all cases except that of indi↵erent fixed points, it is reasonably easy to exhibit a local coordinate on a possibly smaller neighborhood of a fixed point z0 particularly well adapted to f . In this local coordinate, f is linear when z0 is linearly attracting or repelling; in the superattracting case, f is simply a power function when written in the appropriate coordinate. The indi↵erent case is much more difficult: if we write f 0 (z0 ) = e2⇡i✓ , then the behavior of f in a neighborhood of z0 depends crucially on Diophantine properties of ✓.

Linearly attracting and repelling fixed points: local theory Theorem 9.2.2 applies to linearly attracting fixed points (and to repelling fixed points, by replacing f by f 1 ). It is convenient to replace z0 by 0. Theorem in 9.2.2 (Linearization theorem) Let U ⇢ C be a neighborhood of 0, and let f : U ! C be an analytic map with f (0) = 0 and 0 < |f 0 (0)| < 1. Then there exist a neighborhood V ⇢ U of 0 with f (V ) ⇢ V and an analytic map ' : V ! C with '0 (0) = 1 such that '(f (z)) = f 0 (0)'(z) for z 2 V .

9.2.1

In other words, we have the commuting diagram V '

# C

The germ of ' at 0 is unique.

f

! V

f 0 (0)

9.2.2

#'

! C

Proof Write := f 0 (0), and set g(z) := f (z) z. There exist r1 > 0 with Dr1 (0) ⇢ U and a constant C such that |g(z)|  C|z|2 when |z|  r1 . Choose C2 with C22 < | | < C2 < 1, and set r := min(r1 , (C2 | |)/C). Then if |z| < r we have |f (z)|  | z| + C|z|2  | | + Cr |z|  C2 |z|,

9.2.3

hence |f

n

(z)|  C2n |z|.

9.2.4

Set V := { z 2 C | |z| < r }. We can consider the sequence of mappings n 7! 'n : V ! C given by 'n (z) = June 25, 2015

f

n

(z)

n

.

9.2.5

34

Chapter 9.

Dynamics of polynomials

The key issue is that the sequence converges uniformly on V to some map '. If so, ' will be analytic. To see this, consider the associated series: | 'n+1 (z)

'n (z)| = =

(n+1)

f

1 | |n

=

(z)

f

n

n+1

f

(z)

n

n

(z) + g(f

1 |g(f | |n+1

n

n

=

(z))

(z))| 

1 f (f | |n f

n

n

(z))

f

(z)

n

(z) 9.2.6

C|f n (z)|2 C  n+1 | | | |

✓

C22 |z| | |

◆n

.

Since C22 /| | < 1, this shows that the series n 7! 'n converges uniformly on V . Set ' := limn!1 'n . We can compute the power series of ' term by term from '(z) = '0 (z) + '1 (z) = z + '1 (z)

'0 (z) + '2 (z)

z + '2 (z)

'1 (z) + · · ·

'1 (z) + · · · ,

9.2.7

and the computation above shows that |'n+1 (z) 'n (z)| 2 o(|z|), so that only the term '0 (z) = z of the sum contributes to the derivative, and '0 (0) = 1. The mapping ' solves the functional equation 9.2.1. Indeed, '(f (z)) = lim

f

n

(f (z)) n

n!1

=

lim

f

(n+1)

(z) 9.2.8

n+1

n!1

= '(z) = f (0)'(z). 0

Finally, we need to consider the uniqueness of '. If map satisfying equation 9.2.1, then M

'

1

(z) =

f

'

1

(z) =

'

: W ! C is another 1

M (z),

9.2.9

where M stands for “multiplication by ” i.e., by f 0 (0). Note that we are working in the germs of analytic maps, i.e, convergent power series, and ' does admit a (functional) inverse in that space. Thus the power series ' 1 commutes with multiplication by ; it is then immediate that ' 1 is multiplication by some number, and imposing 0 (0) = 1 implies that ' = . ⇤ n

Remark 9.2.3 The idea of looking at the limit limn!1 f n(z) goes by the name of scattering theory. We think of f as a perturbation of the map z 7! z. To conjugate the two, we do f forward a lot of times, and z 7! z backwards the same number of times. If the composition of the two has a limit, it will automatically conjugate the perturbed map to the unperturbed map, as is shown in equation 9.2.8. 4 June 25, 2015

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Fixed points

35

Linearly attracting fixed points: global theory Theorem 9.2.2 is local. We now want to spell out the domain of the maps ' and ' 1 precisely. For this to be possible, the original map f must be a global object; we will restrict ourselves to the case where f is a polynomial, although the results hold for rational functions, entire functions, and even meromorphic functions on C. The global theory for attracting fixed points is surprisingly di↵erent from the global theory for repelling fixed points. Definition in 9.2.4 (Basin of attraction) Let z0 be an attracting fixed point of a polynomial p. The set n o z 2 C lim p n (z) = z0 9.2.10 n!1

is the basin of attraction of z0 , often called its basin.

The basin of an attracting cycle z0 , . . . , zk 1 of a polynomial p is the union of the basins of the fixed points zi for p k . Theorem in 9.2.5 (Linearizing attracting fixed points) Let z0 be a linearly attracting fixed point for a polynomial p, and let U be its basin of attraction. Then the sequence n 7! 'n defined by 'n (z) :=

p n (z) z0 p0 (z0 )n

9.2.11

converges uniformly on compact subsets of U , and the limit is an analytic map ' : U ! C satisfying ' p(z) = p0 (z0 )'(z).

9.2.12

The map ' is the linearizing map for p at z0 . Its inverse is also called the linearizing map at z0 . (When the fixed point is attracting, as it is here, ' has better global properties. When it is repelling, its inverse has better global properties; see Theorem 9.2.8.) The 'n defined in equation 9.2.11 are really the same as those defined in equation 9.2.5, except that the fixed point is z0 , not necessarily 0, and the polynomial p has replaced the analytic function f . Proof We already know that there is a neighborhood V of z0 on which the sequence n 7! 'n converges uniformly to '. Let X ⇢ U be a compact June 25, 2015

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Chapter 9.

Dynamics of polynomials

subset. Then there exists m such that p

m

(X) ⇢ V , and on X we have

p n (z) z0 p n (p m (z)) z0 = lim 0 n n!1 n!1 p (z0 ) p0 (z0 )n+m 1 p n (p m (z)) = 0 lim p (z0 )m n!1 p0 (z0 )n lim

9.2.13 z0

=

⇣ 1 ' p p0 (z0 )m

m

⌘ (z) .

This proves that the sequence n 7! 'n converges uniformly on compact subsets of U . The functional equation is clear. ⇤ We saw in Theorem 9.1.6 one essential way in which the orbits of critical points control the global dynamics of polynomials. Theorem 9.2.6 is another instance of the importance of critical points. Theorem in 9.2.6 Let z0 be an attracting fixed point for a polynomial p of degree 2, and let U0 be the component of the basin U of z0 that contains z0 . Then U0 contains a critical point of p. Proof The restriction p : p 1 (U0 ) ! U0 is proper (the restriction of a proper map to the inverse image of any set is always proper), and so is the further restriction p : U0 ! U0 (the restriction of a proper map to one component of the domain is always proper). If U0 contains no critical point of p, then p : U0 ! U0 is proper and a local homeomorphism, hence a covering map, so it is an infinitesimal isometry for the hyperbolic metric of U0 . Thus |p0 (z0 )| = 1, contradicting |p0 (z0 )| < 1. ⇤ Exercise 9.2.7 Let p be a polynomial with attracting cycle; let U be its basin. Show that at least one of the components of U containing a point of the cycle contains a critical point of p. }

It follows from Theorem 9.2.6 that a quadratic polynomial can have at most one attracting cycle, since every such cycle must attract a critical point, and there is only one to attract. More generally, a polynomial of degree d 2 has at most d 1 attracting cycles. This is pretty amazing: there are lots of cycles, even lots for each period k when k is large, namely all the roots of p k (z) = z, which is a polynomial equation of degree dk . But at most d 1 cycles can be attracting. In fact (though it doesn’t follow from our arguments so far), at most d 1 cycles can be attracting or indi↵erent. All the others are repelling. (See [42], [31], [96], [38].)

Repelling fixed points: global theory Let p be a polynomial satisfying p(z0 ) = z0 and |p0 (z0 )| > 1. The map p has no global inverse, but it does have a unique local inverse f such that June 25, 2015

9.2

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37

f (z0 ) = z0 . There is a map ' that linearizes f , but it is not extendable in any reasonable sense. To consider repelling fixed points, we will be interested instead in the extension of ' 1 , which we will call . Theorem in 9.2.8 (Linearization for repelling fixed points) Let p be a polynomial and let z0 be a point with p(z0 ) = z0 and p0 (z0 ) = with | | > 1. The sequence of polynomials n 7! n defined by ⇣w ⌘ n (w) := p + z 9.2.14 n 0 n converges uniformly on compact subsets of C, and its limit the unique entire function satisfying ( w) = p( (w)) and

0

(0) = 1.

: C ! C is

9.2.15

The map is called the linearizing map at z0 ; it is also called the linearizing coordinate. The proof of Theorem 9.2.8 is left to the reader; it is analogous to the proof of Theorem 9.2.5. Examples 9.2.9 (Linearizing maps for repelling fixed points) 1. Take p(z) = z d . Then z0 = 1 is a repelling fixed point with multiplier d. The map : w 7! ew is the linearizing map: d

p( (w)) = (ew ) = edw = (dw).

9.2.16

In this case the formula = lim n becomes the well-known expression for the exponential, n ⇣ w ⌘d ew = lim 1 + n . 9.2.17 n!1 d 2. Take p(z) = z 2 2. Then z0 = 2 is a fixed point with multiplier p p0 (2) = 4. The map : w 7! 2 cosh w is the linearizing coordinate with 0 (0) = 1: p 2 p p( (w)) = 2 cosh w 2 = 2 2 cosh2 w 1 9.2.18 p p = 2 cosh 2 w = 2 cosh 4w = (4w). 4 Let p(z) = z 2 2. Use the formula ⇣ p z ⌘ 2 cosh z = lim p n 2 + n n!1 4 to compute cosh 1 and cos 1 to four significant digits. } Exercise 9.2.10

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38

Chapter 9.

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Parabolic fixed points and Fatou flowers Let U ⇢ C be a neighborhood of 0; let f : U ! C be an analytic map with a fixed point, which we will put at 0, i.e., f (0) = 0. Recall that 0 is an indi↵erent fixed point if |f 0 (0)| = 1. The theory of indi↵erent fixed points is deep and difficult. Write f 0 (0) = e2⇡i✓ . When ✓ is irrational the big question is whether the fixed point is linearizable: in an epochmaking paper [99], Siegel showed that this is the case for almost every ✓, and much further work by Bruno [21], Herman, Yoccoz, and many others has clarified the situation. But even before Siegel, Cremer [29] had shown that for “sufficiently bad” values of ✓, irrationally indi↵erent fixed points aren’t linearizable. The structure of Kp when a polynomial p has such a fixed point has been analyzed by Perez-Marco and others, but it is still pretty mysterious. By contrast, the local theory for rationally indi↵erent fixed points (i.e., when ✓ = p/q is rational) is quite well understood, largely due to the work of Fatou [42], Ecalle [36], and Voronin [104]. Such fixed points are often called parabolic. We will not need the deeper parts of this theory, but only the more elementary theory of the Fatou flower. Let U ⇢ C be a neighborhood of 0, and let f : U ! C be an analytic map with f (0) = 0 and f 0 (0) = e2⇡ip/q with p and q coprime. If f q is not the identity, the function z 7! f q (z) z has a zero of order 2 at the origin, and the order of that zero is not changed by conjugation, since it is the order of contact of the graph of f q with the diagonal. Proposition 9.2.11 says among other things that the order of the zero is of the form ⌫q + 1. The integer ⌫ is the parabolic multiplicity of the origin for f . Proposition in 9.2.11 There exists a unique integer ⌫, neighborhoods U 0 , 00 U ⇢ U of 0 with f (U 0 ) = U 00 and an isomorphism ' : U 0 [ U 00 ! C such that ⇣ ⌘ ' f ' 1 (w) = e2⇡ip/q w 1 + w⌫q + h(w) with h(w) 2 O(w(⌫+1)q ). Proof Set := e2⇡ip/q . Suppose that in some local coordinate wm we can write f as ⇣ ⌘ m m f : wm 7! wm 1 + am wm + o(wm ) . 9.2.20

We will show that if m is not a multiple of q, we can make a change of variables that eliminates am without introducing new lower-degree terms. m+1 Define wm+1 to satisfy wm = wm+1 ↵wm+1 , so that m+1 m+1 wm+1 = wm + ↵wm + o(wm ).

June 25, 2015

9.2.21

9.2

Fixed points

39

In the local coordinate wm+1 , the map f is written wm+1 7! wm+1 7! (wm+1

m+1 ↵wm+1

⇣ m+1 ↵wm+1 ) 1 + am (wm+1

= wm+1 + (am 7! wm+1 + (am

⌘ m+1 m m+1 m ↵wm+1 ) + o(wm+1 ) )

m+1 m+1 ↵)wm+1 + o(wm+1 ) ⇣ m+1 ↵)wm+1 + ↵ wm+1 + (am

= wm+1 + (am

↵+↵

m

m+1 m+1 )wm+1 + o(wm+1 )

m+1 ↵)wm+1

⌘m+1

9.2.22

Thus, unless m is a multiple of q, we can “cancel” am by setting m ↵ = am /(1 ). Further, once we find a coefficient a⌫q that is not 0, it is easy to make it 1 by scaling the variable. ⇤ Topologically, it is not too hard to understand the behavior of a map f as above if we work in the local coordinate of Proposition 9.2.11. Write ⇣ ⌘ g(w) := e2⇡ip/q w 1 + w⌫q + h(w) with h(w) 2 O(w⌫q+1 ). 9.2.23

To first order, g just rotates w by p/q turns; the next term moves points either further away from the origin or closer to it, depending on whether w⌫q has positive or negative real part. To be more precise, we have the following result, illustrated by Figure 9.2.1 in the case q = 3 and ⌫ = 2. Proposition in 9.2.12 (The Fatou flower: attracting and repelling petals) Let ⌫, p, and q be as in Proposition 9.2.11, and let g be as in equation 9.2.23. For ✏ > 0 sufficiently small, the set { w 2 C | Re 1/w⌫q
1/✏ }

9.2.25

forms ⌫q repelling petals P1 , . . . , P⌫q , circularly labeled so that g 1 maps Pi into Pi ⌫p . All points of the repelling petals have orbits under g 1 tending to 0. Proof Consider the “change of variables” w = ⇣ 1/⌫q , so that ⇣ = w ⌫q . This change of variables isn’t defined in a neighborhood of 0 in the wplane (unless ⌫q = 1), but it is defined in each attracting petal, mapping it to the left halfplane Re ⇣ < 1/✏, and in each repelling petal, mapping it to

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40

Chapter 9.

Dynamics of polynomials Figure 9.2.1 For any n 1 and any R > 0, the region Re 1/wn < R forms n attracting petals arranged around 0 2 C. Here, the n = 6 attracting petals are colored, with the coloring corresponding to the iteration of w 7! e2⇡i/3 w(1 + w6 ); each large dark blue petal is sent to the smaller light blue petal a third of a turn away, and similarly for the gold petals. Since f is locally invertible near 0, there are repelling petals between the attracting petals; these petals, not colored, are attracting for f 1 .

the right halfplane Re ⇣ > 1/✏. The expression g1 of g in this coordinate is remarkably simple: ⇣ ⌘ ⌫q g1 : ⇣ 7! e2⇡ip/q ⇣ 1/⌫q 1 + ⇣ 1 + o(⇣ 1 ) ⇣ (1 + ⇣ 1 + o(⇣ 1 ))⌫q ✓ ✓ ◆◆ ⌫q 1 =⇣ 1 +o ⇣ ⇣ =

=⇣

9.2.26

⌫q + o(1).

The multivalued change of variables we have made is really passing to the q-fold cover of C ramified at 0 and 1; the remainder term o(1) in the final line of equation 9.2.26 is uniformly small in a neighborhood of 1 in the cover, in particular in the halfplanes corresponding to each attracting and repelling petal. We can choose ✏ so small that this remainder term is < ⌫q in each such halfplane. It might seem that formula 9.2.26 then shows that each halfplane maps either inside itself (left halfplanes) or outside itself (right halfplanes). Actually, the map g1 really sends a left halfplane inside the left halfplane in the sheet ⌫q turns away. In any case, the union of the attracting petals maps inside itself, and the union of the repelling petals maps outside itself. ⇤ Remark 9.2.13 The definition of the attracting and repelling petals in Proposition 9.2.12 is not optimal; instead of defining our branches of ⇣ 1/⌫q on a halfplane, we can define them in the complement of a cut (along the positive real axis for the attracting petals and along the negative real axis for the repelling petals). Setting ⇣ := ⇠ + i⌘, we can then define attracting petals for g to correspond to the union ⇢ ⇢ 1 1 ⇠+⌘ < [ ⇠ ⌘< 9.2.27 ✏ ✏ June 25, 2015

9.2

Fixed points

41

because the term o(1) in equation 9.2.26 is uniformly small near 1, in particular small throughout such regions for ✏ sufficiently small. See Figure 9.2.2. Figure 9.2.2 The map g1 of equation 9.2.26 is defined outside some compact set, here represented by the light blue disc, and also outside some cut, here the red line along the positive real axis, since we are describing an attracting petal. The o(1) in that equation is uniformly small near 1, so the green region is mapped into itself.

The virtue of choosing these larger petals is that the attracting and the repelling petals overlap. We will use this in Proposition 9.2.14 below, but it is also central to the Ecalle-Voronin classification of parabolic fixed points up to conjugacy (see Remark 10.12 in Milnor’s book [79]). 4

In particular, we see that if a parabolic fixed point has parabolic multiplicity ⌫, then there are ⌫ distinct cycles of petals around it. Proposition in 9.2.14 Suppose a polynomial P of degree d has a parabolic cycle z1 , . . . , zl of length l, so that (P l )0 (z1 ) is the primitive qth root of unity e2⇡ia/q , of parabolic multiplicity ⌫. Then the parabolic cycle attracts at least ⌫ critical points, so ⌫  d 1. Proof A petal attached to z1 is mapped to a petal attached to z2 , then to a petal attached to z3 , and so forth, until after l moves it is mapped back to a petal attached to z1 and a/⌫q of a counterclockwise turn away from the original petal. After lq moves it is mapped back to itself, having visited only q of the ⌫q petals attached to z1 . Thus the basin of the parabolic point contains ⌫ cycles of components attached to z1 , . . . , zl , each periodic of period lq; see Figure 9.2.3.

Figure 9.2.3 Five Fatou flowers; the petals of the flower at upper right are all attached to z1 . A dark blue petal is mapped inside a petal attached to z2 (the light blue denotes the image of the original dark blue petal). Here we have l = 5, ⌫ = 2, and q = 3.

June 25, 2015

42

Chapter 9.

Dynamics of polynomials

Now we must see that each of these cycles of components must contain a critical point. Let U be one such component, so that P lq maps U to itself; suppose by contradiction that U contains no critical points of p lq , or equivalently, that the cycle of U under P contains no critical point of P . Then P lq : U ! U is an isomorphism. Indeed P

lq

:C

JP ! C

9.2.28

JP

is proper, since it is the restriction of P lq : C ! C to a subset invariant forwards and backwards. So it is still proper after restriction to the component U of C Jp . Further, P lq : U ! U is a local homeomorphism since U contains no critical point of P lq . Thus P lq : U ! U is a covering map, hence trivial since U is a component of the interior of KP , and KP is full, so U is simply connected. Finally, since U is connected, P lq : U ! U is an isomorphism. Let : D ! U be a Riemann mapping; then ↵ :=

1

P

:D!D

lq

9.2.29

is an isomorphism, hence a M¨ obius transformation that must be parabolic lq or hyperbolic, since P has no fixed points in the interior of U . Figure 9.2.4 shows that neither is possible.

lift of P lq (↵ hyperbolic)

z1

2%

lift of P lq (↵ parabolic) 1

Figure 9.2.4 Attracting and repelling petals do not have to be chosen disjoint (see Remark 9.2.13). If you choose them overlapping (here the red repelling petals and the blue attracting petal overlap), then you can find two disjoint (green) regions entirely in one petal at z1 , bounded by a curve joining z1 to itself and invariant by P lq . If this petal were to map to itself under P lq by an isomorphism, these regions would lift to (green) regions in D by the Riemann mapping of the component of KP containing the petal. The map P lq would lift to a hyperbolic or parabolic automorphism of D, and the curves bounding the regions in D must then join the fixed points of the automorphism.

The closed curves 1 and 2 of Figure 9.2.4 both belong to KP , so the regions that they bound do too. The inverse images of these curves in D are disjoint curves that join the fixed point(s) and bound two disjoint regions June 25, 2015

9.2

Fixed points

43

with disjoint boundaries. They are also invariant under ↵; this forces them to be positioned either as drawn on the left (when ↵ is hyperbolic) or as drawn on the right (when ↵ is parabolic). In both cases, this forces @D to be mapped by to z1 , which is impossible. Thus our assumption that U contains no critical points of p lq is false. ⇤

Superattracting fixed points Superattracting fixed points will be essential to our description of the Mandelbrot set. They require a somewhat di↵erent approach than linearly attracting fixed points. The map ' in Theorem 9.2.15 is the B¨ ottcher coordinate of the superattracting fixed point 0 for f . Theorem in 9.2.15 (B¨ ottcher coordinate) Let f (z) = z k (1 + g(z)) be an analytic mapping defined in a neighborhood U ⇢ C of 0, with k 2 and g(z) 2 O(z). There then exist a smaller neighborhood V ⇢ U of 0 and a unique analytic mapping ' : V ! C with '0 (0) = 1 and ('(z))k = '(f (z)). Remark The uniqueness in Theorem 9.2.15 is of a di↵erent nature than in Theorem 9.2.2. For linearly attracting fixed points, there is one linearizing map ' for every nonzero value of '0 (0). In Theorem 9.2.15, by contrast, a map satisfying ( (z))k = (f (z)) and 0 (0) 6= 0 must be = !' with ! k 1 = 1. Thus there are only finitely many such maps, and in fact only one when k = 2, the main case of interest to us. 4 Proof The philosophy is again (see Remark 9.2.3) that to construct a conjugacy between maps F and G, one should try to make the sequence (G n ) 1 F n converge. In this case the conjugacy should be between z 7! z k and f , so we are tempted to “define” ⇣ ⌘1/kn 'n (z) := f n (z) and '(z) := lim 'n (z). 9.2.30 n!1

(In equation 9.2.30, the 1/k power is the “inverse” (G n ) 1 .) This time we don’t have just a convergence problem: we need to specify which kn th root we are considering. In the proof of Theorem 9.2.2, we turned the sequence into a series; here we will turn it into a telescoping infinite product. Thus we write n

2

(f

n

1/kn

(z))

n

f (z)1/k f 2 (z)1/k f n (z)1/k =z· · · · · · · z f (z)1/k f (n 1) (z)1/kn June 25, 2015

1

.

9.2.31

44

Chapter 9.

Dynamics of polynomials

The general term of this product is x ✓⇣ ⌘k ⇣ ⌘◆1/km (m 1) (m 1) m f (z) 1+g f (z) f m (z)1/k = ⇣ ⌘1/km 1 ⇣ ⌘1/km 1 f (m 1) (z) f (m 1) (z) ✓ ⇣ ⌘◆1/km (m 1) = 1+g f (z) 9.2.32

Thus if we can show that there exists r > 0 such that when |z|  r we have |g(f (m 1) (z))| < 1, we can use the principal branch of the root in all terms of our product ⇣ ⌘1/kn 1/k 'n (z) := z · (1 + g(z)) · · · 1 + g(f (n 1) (z)) 9.2.33

to lift the indeterminacy of the root. To accomplish this, first find r1 > 0 and C such that |g(z)| < C|z| for |z|  r1 . Let r2 be the positive root of the equation xk

1

(1 + Cx) = 1,

9.2.34

and set r := min(r1 , r2 , 1/(2C)). Then if |z| < r, we have |f (z)| = |z|k |1 + g(z)|  |z|rk so |f

m

1

(1 + Cr)  |z|,

(z)|  r for all m, and |g(f

(m 1)

(z))|  C|f

(m 1)

(z)| 

9.2.35

C 1 = . 2C 2

9.2.36 1/km

Thus for |z| < r the principal branch of 1 + g(f (m 1) (z)) is well defined for all m. Note that the maximum value of | ln(1+w)| for |w|  1/2 is ln 2, achieved at w = 1/2. The infinite product then converges, since 1 ln 2 ln |1 + g(f (m 1) (z))|  m , m k k P m and the series m ln 2/k converges. The uniqueness is an easy formal power series argument. ⇤ ln |1 + g(f

(m 1)

(z))|1/k

m

=

9.2.37

¨ ttcher coordinates 9.3 Green’s functions, Bo The argument given in the proof of Theorem 9.2.15 is purely local; we need to understand the actual domain of the B¨ ottcher coordinate '. That only makes sense if f is itself a global map. Unlike linearizing coordinates, B¨ ottcher coordinates do not in general extend to the entire basin of attraction of a superattracting fixed point. But there is a Green’s function G that does, at least if you allow the value 1. June 25, 2015

9.3

Green’s functions and B¨ ottcher coordinates

45

The Green’s function of Proposition and Definition 1.5.1 is not defined in dynamical terms. Here we will show that when we are in a dynamical context with a superattracting fixed point, the Green’s function is closely related to the dynamics, by equation 9.3.2. Proposition in and Definition 9.3.1. (Green’s function in dynamics) Let p : C ! C be a polynomial, and suppose that p(0) = 0 and p(z) = z k (1 + g(z)), with k 2 and g(z) 2 O(z) near 0. Let W be the basin of attraction of 0. Then the sequence of functions n 7! Gn : W ! [ 1, 1) defined by 1 ln |p n (z)| 9.3.1 kn converges uniformly on compact subsets of W , with logarithmic poles on the subset Z ⇢ W of points z such that p n (z) = 0 for some n. Moreover, the limit G := limn!1 Gn satisfies Gn (z) =

G(p(z)) = kG(z).

9.3.2

Since the functions Gn take values in [ 1, 1), we give this set the metric of [0, 1), for instance using the exponential. Note that G takes its values in [ 1, 0), and tends to 0 on the boundary of W . Remark In the component of W0 ⇢ W containing the superattracting fixed point 0, the function G is the negative of the Green’s function defined in Chapter 1, for the domain W0 and the base point 0. 4 Proof Let V and ' be as in Theorem 9.2.15, and let 'n be as in equation 9.2.33. Clearly on the domain of 'n we have Gn = ln |'n |. So on any compact subset X ⇢ W such that p m (X) ⇢ V , we have 1 ln |'n (p m (z))|. 9.3.3 km We can take the limit as n tends to infinity, showing that the sequence n 7! Gn converges uniformly on X. The functional equation follows by taking logarithms of '(p(z)) = '(z)k . ⇤ Gn (z) =

For any ⇢ satisfying 0 < ⇢  1, define W⇢ ⇢ W to be the connected component of { z 2 W | G(z) < ln ⇢ } containing 0, and let ⇢0 be the supremum of the ⇢ such that W⇢0 contains no critical point of p other than 0. Proposition in 9.3.2 The B¨ ottcher coordinate ' extends to an analytic isomorphism from W⇢0 to the disc De⇢0 of radius e⇢0 . If W contains no critical point of p other than 0, the map ' is a conformal map from the immediate basin of 0 to the unit disc. June 25, 2015

46

Chapter 9.

Dynamics of polynomials

Proof There exists n sufficiently large so that W⇢n0 is contained in the domain of definition of '. Extend ' successively to W⇢n

1

0

⇢ W⇢n 0

⇢ · · · ⇢ W⇢0 .

9.3.4

! W⇢n0

9.3.5

2

It is enough to carry out the first extension. The restriction p : W⇢n 0

1

is a covering map of degree k ramified exclusively at 0, since W⇢n 0

1

\ Critp = ;,

9.3.6

and since z 7! z k , as a map D⇢n 1 ! D⇢n0 , is also a ramified cover ramified 0 exclusively at 0. There is only one such ramified covering space up to automorphisms, and thus there are precisely k di↵erent maps gi such that the diagram W⇢n 0

p# W⇢n0

1

gi

!

D⇢n 0

1

9.3.7

# z 7! z k

'

!

D⇢n0

commutes; the gi di↵er by postmultiplication by a kth root of 1. One of these gi coincides with ' on W⇢n0 ⇢ W⇢n 1 , giving the desired extension. ⇤ 0

The B¨ ottcher coordinate at infinity for a polynomial When p is a polynomial of degree d 2, the point 1 is a superattractive fixed point. In that case the B¨ ottcher coordinate 'p (with values near 1 rather than 0) can be defined by 1/dn

'p (z) := lim (p n (z)) n!1

,

9.3.8

where the same argument as in the proof of Theorem 9.2.15 shows that the limit exists in a neighborhood of 1, and defines there an analytic function 'p such that d

'p (p(z)) = ('p (z)) .

9.3.9

The function 'p is illustrated in Figure 10.4.2. Exercise 9.3.3 Suppose that p(z) = z d + ad 1 z d 1 + · · · + a0 . Adapt the proof of Theorem 9.2.15 to show that the limit 9.3.8 exists in the region Pd 1 |z| > 1 + i=0 |ai |. } We will be vitally interested in finding a maximal natural domain for 'p ; Proposition 9.3.2 can easily be adapted to the present context. Set ln+ (x) := sup(ln(x), 0). June 25, 2015

9.4

Extending

p

47

Theorem in 9.3.4. (Green’s function at infinity) 1. Let p : C ! C be a monic polynomial of degree d > 1, with critical set Critp . Then the basin of infinity is C Kp , and the limit 1 + n ln |p (z)| n!1 dn exists uniformly on compact subsets of C Kp . Gp (z) := lim

9.3.10

2. The Green’s function Gp can be extended to a continuous subharmonic function on all of C by setting it to be 0 on Kp . 3. Gp satisfies Gp p(z) = dGp (z).

9.3.11

4. Set ⇢ := sup!2Critp Gp (!). Then the B¨ ottcher coordinate 'p can be extended to a conformal mapping { z 2 C | G(z) > ⇢ } ! C

De⇢

5. If Critp ⇢ Kp , then 'p is a conformal map C

Kp ! C

D.

9.3.12

Proof This is an adaptation of Propositions 9.3.1 and 9.3.2. ⇤

9.4 Extending ψ p to S 1 The local connectivity of Julia sets is a central issue in complex dynamics, because (as we see in Chapter 10) “locally connected” is more or less synonymous with “understandable”. Largely due to the work of Yoccoz and Lyubich, the local connectivity of Julia sets of polynomials is understood in considerably greater generality than the cases discussed here, but Theorem 9.4.1 is good enough for our present purposes.1 Suppose that p is a polynomial with Kp connected, so that the map 'p : C Kp ! C D is an isomorphism. The inverse of 'p is the major actor in this subsection, and to lighten notation we define p := 'p 1 . Our main tool for understanding Jp is to extend p to S 1 . (Note that p does not always extend; see Corollary 18.6 in Milnor’s book [79].) 1

A standard result from point-set topology asserts that the image of a locally connected set by a continuous map is locally connected, so if p extends to S 1 , then Jp is locally connected. By a theorem of Caratheodory, the converse is also true: if Jp is locally connected, then p extends. Yoccoz proves local connectivity directly and then uses Caratheodory’s theorem to show that p can be extended. We will construct the extension directly. June 25, 2015

48

Chapter 9.

Dynamics of polynomials

Theorem in 9.4.1 Let p be a polynomial of degree d critical point of p either

2 such that every

1. is attracted to an attracting cycle other than infinity, or 2. has a finite orbit containing a repelling cycle, or 3. is attracted to a parabolic cycle. Then

p

:C

D!C

Kp extends to S 1 .

The hypothesis implies that all the critical points belong to Kp , so Kp is connected. The proof is fairly easy if all critical points are attracted to attracting cycles, more delicate if all critical points are attracted to attracting cycles or land on repelling cycles, and more delicate yet if some critical points are attracted to parabolic cycles. In all three cases, the theme of the proof is the same: we choose some number R > 1 and consider the sequence of maps n 7! n : R/Z ! C given by n (t)

:=

p (R

1/dn 2⇡it

e

)=p

1

(

n 1 (dt)).

9.4.1

It should be clear that the uniform convergence of this sequence implies the theorem. This will require knowing that p 1 is sufficiently contracting for an appropriate metric; finding the metric is the crux of the proof. In the hyperbolic case (case 1), we can do this using an appropriate Poincar´e metric. In the subhyperbolic case (cases 1 and 2), the situation is more complicated; a priori, it looks impossible to find such a metric: any critical point landing on a repelling cycle will belong to the Julia set, so p will be infinitely contracting at such a point, so how can p 1 be contracting? Clearly it can’t for any smooth metric: we will need to introduce singular metrics. One way to introduce singular metrics uses orbifolds; another uses an ad hoc construction where a metric is pieced together by hand. The first method is undoubtedly more elegant, but we will use the second, because we can apply this approach to case 3, where we know of no such elegant method. It is instructive to see the piecing together in a simpler setting. In all cases, choose a number R > 1 and set W :=

z 2 C |'p (z)| < Rd

9.4.2

.

Extending ψp to S1 : the hyperbolic case (case 1) Let Z be the set of attracting cycles; find a neighborhood V0 of Z such that p(V0 ) is relatively compact in V0 . Set Vn := p n (V0 ). Since all the finite critical points are attracted to Z, there is some first Vn that contains all the critical values. Now choose R > 1 and set U := C

Vn

{z 2 C

Kp | |'p (z)|

June 25, 2015

R }.

9.4.3

9.4

Extending

p

49

With this construction, U 0 := p 1 (U ) is a relatively compact subset of U , and p : U 0 ! U is a covering map (see Figure 10.4.5). For (z, ⇠) 2 T U , we denote by |(z, ⇠)|U the length of the vector with respect to the hyperbolic metric of U , and similarly for U 0 . Since p : U 0 ! U is a covering map, |(z, ⇠)|U 0 = |(p(z), p0 (z)⇠)|U . Since U 0 is relatively compact in U , there exists a constant C < 1 such that for all (z, ⇠) 2 T U 0 we have |(z, ⇠)|U  C|(z, ⇠)|U 0 .

9.4.4

|(z, ⇠)|U  C|(z, ⇠)|U 0 = C|(p(z), p0 (z)⇠)|U .

9.4.5

Hence for all (z, ⇠) 2 T U , we have 0

Measured in the metric of U , the map p : U 0 ! U is strongly expanding. Denote by ↵n,t the arc that is the image of ⇢ 7!

p

⇢e2⇡it ,

n+1

R1/d

n

9.4.6

 ⇢  R1/d ,

and denote by ln,t the length of ↵n,t . Write ln := supt2R/Z ln,t . Since p (z d ) = p( p (z)), we have p(↵n,t ) = ↵n 1,td . It follows that P

ln,t  Cln

1,td ,

and hence that ln  Cln

1.

9.4.7

Thus n ln is a convergent series. It follows that the family of mappings (⇢e2⇡it ) converges uniformly as ⇢ de⇢ : R/Z ! U given by ⇢ (t) := creases to 1. This proves Theorem 9.4.1 when all critical points are attracted to attracting cycles.

Extending ψp to S1 : the subhyperbolic case Suppose all critical points of p are either attracted to attracting cycles or are strictly preperiodic, so their orbit is eventually a repelling cycle. Let U1 ⇢ C be the set U constructed in equation 9.4.3, such that U10 := p 1 (U1 ) is relatively compact in U1 and contains all the critical points attracted to attracting cycles. The map p : U10 ! U1 will have critical points (unless we actually are in case 1); let Critp be the set of its critical points, and set [ Pp := pn (Critp ). 9.4.8 n>0

Our hypotheses imply that Pp is finite. Set U := U1

Pp

and U 0 := U10

p

1

(Pp );

9.4.9

with these definitions, U 0 ⇢ U and p : U 0 ! U is a covering map, so for (z, ⇠) 2 T U 0 , we have |(z, ⇠)|U < |(z, ⇠)|U 0 =

p(z), p0 (z)⇠

June 25, 2015

U

.

9.4.10

50

Chapter 9.

Dynamics of polynomials

But U 0 is not relatively compact in U , so unlike equation 9.4.5, there is no constant C < 1 in this inequality.2 In fact, p cannot be strongly expanding near a critical point for any smooth metric, and to use a proof like the one for the hyperbolic case, we will have to allow more general metrics. As a model, consider the mapping z 7! w = z 2 . It is of course not expanding near 0 (it is in fact infinitely contracting) for any choice of smooth metrics in domain and codomain, but if we use the metrics |dw| |dz| in the domain and m p , m > 1 in the codomain, 9.4.11 2 |w| p the map is expanding by the factor m. The metric |dw|/2 |w| is a perfectly good metric; in fact, it is the metric of a Euclidean cone with cone angle ⇡. The origin is at finite distance from other points of C, just as the vertex is at finite distance from the other points of the cone. We will piece together several metrics: the Poincar´e metric of the appropriate analogue of the region U above, and singular metrics |dw|/|w|1/⌫ for appropriate integers ⌫ in discs around the points of the orbits of the strictly preperiodic critical points. Define the function ⌫ : U ! N to be ⌫(z) :=

p

lcm

n (y)=z

degy pon .

9.4.12

It should be clear that ⌫(z) = 1 implies z 2 / Pp , and that ⌫ is well defined: a point of Critp can appear at most once in any orbit, however long, so Y ⌫(z)  degz p. 9.4.13 z2Critp

This “orbifold structure” on U is discussed in Proposition 10.1.8, where the situation is complicated by the possible presence of superattracting cycles, forcing us to accept 1 as a value of ⌫. Lemma 9.4.2 There exist 1. a number C < 1 2. disjoint neighborhoods Dw of the points w 2 Pp [ Critp that satisfy p(Dw ) ⇢ Dp(w) 3. local coordinates ⇣w in each Dw such that with respect to the metric µw :=

|d⇣w | , |⇣w |1 1/⌫(w)

2

This is just as well, since if there were such a constant, the to a limit in U , and they don’t. June 25, 2015

9.4.14 n

would converge

9.4

Extending

p

51

the map p : Dw ! Dp(w) is expanding by at least 1/C. More precisely, for all z 2 (Dw \ p 1 (Dp(w) )) {w} and all ⇠ 2 Tz Dw , |(z, ⇠)|µw  C p(z), p0 (z)⇠

µp(w)

.

9.4.15

Proof of Lemma 9.4.2 Use linearizing coordinates at the points of the postcritical cycles. Of course the expression of p in these coordinates is linear. If a cycle has length q and multiplier , we can normalize the linearizing coordinates ⇣w so that all Dw are unit discs for the coordinate ⇣w , and the expression of p in these coordinates is ⇣w 7! 0 ⇣p(w) for some number 0 satisfying ( 0 )q = . The points of the cycle all have the same orbifold weight; if they have orbifold weight ⌫, then with these coordinates we find an expansion under p by | |1/q⌫ . Denote by (Z) the multiplier of Z and set 1 1/q⌫ := min (Z) . 9.4.16 cycles Z⇢P C of length q

Now pull these coordinates back to discs around the strictly preperiodic points of Pp [Critp , extracting appropriate roots as needed. Suppose Dp(w) and ⇣p(w) are chosen, and to lighten notation write dw := degw p. We will show how to choose ⇣w so that if p : Dw ! Dp(w) is written dw ⇣p(w) = a⇣w

for some a 2 R,

9.4.17

the map is appropriately expanding. The condition that p should be expanding by 1/C is 1 1 . C |⇣w |1 1/⌫(w)

adw |⇣w |dw 1 dw 1 1/⌫(p(w)) |a⇣w |

After various cancellations this gives

1 . C This inequality is true in the unit disc with the origin omitted if a1/⌫(p(w)) dw |⇣w |(dw /⌫(p(w))) a = (Cdw ) since dw ⌫(p(w))

1/⌫(w)

⌫(p(w))

,

1  0. ⌫(z)

9.4.18

9.4.19 9.4.20 9.4.21

(At the origin, the left side may have a pole, which presents no problem.) ⇤ Lemma 9.4.2 Denote by µU the Poincar´e metric on U , and by µD the metric on [ D := Dw that is µw on each Dw . w2Pp [Critp

June 25, 2015

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Chapter 9.

Dynamics of polynomials

Lemma 9.4.3 There exists a constant M such that for the metric µ := inf{µU , M µD },

9.4.22

the polynomial p is strongly expanding, in the sense that there exists a constant C 0 < 1 such that for any rectifiable path ⇢ U 0 we have Lengthµ ( )  C 0 Lengthµ p( ) .

Proof of Lemma 9.4.3 Start by defining M . Clearly p compact closure in U 0 , so we may set ⇣ ⌘ µD p(z), p0 (z)⇠ 1 := C inf . M µU (z, ⇠) z2p 1 (D) D,

9.4.23

1

(D)

D has

9.4.24

⇠2Tz U

Taking M larger if necessary, we may assume that the subset Y ⇢ U 0 where CµU  M µD has compact closure in U . On Y the map p is expanding for µU by some constant 1/C 00 > 1. Note that Y contains all points where the minimum defining µ is realized by µU . We must show that every vector (z, ⇠) 2 T U 0 is expanded under p by a fixed factor (in fact, by C 0 := min(1/C, 1/C 00 )). There are four cases to consider, depending on whether the minimum at z and the minimum at p(z) is realized by µU or M µD : • If at both z and p(z) the minimum is realized by M µD , the result follows from the choice of µD , as asserted by Lemma 9.4.2; in that case, p is expanding by 1/C. • If at both z and p(z) the minimum is realized by µU , then z is in Y and p is expanding by at least 1/C 0 . • If the minimum at z is realized by µU and the minimum at p(z) is realized by M µD , there are two possibilities. If z 2 p 1 (D) D, the choice of M guarantees that p is expanding by at least 1/C. If both z and p(z) are in D, then |(z, ⇠)|U  |(z, ⇠)|MµD  C|(p(z), p0 (z)⇠)|M µD .

9.4.25

|(z, ⇠)|MµD  |(z, ⇠)|U  C 0 |(p(z), p0 (z)⇠)|U .

9.4.26

• The case where the minimum at z is realized by M µD and the minimum at p(z) is realized by µU again subdivides into two. If z 2 Y , then If z 2 / Y , then

|(z, ⇠)|MµD  C|(z, ⇠)|U  C|(p(z), p0 (z)⇠)|U .

9.4.27

Thus in all cases the map p is expanding by at least a factor of 1/C . ⇤ 00

We can now apply the same argument as in equations 9.4.6 and 9.4.7, to prove that the sequence n 7! n converges. ⇤ Subhyperbolic case June 25, 2015

9.4

Extending

p

53

Extending ψp to S1 : the parabolic case Now we add the possibility that some critical points are attracted to parabolic cycles. The proof is elaborate, but even in the simplest case where p(x) = x2 + x I do not know of a simpler argument. Let p be a polynomial such that all critical points are either attracted to attracting cycles, or have finite orbits ending in repelling cycles, or are attracted to parabolic cycles. As above, we will define an appropriate metric on an appropriate set U for which p is expanding, but we will not achieve the strong expansion needed for geometric convergence of the n . The n do converge, but more slowly than geometrically, and the argument uses a variant of the Banach fixed point theorem that I have not seen elsewhere. To simplify matters, replace p by an appropriate iterate, so that all the parabolic cycles consist of fixed points with derivative 1; denote by Y the set of these fixed points. Remark We lose nothing by making this modification: the Julia sets of p and of p k coincide. If we had wanted to, we could have made all attracting cycles fixed also, but that wouldn’t have much simplified the proof. 4 Lemma 9.4.4 1. There exist a neighborhood Vy of each fixed y 2 Y , a local coordinate ⇣ on Vy , and an integer m 1 such that in this local coordinate p is written ⇣ ⌘ fy (⇣) = ⇣ 1 + ⇣ m + O(⇣ m+1 ) . 9.4.28 2. There exists ⇢ > 0 such that in the subset of Vy n o Wy+ := |⇣ m |  ⇢, Re (⇣ m ) > 0 ,

the map fy satisfies |fy0 | > 1, whereas the compact subset n o Wy := |⇣ m |  ⇢, Re (⇣ m )  0

9.4.29

9.4.30

satisfies

p(Wy ) ⇢ W y [ {y}.

9.4.31

Proof of Lemma 9.4.4 The number m is the order of contact of the graph of p and the diagonal in C ⇥ C at (y, y). (If m = 0, the curves intersect transversely, m = 1 means an ordinary tangency, etc.) Clearly for some c 6= 0 we have p(y + u) = y + u + cum+1 + O(um+2 ). June 25, 2015

9.4.32

54

Chapter 9.

Dynamics of polynomials

If we set ⇣ := au with am = c, it is easy to see that the expression of p in the coordinate ⇣ is f (⇣) = ⇣ 1 + ⇣ m + O ⇣ m+2 ;

9.4.33

we write f rather than fy because y is fixed in this discussion. Since f 0 (⇣) = 1 + (m + 1)⇣ m + O(⇣ m+1 ), clearly |f 0 (⇣)| > 1 if Re (⇣ m ) > 0 and |⇣| > 0 is so small that the term O(⇣ m+1 ) is negligible.

To see that p(Wy ) ⇢ W y [ {y}, it is easiest to make the change of variables w = 1/⇣ m and to study the situation near infinity. The change of variables should be understood as occurring in a neighborhood of each component of Wy , mapping it to the region ⇢ 1 Re w  0, |w| . 9.4.34 ⇢ The expression of f in the variable w is of the form w 7! w m + O(w1/m ); clearly by taking ⇢ sufficiently small, f maps this region into its interior (but maps 1 to itself). ⇤ Lemma 9.4.4 There exists n such that [y p n (Wy ) contains all the critical values attracted to the parabolic cycles. Recall that in case 1 (the hyperbolic case), we constructed a set U as the complement of neighborhoods of the attracting cycles. This subset was called U1 in the subhyperbolic case, and a further subset of U1 , called U , was constructed by removing Pp from U1 (see equation 9.4.9). Now we denote by U2 the set called U in case 2. Set [ U := U2 p n (Wy ) and U 0 := p 1 U. 9.4.35 y

Once more we have U ⇢ U , and p : U 0 ! U is a covering map, hence expanding for the Poincar´e metric µU of U ; once again U 0 is not relatively compact in U , so there is no constant of expansion. Recall that all the parabolic cycles consist of fixed points with derivative 1, and that the set of these fixed points is denoted Y . We will cobble together a metric on neighborhoods of the points y 2 Y . Define B := [y Wy+ , and denote by µB the restriction of the metric |d⇣y |. Choose a number M satisfying 0

sup

M> z2p

1 (B)

B

µB

µU (z, ⇠) (p(z), p0 (z)⇠

9.4.36

and define µ := inf{µU , M µB },

9.4.37

which is an infinitesimal metric on the space A := U [ B. With the associated path metric dµ , the set A becomes a complete metric space. For this June 25, 2015

9.4

Extending

p

55

metric dµ , the polynomial p is expanding, but not strongly expanding, and we need a di↵erent argument. We will use the following rather surprising generalization of the Banach fixed point theorem, which deserves to be better known. The Banach fixed point theorem is the case where h(s) = Cs for some C < 1. Theorem in 9.4.5 (Generalized Banach fixed point theorem) Let (X, d) be a complete metric space, and let h : R+ ! R+ be a continuous increasing function such that h(s) < s for all s > 0 and

lim s

s!1

h(s) = 1.

Let f : X ! X be a map such that for all x, y 2 X we have ⇣ ⌘ ⇣ ⌘ d f (x), f (y)  h d(x, y) .

9.4.38

Then f has a unique fixed point z 2 X, and for any x0 , the sequence x0 , x1 = f (x0 ), . . . , xn+1 = f (xn ), . . . converges to z. Proof Clearly if f has a fixed point, it is unique, and clearly if the sequence x0 , x1 = f (x0 ), . . . , xn+1 = f (xn ), . . . converges, then the limit is a fixed point. So it is enough to prove that the sequence converges. Set l := d(x0 , x1 ) and find L such that L h(L) > l. Let us see by induction that for all n 1 we have d(x0 , xn )  L. Indeed, it is true when n = 1, since L > l. Suppose it is true for n. Then (using equation 9.4.38) d(xn+1 , x0 )  d(xn+1 , x1 ) + d(x1 , x0 )  h(d(xn , x0 )) + l  h(L) + l < L. Now for n and m we have

d(xn+m , xn )  h n (d(xm , x0 ))  h n (L).

9.4.39

The sequence n 7! h n (L) tends to 0, since it is a decreasing sequence of positive real numbers, so it must converge, and the limit must be a fixed point of h, and hence must be 0. Thus our sequence is Cauchy, hence convergent. ⇤ Recall from equation 9.4.1 the definition of n (t)

:=

p (R

n

1/d

n:

e2⇡it ).

9.4.40

To apply Theorem 9.4.5, we need to specify a complete metric space X and a map f : X ! X satisfying the hypotheses. There are various choices, such as parametrized curves in appropriate homotopy classes, but the simplest is to take X to be the closure of the sequence n 7! n in the uniform topology for the metric dµ on A; see equation 9.4.37. Lemma 9.4.6 implies that the map n 7! n+1 extends to the closure, and that the June 25, 2015

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extension satisfies the hypotheses of Theorem 9.4.5. Let us denote by deµ the metric induced on X by the metric dµ on A. Lemma 9.4.6 Let h : R+ ! R+ be the function defined by h(s) := sup Lengthµ ( 0 ),

9.4.41

where the supremum is taken over all components 0 of p 1 ( ), where ⇢ A is of length  s. Then h is a continuous increasing function satisfying h(s) < s for all s > 0, and lims!1 s h(s) = 1. Proof Clearly h is continuous and increasing, and for all s > 0 it satisfies 0 < h(s) < s. Choose s0 > 0 and let k be an integer. Then we have h(ks0 )  kh(s0 ). It easily follows that lims!1 s h(s) = 1. ⇤ Lemma 9.4.6 It should be clear that deµ (

n+1 , n )

⇣  h deµ (

⌘ .

n, n 1)

9.4.42

So the map n 7! n+1 extends to X, and satisfies the hypotheses of Theorem 9.4.5, showing that the sequence n 7! n converges uniformly. This completes the proof of Theorem 9.4.1 ⇤

Components of the interior of Kp Our next result, Proposition 9.4.7, is a weak version of a much better result: Sullivan’s no-wandering domain theorem, discussed in Appendix C7.1. But it is easy to get with our present tools, so we present it anyway. Proposition in 9.4.7 Let p be a polynomial with Kp connected, such that all critical points are either attracted to attracting or parabolic cycles, or have finite orbits. Then Kp consists of the basins of attraction of the attracting and parabolic cycles. If there are no attracting or parabolic cycles, then Kp has empty interior. Proof The key issue is that p is analytic, so its Jacobian is |p0 |2 , and the metrics µ that we constructed in the proof of Theorem 9.4.1 are conformal. Let us write the infinitesimal metric µ as µ(z)|dz|, and consider the element of area (µ(z))2 |dz|2 . Then the statement that p is expanding becomes µ(p(z))|p0 (z)|

µ(z),

so

2

µ(p(z)) |p0 (z)|2

2

µ(z) ,

9.4.43

so areas are also expanded. Areas of all open sets are in fact strictly expanded, since µ p(z) |p0 (z)| > µ(z) June 25, 2015

9.4.44

9.4

Extending

p

57

except at finitely many points: when z is an element of a parabolic cycle or a pole of µ. Let U be as defined in equation 9.4.35. Suppose there is a component of K p whose points are not attracted to an attracting or parabolic cycle. All such components are subsets of p 1 (U ), which has finite area, so there must be one, say V , with maximal area. The map p : V ! p(V ) is an analytic isomorphism: it is proper and a local homeomorphism, with simply connected codomain and connected domain. So the µ-area of p(V ) is strictly greater than the µ-area of V . This is a contradiction. ⇤

The Caratheodory loop Let p be a monic polynomial of degree d 2. If Kp is connected, the 1 map p := ('p ) : C D ! C Kp is an analytic isomorphism satisfying d p (w ) = p( p (w)). The external ray of Kp at angle t is the set Rp (t) :=

2⇡it ), p (re

1 < r < 1.

9.4.45

Clearly the external ray at angle t is mapped by p to the external ray at angle dt. Suppose that Kp is also locally connected. Then 'p extends continuously to S 1 = @D: we call this extension p (t)

=

2⇡it ) p (e

9.4.46

the Caratheodory loop of p (the main actor of Section 10.4). It is a continuous surjective map R/Z ! Jp satisfying p (dt) = p( (t)). Theorem in 9.4.8 Let p be a polynomial with Kp connected and locally connected, and let z0 2 Jp be a fixed point. Then p 1 (z0 ) is a finite set, and all the elements of p 1 (z0 ) are periodic of the same period under t 7! (deg p)t. Proof Set d := deg p and denote by fd : R/Z ! R/Z the map t 7! dt. The set X := p 1 (z0 ) is a closed subset of R/Z, and fd is injective on X. Indeed, if two di↵erent external rays of Kp land at z0 , and are both mapped to the same ray, then there are points on these rays arbitrarily close to z0 that are mapped to the same point. But |p0 (z0 )| 1, so p is a local di↵eomorphism near z0 , so this cannot happen. Clearly fd (X) is a subset of X, since the image of a ray landing at z0 is a ray landing at z0 . Thus there is a map g : fd (X) ! X given by fd 1 . Moreover, g is continuous: a closed subset C ⇢ X is compact, so g 1 (C) = fd (C) is compact. June 25, 2015

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Since g is continuous on a compact set, it is uniformly continuous: there exists ✏ > 0 such that |x y| < ✏ implies |g(x) g(y)| < 1/(2d); we may suppose ✏ < 1/2. Cover fd (X) by n intervals I1 , . . . , In each of length < ✏. Then only one component of fd 1 (X) can contain any points of X; points of distinct components are at least 1 d

✏ 1 > d 2d

9.4.47

apart. Thus X is covered by n intervals of length ✏/d; of course these intervals cover fd (X) also, so we can repeat the argument, to cover X by n intervals of length ✏/d2 , and so forth. Thus X can be covered by n intervals of length at most ✏/dm for every m > 0, and this implies that X consists of at most n points. The corresponding rays are arrayed in circular order around z0 ; it is then easy to see that they are all periodic of the same period and are rotated by ↵/ , forming some number k of cycles. ⇤ Exercise 9.4.9 Call the regions bounded by consecutive rays sectors. Show that each cycle of sectors must contain a critical point. } Thus the number k of cycles of rays at a fixed point is at most the number of critical points of p; in particular, it is at most d 1. For quadratic polynomials there is at most one cycle at a fixed point. The fixed point z0 must be repelling or indi↵erent; it can’t be both attracting and an element of Jp . If it is indi↵erent, the multiplier must be a root of unity. This is not so easy to see; it requires knowing that otherwise it is either linearizable, in which case z0 2 / Jp , or not linearizable, in which case Jp is not locally connected.

9.5 External rays at rational angles land Let p be a monic polynomial with Kp connected, and let of the B¨ ottcher coordinate. If lim

r&1

2⇡i✓ ) p (re

= u,

p

be the inverse 9.5.1

we say that the external ray of Kp at angle ✓ lands at u. If the limit exists, we may say that the ray lands, without specifying where it lands. If Kp is locally connected, so that the Caratheodory loop exists, all external rays land. By Fatou’s theorem (see for instance theorem 11.20 in [93]), even if Kp is not locally connected, almost every ray lands. But this gives no information about whether any specific ray lands. June 25, 2015

9.5

Rational rays land

59

Proposition in 9.5.1 (Rays at rational angles land) Let p be a monic polynomial of degree d with Kp connected; let p : C D ! C Kp be the inverse of the B¨ ottcher coordinate. Then for every ✓ 2 Q/Z, the limit z✓ := lim

r&1

p

re2⇡i✓

exists.

9.5.2

If ✓ is periodic of period exactly m, i.e., m > 0 is the smallest integer such that dm ✓ = ✓, then z✓ is periodic of period k dividing m. Let ↵ := (p k )0 (z✓ ) be the multiplier of the cycle. Then either |↵| > 1 (the cycle is repelling), or ↵ is a primitive (m/k)th root of unity. Proof If ✓ 2 Q/Z, then the sequence ✓, d✓, d2 ✓, . . . eventually repeats, and since an inverse image of a ray that lands also lands, it is enough to consider the case ✓ periodic under ✓ 7! d✓. Suppose dm ✓ = ✓, so the ray at angle ✓ is fixed under the mth iterate pe := p m of p; then pe is of degree d˜ := dm . We will first show that a fixed ray for pe lands at a fixed point, which is either repelling or indi↵erent with derivative 1. Note that the inverse B¨ ottcher coordinate p : C D ! C Kp for p is also the inverse B¨ ottcher coordinate for pe. Set n o 2⇡i✓ Il := 1/d˜l  log r  1/d˜l 1 p re 9.5.3 ⇣ l ⌘ ˜ zl := Il \ Il 1 = p ed e2⇡i✓ .

Then pe maps Ik homeomorphically to Ik 1 (this is where we use that the ray at angle ✓ is fixed by pe), and also isometrically if we use the Poincar´e metric of C Kp . Thus these Il all have the same Poincar´e length, and their Euclidean length tends to 0 as l ! 1. Let u be a point where some subsequence zli accumulates. Then the Euclidean distance between zli and p(zli ) tends to 0, so u = pe(u). But the set of accumulation points of the ray is connected, and pe has only finitely many fixed points, so the ray lands at a fixed point z✓ . We still need to show that if z✓ is not repelling (and hence obviously indi↵erent), then it must have multiplier 1; this is the hardest part of the argument. Set := p˜0 (z0 ), so that in particular = (p m )0 (z0 ) = ↵m/k , and suppose | | = 1. Write = ei' , and write the dynamical variable z in polar coordinates centered at z✓ , i.e., set z := z✓ + ⇢ei! . In these coordinates, pe is written (⇢, !) 7! (⇢0 , ! 0 ) with ⇢0 = ⇢ + o(1) and ! 0 = ! + ' + 2⇡n + o(1) for some integer n; since the angle ' was only defined up to multiples of 2⇡, we may choose it so that n = 0. See Figure 9.5.1. June 25, 2015

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Dynamics of polynomials

!

Figure 9.5.1

!

Top: The map

(⇢, !) 7! (⇢, ! + ').

!0

Bottom: In polar coordinates, the map pe is approximately

!0 + '

(⇢, !) 7! (⇢, ! + ').

⇢

If ' 6= 0, this guarantees that the fixed point z✓ of pe has a neighborhood that is mapped to a strictly larger neighborhood, so the multiplier = pe0 (z✓ ) satisfies | | > 1.

⇢ pe

Suppose by contradiction that ' 6= 0. Then one lift of our fixed ray will be a simple arc tending to infinity, asymptotic to the !-axis ⇢ = 0, negative or positive depending on the sign of '. Assume that ' < 0 (the case ' > 0 is identical). In the original coordinates, this means that the ray spirals to z✓ counterclockwise (remember that under pe, points on the ray iterate away from z✓ , towards infinity). For !0 sufficiently large, a segment 0  ⇢  ⇢0 of the line ! = !0 will meet the ray first at (⇢0 , !0 ), and the image of that segment will be very nearly a segment of the line ! = !0 + ' < !0 . Thus the region in the z-plane corresponding to the region bounded by the ✓-axis, the lift of the ray, and where ! > !0 , maps to the strictly larger region corresponding to that bounded by the image of the segment, the ✓-axis, and the lift of the ray. Thus | | > 1, contradicting our assumption that | | = 1. Thus ' = 0, so = 1. Now go back to the original p of degree d; the above shows that z✓ is periodic under p of some period k dividing m, and that (p k )0 (z✓ ) is a (m/k)th root of unity. It remains to show that it is a primitive (m/k)th root of unity. Suppose not; then (p k )0 (z✓ ) is a root of unity of some order n dividing m/k, so there exists an integer l such that nkl = m. At z✓ , the polynomial p k has a Fatou flower with ⌫n petals for some ⌫ 1, and p nk maps each repelling petal to itself. The external ray at angle ✓ must approach z✓ in some repelling petal, and thus the image of that ray by p nk approaches z✓ in the same petal, but it is not the same ray: it is either clockwise or counterclockwise from the ray at angle ✓. So the images of the ray at angle ✓ by p

nk

,p

2nk

,p

3nk

,...

9.5.4

are circularly ordered within one repelling petal, and are all distinct. This contradicts the statement that the ray at angle ✓ is periodic of period m = nkl. ⇤ Proposition 9.5.1 June 25, 2015

10 Rational functions We now come to the second of Thurston’s theorems treated in this book: the topological characterization of rational functions, Theorem 10.1.14.

10.0 Introduction The combinatorial aspect of iteration theory for rational functions concerns postcritically finite maps: maps whose critical points have finite orbits. Think for instance of f (z) = z 2

1,

10.0.1

with critical points 1, 0 and critical orbits 1 7! 1, 0 7! g(z) = z + i, 2

with critical points 1, 0 and critical orbits 1 7! 1,

0 7! i 7!

1 + i 7!

i 7!

1 + i.

1 7! 0, or of

10.0.2

10.0.3

Theorem 10.1.14 answers essentially all questions about such mappings; more precisely, it reduces them to (sometimes difficult) topological problems. It is easy to see what a topological postcritically finite map should be: an orientation-preserving ramified covering map f : S 2 ! S 2 such that all the ramification points have finite orbits. We call such maps Thurston mappings. The polynomials f and g above are Thurston mappings, but not “interesting ones” in the present context, since they are already rational functions. To find an interesting one, consider D g, where D is the Dehn twist around a simple closed curve on C {i, 1 + i, i}, for instance one of the curves 1 or 2 shown in Figure 10.0.1. Thurston’s theorem answers the question: when does such a Thurston mapping “look like” a rational function? This is part of the philosophy that sometimes a “floppy” topological object has a natural rigid geometry. Here the floppy topological object is a Thurston mapping and the rigid object is a rational function. A Thurston mapping is “floppy” in part because a priori S 2 carries no particular complex structure. But the main source of floppiness will become clear when we say exactly what “looks like” means: f is not required to be June 25, 2015

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conjugate to a rational function, but only conjugate to a rational function “up to homotopy” rel the orbits of the critical points. Why the caveat that Thurston’s theorem answers “essentially” all combinatorial questions about rational maps? Although the theorem says that either f looks like a rational function, or there is a purely topological obstruction, these obstructions are quite difficult to understand, and almost every successful attempt to classify the obstructions is a major theorem in its own right. Theorems 10.3.1 and 10.3.3 are examples of such theorems.

1+i

Figure 10.0.1 Let g be the function g(z) = z 2 + i of equation 10.0.2, with postcritical set {1, i, 1+i, i}. Let D i be the Dehn twist around a simple closed curve i in the complement of the postcritical set. We can ask whether the maps D 1 g and D 2 g look like rational functions (in this case necessarily a polynomial, since 1 is a fixed critical point). Even with Thurston’s theorem, the question is difficult to answer. The solution was given by L. Bartholdi and V. Nekrashevych in 2005 in a 41-page paper [10].

i

1

2

0

i

This chapter has the following structure. Section 10.1 states the theorem and develops the necessary vocabulary: Thurston map, Thurston equivalence, Thurston obstruction. It also gives an outline of the proof. Sections 10.2–10.5 are devoted to one extended example: quadratic polynomials. We define a family of Thurston maps in Section 10.2; we analyze their Thurston obstructions in Section 10.3; in Section 10.4 we analyze the corresponding polynomials in some cases where there are no Thurston obstructions. In Section 10.5, we use these polynomials as “organizing centers” in a description of the Mandelbrot set. Finally, the proof of the theorem is given in Sections 10.6–10.11. Sections 10.4 and 10.5 rely on results from Chapter 9.

10.1 Thurston mappings Let S 2 be the topological sphere (thus it is floppy; we are not thinking of the sphere with any particular metric or analytic structure). Let f : S 2 ! S 2 be an orientation-preserving ramified covering map, called a branched map for short. Let Crit f be the set of critical points of f (points at which f is June 25, 2015

10.1

Thurston mappings

63

not a local homeomorphism), and define the postcritical set Pf by Pf :=

[

f

n

(Crit f ).

10.1.1

n>0

Note that n is strictly greater than 0; critical points are postcritical only if they are in the orbit of a critical point, for instance by being periodic. Definition in 10.1.1 (Thurston map) A branched map f : S 2 ! S 2 of degree d 2 whose postcritical set Pf is finite is called a Thurston map.

Exercise 10.1.2

Show that a branched map of degree d > 0 has (2d 2)

critical points, counted with multiplicity. } Thurston’s theorem on the topological characterization of rational functions applies only to Thurston mappings. In Section 10.2 we show how to generate many Thurston maps. All our examples will be topological polynomials, which are much easier to deal with than general Thurston maps. Definition in 10.1.3 (Topological polynomial) A Thurston mapping f is a topological polynomial if there exists a point 1 2 S 2 such that f 1 (1) = {1}. The point 1 is then necessarily a critical point of f , and the local degree

of f at 1 is the same as the degree d of f . Exercise 10.1.4

Let f be a topological polynomial, and D ⇢ S 2

{1}

be a topological disc, with @D \ f (Crit f ) = ;. Show that every component of f

1

(D) is a topological disc.

}

The next exercise is quite surprising at first. If f has degree d, then f has degree d , hence (2d k

k

k

2) critical points, counted with multiplicity.

One might expect the postcritical set to grow also, but it doesn’t. Exercise 10.1.5

Show that if f : S 2 ! S 2 is a Thurston map with

postcritical set Pf , then for all k

1 we have Pf = Pf k . }

June 25, 2015

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Definition in 10.1.6 (Thurston equivalence) Two Thurston maps f and g are Thurston equivalent if there exist homeomorphisms ', '0 : S 2 ! S 2 such that 1. ' and '0 coincide on Pf 2. The following diagram commutes: S2 #f S2

'0

S2 #g ' ! S2 !

10.1.2

3. ' is isotopic to '0 rel Pf Part 3 means that there exists a path t 7! 't , t 2 [0, 1] in the space of homeomorphisms from the sphere to itself, with '0 = ' and '1 = '0 , such that all 't coincide with ' on Pf . Thurston equivalence is a “floppy” notion of conjugacy: two Thurston equivalent maps are “conjugate up to isotopy rel Pf ”. The justification of this definition is that it makes Theorem 10.1.14 true. Since diagram 10.1.2 commutes, '0 necessarily sends Crit f to Crit g, hence Pf to Pg . Using ' 1 and ('0 ) 1 , we see that this is indeed an equivalence relation. Before stating Theorem 10.1.14, we need to define 2-dimensional orientable orbifolds, which will allow us to define the orbifold of f . Definition in 10.1.7 (Orbifolds and their Euler characteristics) 1. A 2-dimensional orbifold (X, ⌫) is an oriented surface X together with a function ⌫ : X ! N⇤ [ {1} that assigns 1 to all but a discrete set of points. 2. The Euler characteristic of a compact 2-dimensional orbifold (X, ⌫) is ◆ X✓ 1 (X, ⌫) := (X) 1 . 10.1.3 ⌫(x) x2X

3. The orbifold (X, ⌫) is hyperbolic if (X, ⌫) < 0.

This definition is not transparent. We give a more conceptual presentation in Chapter 11. Note that a discrete subset of a compact space is finite, and only the finitely many points x 2 X satisfying ⌫(x) > 1 contribute to the sum in equation 10.1.3. In this chapter we are concerned only with the case X = S 2 , so that ◆ X✓ 1 2 (S , ⌫) = 2 1 . 10.1.4 ⌫(x) 2 x2S

June 25, 2015

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Thurston mappings

65

1 1 Note that if ⌫(x) > 1, then 1 ⌫(x) 2 . Thus if the cardinality of the set of points where ⌫(x) > 1 is bigger than 4, the orbifold (S 2 , ⌫) is hyperbolic, so almost all examples are hyperbolic. To define the orbifold of a Thurston mapping, we will need the following proposition. We denote by degy f the local degree of f at y.

Proposition in 10.1.8 Let f : S 2 ! S 2 be a Thurston map with postcritical set Pf . There exists a smallest function ⌫f : S 2 ! N [ {1} such that 1. ⌫f (x) = 1 if x 2 / Pf .

2. For all x 2 S 2 and all y 2 f of (degy f )(⌫f (y)).

1

(x), the “weight” ⌫f (x) is a multiple

Proof If x 2 / Pf , set ⌫f (x) := 1. If x is part of a cycle1 that contains a critical point, set ⌫f (x) := 1. Otherwise, set ⌫f (x) :=

lcm

y2Crit f,f

k (y)=x

degy (f

k

),

10.1.5

where lcm is the least common multiple. If x is periodic, there are infinitely many such k. But if k1 and k2 are two such numbers, then degy (f

k1

) = degy (f

k2

),

10.1.6

since x belongs to a cycle containing no critical points. We leave it to the reader to check that this works. ⇤ Definition in 10.1.9 (Orbifold of a Thurston mapping) The orbifold of a Thurston mapping f is Of := (S 2 , ⌫f ), where ⌫f is the function defined in Proposition 10.1.8. Examples 10.1.10 (Orbifolds of some Thurston maps) 1. If f (z) = z d with |d|

2, then

Crit f = Pf = {0, 1}, ⌫f (0) = ⌫f (1) = 1, ✓ ◆ ✓ ◆ 1 1 (Of ) = 2 1 1 = 0. 1 1

2. If f (z) = z 2

2, then Crit f = {0, 1}, Pf = { 2, 2, 1}. Moreover,

⌫f ( 2) = ⌫f (2) = 2, ⌫f (1) = 1, ✓ ◆ ✓ ◆ ✓ 1 1 (Of ) = 2 1 1 1 2 2 1

10.1.7

1 1

◆

= 0.

In this chapter, a cycle is a periodic orbit; see Appendix 9. June 25, 2015

10.1.8

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3. If f (z) = z 2

1, then Crit f = {0, 1}, Pf = {0, 1, 1}. Moreover, ⌫f (0) = ⌫f ( 1) = ⌫f (1) = 1 ✓ ◆ 1 (Of ) = 2 3 1 = 1. 1

10.1.9

4. If f (z) = z 2 + i, then Crit f = {0, 1}, Pf = {i, 1 + i, i, 1}, and ⌫f (i) = ⌫f ( 1 + i) = ⌫f ( i) = 2, ⌫f (1) = 1 ✓ ◆ ✓ ◆ 1 1 1 (Of ) = 2 3 1 1 = . 2 1 2

10.1.10

The third and fourth examples have hyperbolic orbifolds. 4 Thurston maps with nonhyperbolic orbifolds are all related to trigonometric and elliptic functions; see Appendix C6. For instance, example 2 above is related to the trigonometric formula 2 cos 2✓ = (2 cos ✓)2 2.

Thurston linear transformations and f -stable multicurves We will denote the space S 2 Pf by (S 2 , Pf ) when we want to think of the elements of Pf as marked points, rather than removed points (see Figure 10.6.1). A multicurve on (S 2 , Pf ) is a set of disjoint, nonhomotopic simple closed curves on S 2 Pf such that both components of the complement of every curve 2 contain at least two points of Pf . Recall from Section 8.2 that such curves are called nonperipheral. Exercise 10.1.11 Show that a multicurve |Pf | 3 elements. }

on (S 2 , Pf ) has at most

Definition in 10.1.12 (f -stable multicurve and its Thurston linear transformation) Let f : S 2 ! S 2 be a Thurston mapping. A multicurve on (S 2 , Pf ) is f -stable if for every 2 , each component of f 1 ( ) is either homotopic rel Pf to an element of or peripheral. The Thurston linear transformation f : R ! R associated to an f -stable multicurve is defined by ! X X 1 f ([ ]) := [ ], 10.1.11 deg(f |⌘ : ⌘ ! ) 1 2

components ⌘ of f ( ) homotopic rel Pf to

where [ ] 2 R is the basis vector corresponding to

2 .

Note that in defining the Thurston linear transformation, peripheral components of f 1 ( ) don’t count. June 25, 2015

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Thurston mappings

67

For an example of an f -stable multicurve, see Figure 10.3.1; setting up an example here would take us too far afield. For an explanation of the meaning of the Thurston linear transformation f , look at Figure 10.8.1 and its caption. If we label the elements of as 1 , . . . , n , then f can be written as an n ⇥ n matrix (ai,j ), called the Thurston matrix , where f ([ i ]) =

n X

aj,i [ j ],

j=1

with ai,j =

X

1

components ⌘ of f ( homotopic rel Pf to

j)

1 deg(f |⌘ : ⌘ !

i

j)

.

10.1.12

Definition in 10.1.13 (The eigenvalue ) The matrix of the Thurston linear transformation f has nonnegative entries, hence has a real and nonnegative leading eigenvalue by the Perron-Frobenius theorem (see Appendix C1). This eigenvalue is denoted . We are now in a position to state Thurston’s theorem. Theorem in 10.1.14 (Topological characterization of rational functions) Let f : (S 2 , Pf ) ! (S 2 , Pf ) be a Thurston mapping with hyperbolic orbifold. Then f is Thurston equivalent to a rational function if and only if for every f -stable multicurve on (S 2 , Pf ) the leading eigenvalue satisfies < 1. If f is Thurston equivalent to a rational function g, then g is unique up to conjugation by an automorphism of the Riemann sphere P1 . Definition in 10.1.15 (Thurston obstruction) An f -stable multicurve 2 on (S , Pf ) with leading eigenvalue satisfying 1 is called a Thurston obstruction. Theorem 10.1.14 gives a purely topological characterization for when a Thurston mapping is equivalent to a rational function. But the hypothesis concerns all f -stable multicurves – a large and mysterious set, quite difficult to work with. It is usually an important theorem in itself to verify that some class of Thurston mappings satisfies the hypothesis.

Outline of the proof The proof of Theorem 10.1.14 is long and elaborate; it starts in Section 10.6 and ends at the end of Section 10.11. It follows the proof in [33], inspired by lectures given by Thurston in 1983. In this outline, note the similarities with the proof of the classification theorem, Theorem 8.1.3: both use the Mumford compactness theorem and the gap principle to exhibit an “obstruction June 25, 2015

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multicurve” when a sequence in Teichm¨ uller space does not converge. The Teichm¨ uller spaces we will consider will be Teichm¨ uller spaces for surfaces of genus 0, i.e., for the sphere with marked points. In Section 10.6 we define an analytic mapping f : T(S 2 ,Pf ) ! T(S 2 ,Pf ) , such that a fixed point of f corresponds to a solution of our problem, i.e., it corresponds to a conjugacy class of rational functions that are Thurston equivalent to f . This map f is called the Thurston pullback map. In Section 10.7 we compute the derivative of f and its norm; we show that kD f 2 (⌧ )k < 1, and we use this to prove Thurston rigidity: when f has a fixed point, it is unique. In Section 10.8 we prove that the criterion < 1 is necessary by showing that f cannot have a fixed point if there is an f -stable multicurve with 1. This part of the proof is well worth studying with care, because it illustrates what the Thurston linear transformation is measuring. The hard part of the proof, which takes up the remaining three sections of the chapter, is sufficiency: the existence of an equivalent rational function. It comes from the contracting mapping fixed point theorem. The obvious way to find a fixed point of a map is to start iterating the map somewhere and see if the resulting sequence converges. Thus choose ⌧0 2 T(S 2 ,Pf ) , and define the sequence i 7! ⌧i by ⌧i+1 := f (⌧i ); we will show that either this sequence converges or f admits a Thurston obstruction. In Section 10.9 we show that the norm kD f (⌧ )k of the derivative depends only on the projection of ⌧ in moduli space. It then follows that either the sequence i 7! ⌧i converges in Teichm¨ uller space, or the projections of the ⌧i in moduli space leave every compact subset of moduli space. Let ⌧i be represented by 'i : (S 2 , Pf ) ! P1 . If the projections of the ⌧i in moduli space diverge, the Riemann surfaces P1 'i (Pf ) must develop arbitrarily short geodesics by the Mumford compactness theorem. It is quite easy to show that the set of short geodesics forms an f -invariant multicurve; this uses the same gap argument used to prove the classification of homeomorphisms (Theorem 8.4.1). We may not have proved the theorem, but we have a candidate for a Thurston obstruction, and we need to study its Thurston linear transformation. Now comes the key point: one can make a “caricature” of the point ⌧i of Teichm¨ uller space by simply remembering the vector whose entries are inverses of the lengths of the short curves, i.e., the lengths of the elements of . Moreover, we can understand the action of f within this caricature: the transformation f is approximately given by the action of the Thurston linear transformation f on this vector. In particular, if < 1, the inverses get smaller and the lengths get bigger. This means that the region of June 25, 2015

10.2

Spiders

69

Teichm¨ uller space where these curves are short is repelling, and the iteration cannot enter it. Thus in order to develop short curves, we must have 1. The inequalities in Proposition 10.10.3 justify the caricature; they are the essence of the proof, which is completed in Section 10.11. The next four sections are devoted to examples showing the immense power of Theorem 10.1.14.

10.2 Thurston mappings associated to spiders In this section, we focus on Thurston mappings related to polynomials. More specifically, we will define Thurston mappings f✓ associated to every rational angle ✓ 2 Q/Z. This will involve defining spiders and extended spiders. This section and the next are largely extracted from [11]. Let ✓ 2 Q/Z be a rational angle, written in lowest terms ✓ = p/q. The Thurston map f✓ we will describe is a bit di↵erent when q is odd and when q is even. This slight di↵erence has important consequences: we will see in Theorem 10.3.8 that there never are Thurston obstructions when q is odd. Remark 10.2.1 In Sections 10.2–10.5, which concern quadratic polynomials, we will be constantly dealing with angles. When iterating polynomials of any degree, the reasonable way to measure angles is in turns, not radians, i.e., as elements of R/Z, not elements of R/2⇡Z. For quadratic polynomials, the natural operation on angles is angle doubling. Thus writing our angles in base 2 simplifies arithmetic, just as it is easy to multiply numbers by powers of 10 in base 10. 4 If q is odd, ✓ is periodic under angle doubling, of some period k: there exists k > 0 such that 2k ✓ = ✓. If q is even, ✓ is preperiodic: only after a certain number of doublings (the preperiod ) does the angle become periodic. Equivalently, if q is odd, the binary representation of ✓ repeats from the binary point; for instance, if ✓ = 15 , we have ✓ = .0011 = .0011 0011 0011 . . . ; thus ✓ = 15 has period 4 under angle doubling. If q is even, for instance, 9 if ✓ = 56 , we have ✓ = .001 010; both the preperiod and the repeating (periodic) block have length 3. We denote by Qodd /Z the elements of Q/Z represented by p/q with p, q coprime and q odd, and by Qeven /Z those with p, q coprime and q even. In both cases, set ✓m := 2m

1

✓,

10.2.1

so that ✓1 = ✓. Note that if q is even, then ✓m 6= ✓1 for all m > 1: the binary expansion is not periodic, so doubling never comes back to where we started. When q is odd, then ✓k+1 = ✓1 when k is the length of the period of the repeating binary expansion. June 25, 2015

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The ✓-spider for q even Set X0 := 0,

Xm := e2⇡i✓m for m

1.

10.2.2

The set {Xm , m 1} is finite, with cardinality k equal to the total length of the preperiod and the periodic block in the binary expansion of ✓. 9 For instance, if ✓ = 56 , the cardinality of {Xm , m 1} is 6; the set consists of X1 = e2⇡i(.001010) , X4 = e2⇡i(.010) , For m

X2 = e2⇡i(.01010) ,

X5 = e2⇡i(.100) ,

7, we have Xm = Xm

3,

X3 = e2⇡i(.1010)

X6 = e2⇡i(.001) ,

X7 = X4 .

10.2.3

so these are the only elements of the set.

Definition in 10.2.2 (The ✓-spider for q even) For ✓ 2 Qeven /Z, the ✓-spider S✓ ⇢ C is the set [ S✓ := { rXm | r 1 } [ {1}. 10.2.4 | {z } m 1

mth leg of spider

The subset Lm (✓) := { rXm | r 1 } is the mth leg of the spider. The 9 ✓-spider for ✓ = 56 is shown in Figure 10.2.1, left; it has six legs (maybe it should be an insect). When there is no ambiguity, we will drop the argument ✓ and write simply Lm . L4

L2

L1

L6 ✓00

X2

X1 X6

X4 = X7

M✓

X1 X4 = X7

B✓

X6

✓

✓0

X0

X5

X5

L5

X2

A✓ X3

X3

L3 Figure 10.2.1 Left: The spider S9/56 . The legs begin at infinity, which is the “body” of the spider. The (unit) circle drawn above is not part of the spider; it just sets the scale. (See also Figure 10.2.3, where the “body” of the spider is e 9/56 ; we have drawn the angle ✓ and its visible.) Right: The extended spider S 0 00 half angles ✓ and ✓ . Legs on one side of the green dividing line M✓ are gold; the single leg on the other side is red. (The region to one side of M✓ is labeled A✓ , the other B✓ ; we use A✓ and B✓ in Definition 10.3.2 of the kneading sequence.) June 25, 2015

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Spiders

71

In order to define the Thurston mapping f✓ we must first define the e ✓ , illustrated by Figure 10.2.1, right. Like all angles, ✓ has extended spider S two halves; if ✓ = p/q then the halves are ✓0 = p/(2q) and ✓00 = (p+q)/(2q).2 Call the line n o n o 0 00 M✓ := te2⇡i✓ | t 2 R = te2⇡i✓ | t 2 R 10.2.5 the dividing line, and label X0 = 0 the “middle” of M✓ . Define the extended e ✓ := S✓ [ M✓ . ✓-spider to be S

The ✓-spider for q odd

Let k > 0 be the smallest integer such that 2k ✓ = ✓ in Qodd /Z; the integer k is the period of ✓ under angle doubling. For instance, if ✓ = 15 , then k = 4. m 1 Indeed, 24 · 15 = 16 ✓; set 5 , which equals 1/5 in Q/Z. Recall that ✓m := 2 Xm :=

⇢

e2⇡i✓m

for m = 1, . . . , k

0

if m = 0 or m = k.

Definition in 10.2.3 (The ✓-spider for q odd.) periodic of period k. The ✓-spider S✓ ⇢ C is 0 1 [ S✓ := @ { rXm | r 1 }A [ re2⇡i✓k | r | {z 1mk 1

1

10.2.6

Let ✓ 2 Qodd /Z be 0

extends leg Lk to 0

}

[ {1}. 10.2.7

As above, we call the path in S✓ leading from infinity to Xm the mth leg of the spider, written Lm (✓) or simply Lm . The ✓-spiders for q even and q odd di↵er in the middle term of equation 10.2.7, which extends the kth leg to 0. As before, we denote by ✓0 and ✓00 the two halves of ✓ and call n o n o 0 00 M✓ := te2⇡i✓ | t 2 R = te2⇡i✓ | t 2 R 10.2.8

the dividing line, and label X0 = 0 the “middle” of M✓ , which is already labeled Xk . Note that the leg Lk is half the dividing line M✓ . Define the e ✓ = S✓ [ M✓ , illustrated at right in Figure 10.2.2. extended spider S 2

When angles are measured in turns, as they are here, doubling each half gives either the original angle or the original angle plus an integer. June 25, 2015

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L1 L2

X1

X2 X0 = X4

L4

✓00 X2

✓

M✓ X1

✓

X0 = X4

X3

✓0

X3

L3 Figure 10.2.2 Left: The spider S1/5 , with period k = 4 (i.e., 1/5 has period e 1/5 = S1/5 [ M1/5 . The 4 under angle doubling). Right: The extended spider S kth leg L4 extends to 0: it is half the dividing line M✓ .

f The map f✓ and the Thurston spider map f ✓

Now we will define a map f✓ from the extended spider to the spider, which is essentially angle doubling, with the radial coordinate slightly adjusted. Such a map is not a Thurston map, which has to be defined on the sphere, but we will show that f✓ has an extension fe✓ to the sphere. This extension is unique up to Thurston equivalence, and therefore something to which Theorem 10.1.14 can be applied. e ✓ ! S✓ by If ✓ 2 Qeven /Z, we define f✓ : S 8 2it e ✓ M✓ if reit 2 S > < re f✓ (reit ) := (r + 1)e2it if reit 2 M✓ , r > 0 10.2.9 > : X1 if r = 0.

If ✓ 2 Qodd /Z, so that x1 is periodic of period k under angle doubling, e ✓ ! S✓ by we define f✓ : S 8 2it e ✓ (M✓ [ Lk 1 (✓)) re if reit 2 S > > > 2it it < (r + 1)e if re 2 M✓ , r > 0 f✓ (reit ) := 10.2.10 2it > (r 1)e if reit 2 Lk 1 (✓) > > : X1 if r = 0 In addition, f✓ (1) = 1. These formulas, for q even and q odd, are cooked up so that f✓ maps the mth leg Lm (✓) ⇢ S✓ homeomorphically to the next leg Lm+1 (✓) and folds the entire line M✓ at X0 = 0, mapping the resulting closed half-line homeomorphically onto L1 (✓). See Figure 10.2.3. June 25, 2015

10.2

X5

f✓

L1

X1 X6

73

1

1

X2 X4

Spiders

M✓

X5 X2 X

4

L1

X1 X6

X3

X0

e✓ ! Figure 10.2.3 Here ✓ = 9/56. In both cases (even and odd), the map f✓ : S S✓ goes from the extended spider (left) to the spider (right). We have only drawn half of the extended spider. This figure is essentially the same as Figure 10.2.1, with the sides reversed. Seen on the sphere, the dividing line M✓ becomes the (green) great circle shown at left; it goes through 0 = X0 and through 1. It gets folded over itself and mapped to the leg L1 at right. The leg L1 at left gets mapped to the leg at right connecting 1 and X2 , and so on.

Theorem in and Definition 10.2.4 (The Thurston spider map) The e ✓ ! S✓ can be extended to a Thurston map fe✓ : S 2 ! S 2 of map f✓ : S degree 2, with critical set Crit fe✓ = {0 = X0 , 1} and postcritical set Pf˜✓ = {X1 , . . . , Xk , 1}. This extension, called the Thurston spider map, is unique up to Thurston equivalence; it is a topological polynomial. The dynamics on the postcritical set is represented by Figure 10.2.4. X1 X0

X1

X0 = Xk

Figure 10.2.4 The set {Xm , m 1} defined in equations 10.2.2 and 10.2.6, together with infinity, forms the postcritical set Pfe✓ of fe✓ . Left: When q is odd, the critical point X0 of the Thurston spider map fe✓ is periodic, hence postcritical. Right: When q is even, the critical point is preperiodic and not postcritical.

We will need Proposition 10.2.5 to prove Theorem 10.2.4. It is proved in Appendix C2. June 25, 2015

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Proposition in 10.2.5 (Alexander trick) Let D be the closed disc, and let g : @D ! @D be a homeomorphism. Then g extends to a homeomorphism ge : D ! D, for instance by the radial extension ge(rei✓ ) = rg(ei✓ ). Two such extensions ge0 and ge1 are isotopic by a family of homeomorphisms get , 0  t  1, such that get (x) = g(x) for x 2 @D and 0  t  1.

Proof of Theorem 10.2.4 We will get a Thurston map fe✓ by extending f✓ . Consider one of the closed halfplanes H bounded by M✓ (including the point at 1), as shown in Figure 10.2.3, left. Cut it along the legs contained in it, as shown in Figure 10.2.5, left. 1

1

H

fe✓

X5 X2 X

4

X1 X6 M ✓

X2 X

4

X1 X6

X3

X0

Figure 10.2.5 Here we have cut along the legs. At left, where the slits look white, we are looking through the halfplane, with nothing on the other side; where they are light blue, we are looking through to the “back” of the halfplane. The split halfplane and the split plane are both homeomorphic to discs. Since X0 (on the dividing line M✓ ) is a critical point of fe✓ , the point X1 is the corresponding critical value.

Now consider the extended plane C cut along the lines that are images under f✓ of legs in H, including the leg leading to X1 , which is the image of the dividing line M✓ . The slit halfplane and the slit plane are both homeomorphic to discs. Moreover, f✓ induces a homeomorphism of their boundaries; the only places where this is not obvious is at the points corresponding to 1, and there it requires knowing that the circular order of the legs in H coincides with the circular order of their images. This follows from the fact that the doubles of any collection of angles contained in a halfcircle appear with the same circular order in the whole circle. (This is why we cut along M✓ .) Thus by Proposition 10.2.5, the map induced by f✓ extends to a homeomorphism fe✓ of the interiors, unique up e ✓ . Putting together the homeomorphisms for the two to isotopy rel H \ S halfplanes gives the Thurston map. ⇤ June 25, 2015

10.3

Obstructions and Levy cycles

75

10.3 Thurston obstructions for spider maps and Levy cycles We have now created a large and interesting collection of Thurston mappings. In fact, we have constructed all the postcritically finite quadratic polynomials, although this is far from obvious; most of this section and the next two are devoted to exploring this claim. In the process we will see that we are well on our way to justifying a (still conjectural) description of the Mandelbrot set. We will also explain the redundancy in our description, i.e., when di↵erent extensions fe✓ are Thurston equivalent to each other. But to put any teeth into the description above we need to understand when the mappings fe✓ admit Thurston obstructions. For the Thurston spider maps fe✓ , as opposed to more general Thurston maps, this can be done in complete generality. But even here it is a major result. Theorem in 10.3.1. (A weak form of the Levy-Berstein theorem) When ✓ is in Qodd /Z, the spider map fe✓ has no Thurston obstruction.

We will prove this result when we prove the stronger Theorem 10.3.8. When q is even, there sometimes are obstructions. For instance, when ✓ = 9/56, the map fe✓ has a Thurston obstruction (see Figure 10.3.1). To describe exactly when Thurston obstructions arise, we need to define the kneading ✓-sequence of a point Xi in the postcritical set Pfe✓ . For ✓ 6= 0, the dividing line M✓ cuts the circle into two semicircles. Denote by A✓ the semicircle that contains 1 = e2⇡i 0 , and by B✓ the other. Then X1 = e2⇡i✓ is necessarily in the semicircle B✓ ; see Figure 10.2.1. Definition in 10.3.2 (Kneading sequence) The ✓-kneading sequence

Let t be an angle in R/Z.

S✓ (t) = S✓0 (t), S✓1 (t), S✓2 (t), S✓3 (t), . . . is the sequence of A’s and B’s defined by n ⇢ A if e2⇡i2 t 2 A✓ n S✓ (t) := n B if e2⇡i2 t 2 B✓ .

10.3.1

10.3.2

The ✓-kneading sequence S✓ (t) is both the ✓-kneading sequence of the angle t and the ✓-kneading sequence of the point e2⇡it . A particularly important ✓-kneading sequence is S✓ (✓), which is both the ✓-kneading sequence of ✓ and the ✓-kneading sequence of e2⇡i✓ = X1 , the critical value of fe✓ . We will call S✓ (✓) the kneading sequence of the Thurston map fe✓ . June 25, 2015

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Historical remark Let I ⇢ R be an interval. Kneading sequences are traditionally defined for maps f : I ! I with a unique minimum or maximum; such maps are said to be unimodular 3 . In this context, the kneading sequence of a point x 2 I is a sequence of symbols A, B where the nth symbol is A if f n (x) is to the right of the critical point and B if it is to the left. The kneading sequence of the unimodular map is the kneading sequence of the critical value. Our use of the term “kneading sequence” generalizes this usage. 4 For ✓ 2 Qeven /Z, the sequence S✓ (✓) is unambiguous, since Xk , k > 0 is never on M✓ .4 (If ✓ 2 Qodd /Z, there is an unambiguous way to assign Xk to either A or B. It will be given in the discussion preceding Figure 10.4.10.) For ✓ 2 Qeven /Z, the kneading sequence of each Xi is eventually periodic, with the length of the repeating block dividing the length of the repeating block in the binary expansion of ✓, but not necessarily equal to it. For instance, for ✓ = 9/56, the kneading sequences are S✓ (X1 ) = BBA B,

S✓ (X2 ) = BA B,

S✓ (X3 ) = A B,

S✓ (X4 ) = S✓ (X5 ) = S✓ (X6 ) = B,

10.3.3

since X3 is in A✓ and all other Xm , m 1, are in B✓ . Note that X4 , X5 , X6 share the same kneading sequence B := BBB . . . Theorem 10.3.3 asserts that this is the condition leading to Thurston obstructions. The theorem is proved at the end of this section. Figure 10.3.1 illustrates Theorem 10.3.3 for ✓ = 9/56, shown earlier in Figure 10.2.1. Theorem in 10.3.3 If ✓ is in Qeven /Z, then the Thurston spider map fe✓ admits no Thurston obstruction if and only if all the distinct postcritical points Xm have distinct kneading sequences. If two or more distinct Xm share a kneading sequence, they can be separated from the others by a simple closed curve that intersects each corresponding leg transversally at a single point, and does not intersect any other leg. The simple closed curves corresponding to shared sequences together form a Thurston obstruction. Examples 10.3.4 (Thurston obstructions for q even) The cases ✓ = 9/56 (Figure 10.3.1) and ✓ = 715/2016 both admit Thurston obstructions. 3

This terminology is unfortunate: “unimodular” usually means “of determinant 1”. 4 Recall from the discussion at the beginning of Section 10.2 that “if q is even, then ✓m 6= ✓1 for all m > 1 . . . doubling never comes back to where we started.” But if a point Xm , m > 0 were on M✓ , angle doubling would take it to X1 . June 25, 2015

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Obstructions and Levy cycles

77

L1

"

X2 X4 = X7

X1

X6

fe✓

(fe✓ )

X5

X2

"

X4 = X7 1(

)

X1 X6

B✓

!

A✓ X5

X3

M✓

(fe✓ )

X3

1(

)

Figure 10.3.1 The case ✓ = 9/56. Left: Since X4 , X5 , X6 share the kneading sequence B, Theorem 10.3.3 says that = { } is a Thurston obstruction. Right: The inverse image fe✓ 1 ( ) has two components, one on each side of M✓ . (Since does not intersect the leg L1 , shown in dark blue on left, its inverse image does not intersect M✓ .) Each component covers with degree 1. One component is homotopic to ; the other is peripheral, since it surrounds only X3 . Therefore is an fe✓ -stable multicurve (Definition 10.1.12), and the Thurston linear transformation is the 1 ⇥ 1 matrix 1, with leading eigenvalue = 1.

Now consider ✓ = 715/2016. The point X1 corresponds to ✓ = ✓1 ; it is in the halfcircle B✓ , which, if we go counterclockwise from angle 0, begins at ✓/2 = .001011 010110 and ends at (✓ + 1)/2 = .101011 010110. Doubling ✓1 gives ✓2 = .1011 010110; in base 2 we move the binary point to the right. Since .1011 > .1010, the point X2 is in the semicircle A✓ , and so on. This leads to Table 10.3.2. angle ✓1 ✓2 ✓3 ✓4 ✓5 ✓6 ✓7 ✓8 .. . ✓11 ✓12

angle in base 2 = = = = = = = = = =

✓-symbol

.01011 010110 .1011 010110 .011 010110 .11 010110 .1 010110 .010110 .101100 .011001 .. .

B A B A B B A B .. .

.001011 ✓6

A B

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Table 10.3.2. Here ✓ = 715/2016. The point X1 corresponds to ✓ = ✓1 . Doubling ✓1 gives ✓2 ; in base 2 we move the binary point to the right.

Thus X6 , X8 , X10 share the kneading sequence BA, and X7 , X9 , X11 share the kneading sequence AB. This leads to a Thurston obstruction consisting in the two curves drawn in Figure 10.3.3. ✓/2

✓ X1 X8 X3

X6 = X12

X11

B✓ 1

2

A✓ X10 X5

X7 X2

X9

X4

Figure 10.3.3 Here ✓ = 715/2016. The orange curves 1 and 2 form a Thurston obstruction. The inverse image of 1 consists of two curves, one homotopic to 2 , and the other peripheral; similarly, the inverse image of 2 consists of two curves, one homotopic to 1 , the other peripheral. Thus the Thurston matrix is  0 1 1 0 (see equation 10.1.12), with leading eigenvalue 1. 4

(✓ + 1)/2

Levy cycles and Thurston obstructions Before embarking on the proofs of Theorems 10.3.1 and 10.3.3, we will describe a special kind of Thurston obstruction, associated to a Levy cycle. Levy cycles are considerably easier to deal with than general Thurston obstructions. Thurston mappings with Thurston obstructions but no Levy cycles do exist [102], but Levy cycles are adequate for our present purposes: if fe✓ has a Thurston obstruction, then it has a Levy cycle. Definition in 10.3.5 (Levy cycle) Let f : (S 2 , Pf ) ! (S 2 , Pf ) be a Thurston mapping. A circularly ordered multicurve = { 0, . . . ,

n 1, n

=

0}

10.3.4

on (S 2 , Pf ) is a Levy cycle for f if for every i at least one component 0 1 ( i ) is homotopic to i 1 in S 2 Pf , and f : i0 1 ! i has i 1 of f degree 1. Exercise 10.3.6 is an easy special case of one direction of Theorem 10.1.14. June 25, 2015

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79

Exercise 10.3.6 Prove that if is a Levy cycle for a Thurston mapping f , then f is not Thurston equivalent to a rational function. Hint: Suppose that f is Thurston equivalent to a rational function g, i.e., that there exist homeomorphisms ', '0 such that the diagram S2 #f S2

'0

P1 #g ' ! P1 !

10.3.5

commutes, with ' and '0 coinciding on Pf and isotopic rel Pf . Then, for each i, the curves '( i ) and '0 ( i ) are homotopic in P1 Pg ; replace them by the geodesic ˜i in their common homotopy class. Now show that the component of g 1 ( ˜i ) homotopic to ˜i 1 is strictly shorter than ˜i 1 for the hyperbolic metric of P1 '(Pf ). This is a contradiction. }

Some Levy cycles are relatively easy to understand: the degenerate Levy cycles. These consist of a multicurve C = 0 , . . . , n 1 , n = 0 together with disjoint discs D i ⇢ S 2 bounded by i for each i, such that one component of f 1 (D i ) is a disc Di0 1. whose boundary is homotopic to i 1 in S 2 Pf 2. that contains the same postcritical points as D i 3. such that f : Di0 ! D i is a homeomorphism.

1

We will prove that a Thurston obstruction implies the existence of a degenerate Levy cycle not just for spider maps but in the more general setting of topological polynomials; see Definition 10.1.3. Credit for this result goes to Silvio Levy, Mary Rees, Tan Lei, and Israel Berstein. Theorem in 10.3.7 (Levy) If a topological polynomial f has a Thurston obstruction, then it has a degenerate Levy cycle. Proof of Theorem 10.3.7 Let be a minimal Thurston obstruction: a Thurston obstruction such that if a submulticurve 0 ⇢ is f -stable and 0 6= , then 0 < 1. Write = E t N , where 2

N

()

lim f

m

m!1

([ ]) = 0.

10.3.6

A curve in E is said to be an essential curve; a curve in N is nonessential. Note that N is an f -stable multicurve (although it is not a Thurston obstruction), but usually E is not: if 2 E , some nonperipheral components of f 1 ( ) may well belong to N . Since is a Thurston obstruction, we see that E 6= ;: under iteration m of f N all basis vectors of R N converge to 0, so limm!1 f N = 0 and so 1, so 6= N . Note also that for every 2 E , at least N < 1. But one component of f 1 ( ) belongs to E . June 25, 2015

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Every simple closed curve on S 2 Pf bounds two discs in S 2 ; we denote by D the one that does not contain 1. Using these discs, we can define i ⇢ E to be the multicurve formed of innermost essential curves: 2 i if D contains no other curve of E . We will show that i is a degenerate Levy cycle, which will prove the result. Indeed, associate to every element of 2 i the set I ⇢ E of elements homotopic in C Pf to at least one component of f 1 ( ). We will show that I is a singleton, and its unique element is innermost. Let 2 i . There exists ⌘ 2 E homotopic to some component of f 1 ( ). Suppose that ⌘ is not innermost; there then exists 2 E with ⇢ D⌘ ; see Figure 10.3.4. f

1

( )

↵0 x1

D

x3

D⌘

⌘

f !

y3

↵00 y1

x2 y2

D

Figure 10.3.4 If the curve (on the right) is innermost in E , and ⌘ 2 is homotopic to one component of f 1 ( ) (the blue curve at left), then ⌘ must be innermost also. Indeed, if x1 , x2 , x3 are all in the disc bounded by the component of f 1 ( ) homotopic to ⌘, then they are contained in D⌘ (the disc bounded by the green curve at left), and if another element 2 separates x3 from x1 and x2 , then f (D ) must contain y1 and y2 but not y3 , where yi = f (xi ). But this forces f ( ) to be homotopic to a curve in E disjoint from . Two candidates, ↵0 and ↵00 , are drawn on the right. Both contradict our hypotheses: D↵0 contains y3 , and D↵00 is contained in D , contradicting the hypothesis that is innermost. The two unlabeled marked points at left are there to guarantee that ⌘ is nonperipheral.

By minimality, is homotopic to some component of f 1 (↵) for some curve ↵ 2 E . We can’t say much about the exact positions of the curves, because homotopies are allowed, but we can speak of the points of Pf contained in the corresponding discs. The disc D must contain at least two elements x1 , x2 2 Pf \D⌘ ; at least one other element x3 2 Pf \D⌘ must not be contained in D . Then f (x1 ), f (x2 ), f (x3 ) all belong to D , and D↵ must contain f (x1 ) and f (x2 ) and cannot contain f (x3 ). This forces ↵ to be in the disc D , contradicting our assumption that is innermost. This shows that all elements of I are innermost, so the map 7! I is a map i ! P( i ), where P denotes the power set. It is easy to see that I \I = ; if 6= . By trivial set theory, this shows that I is a singleton, so June 25, 2015

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81

the map 7! I induces a permutation of i . This permutation is circular by minimality; also by minimality, we can now see that E = i . Thus if is a minimal Thurston obstruction for a postcritically finite topological polynomial, the curves of E can be numbered circularly 0 , 1 , . . . , m 1 , in such a way that the discs D i are disjoint, and exactly one component ⌘i of f 1 ( i ) is homotopic in S 2 Pf to i 1 . We still need to compute the degree di of f |⌘i : ⌘i ! i . Number the elements of N with indices m, so that f becomes a matrix of the form 2 0 3 1/d 0 ... 0 f =



A C

0 , D

6 0 6 . where A = 6 6 .. 4 0 1/d0

0 .. . 0 0

1

1/d2 .. . 0

... .. . 0 ...

0 .. .

1/dm 0

1

7 7 7 7 5

10.3.7 The eigenvalues of D are those of f N , so they all have absolute value < 1. The characteristic polynomial of f is ✓ ◆ 1 m det ( I A) det( I D) = det( I D). 10.3.8 d0 · · · dm 1 Since the leading eigenvalue of f is 1, we see that d0 = · · · = dm 1 = 1, so E is a Levy cycle. This Levy cycle is degenerate, since one component of f 1 (D i ) is a disc containing the same postcritical points as D i 1 . ⇤ Theorem 10.3.7

The Levy-Berstein theorem We can now prove a stronger theorem than Theorem 10.3.1. Theorem in 10.3.8 (Levy-Berstein theorem) If f is a postcritically finite topological polynomial such that all elements of Pf have critical points in their forward orbits, f has no Thurston obstruction. In particular, if ✓ = p/q with q odd, then fe✓ admits no Thurston obstruction.

Proof Suppose is a Thurston obstruction, so that (by the proof of Theorem 10.3.7) E is a degenerate Levy cycle. We can number the elements of E as 0 , . . . , m 1 , m = 0 , bounding disjoint discs D0 , . . . , Dm 1 . Moreover, f (Pf \ Di ) = Pf \ Di+1

for all i = 0, . . . , m

1.

10.3.9

In particular, at least one of the Di must contain a critical point, and then the component of f 1 ( i+1 ) homotopic to i cannot map with degree 1 to i+1 . This is a contradiction, so there is no Thurston obstruction. ⇤ June 25, 2015

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Proof of Theorem 10.3.3 Suppose ✓ 2 Qeven /Z and that fe✓ admits a Thurston obstruction, hence admits a degenerate Levi cycle = { 0, . . . ,

` 1, `

=

0 },

10.3.10

with i bounding discs Di ⇢ C. The union [i Di must contain all the periodic points of P✓ and none of the strictly preperiodic points. Indeed, the forward orbit of any postcritical point in [i Di stays in [i Di , so the periodic postcritical points are in [i Di . Let D0 , bounded by 0 , contain the first periodic postcritical point. The map fe✓ must map 0 to its image with degree 1. Hence fe✓ 1 ( 1 ) must two components, one homotopic to 0 , and one containing the last strictly preperiodic point and no other postcritical points. Thus this component is peripheral, and the Levi cycle is a Thurston obstruction. The same analysis shows that the curves of can be homotoped in C Pfe✓ so that they do not intersect the legs leading to any strictly preperiodic points. Indeed, if is put in minimal position (see Definition 3.3.10), then it will intersect each such leg in some number of intervals, but the number of such intervals is non-increasing under inverse images. In fact, if this number is positive it will strictly decrease, since somewhere (on the leg L1 if nowhere else) the inverse limit of a leg will not lead to a postcritical point, or it will strictly decrease. In particular, the curves of will not intersect the leg L1 , so each curve of lies entirely on one side of the dividing line, hence all the Xj in a given Di have the same kneading sequence. Conversely, suppose two distinct periodic postcritical points have the same kneading sequence. Then, since there is only one critical point, hence one postcritical orbit, every periodic postcritical point has the same kneading sequence as some other periodic postcritical point. Construct the curves =

0, . . . , ` 1, `

=

0

10.3.11

as in Definition 10.3.5, numbered so that the disc D0 bounded by 0 contains the first periodic postcritical point. A first thing to see is that actually is a multicurve. Certainly its curves are nonperipheral, and the caption of Figure 10.3.5 explains why they can’t intersect. June 25, 2015

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Polynomials with superattracting cycles

Figure 10.3.5 The curves i and j cannot intersect as shown at left: the points inside i have a di↵erent kneading sequence from those inside j , so for some p the components of fe✓ p ( i ) and fe✓ p ( j ) that belong to will be on opposite sides of the dividing line, hence disjoint. Hence they will still be disjoint by fe✓ ` , but they must be homotopic in C Pfe✓ to the curves we started with.

j

i

83

The hypothesis that the periodic postcritical points are in one of the discs Di implies that the multicurve is fe✓ -invariant, and in fact is a Levy cycle. ⇤ Theorem 10.3.3

10.4 Julia sets of quadratic polynomials with superattracting cycles By Theorem 10.3.8, a Thurston map fe✓ admits no Thurston obstructions when ✓ 2 Qodd /Z. Hence each such map fe✓ is equivalent to a quadratic polynomial p✓ , unique up to conjugation by an element of Aut C, i.e., by an affine map z 7! az + b with a 6= 0. The object of this section is to understand the dynamics of p✓ for ✓ 2 Qodd /Z. The only thing we know about p✓ is that there exist homeomorphisms ', '0 : S 2 ! P1 that agree on Pf˜✓ , such that the diagram S2 # f˜✓ S2

'0

! '

!

P1 # p✓ P1

10.4.1

commutes and ' is isotopic to ' rel Pf˜✓ . Since X1 , X2 , . . . form the postcritical set of f˜✓ and X0 is one of its critical points, this implies that 0

x0 := '(X0 ), . . . , xk

1

:= '(Xk

1 ), xk

:= '(Xk ) = '(X0 ) = x0

10.4.2

forms a superattracting cycle of period k; see Definition 9.2.1. (The multiplier of the cycle is 0, by the chain rule, since the critical point x0 of p✓ is one point of the cycle.) Corollary 10.4.14 says that all quadratic polynomials with superattracting cycles are of the form p✓ for an appropriate ✓ 2 Qodd /Z. At the moment we will assume that we have a quadratic polynomial p with a superattracting cycle, without knowing that it is of the form p✓ . Understanding the dynamics of any polynomial p is essentially the same thing as understanding the geometry of the corresponding filled Julia set Kp :=

z 2 C the sequence z, p(z), p 2 (z), . . . is bounded June 25, 2015

.

10.4.3

84

Chapter 10.

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The Julia set Jp is the boundary of Kp . The sets Kp and Jp are discussed further in Appendix 9. We will write K✓ and J✓ when p is known to be of the form p✓ for ✓ 2 Qodd /Z.

Figure 10.4.1 shows two filled Julia sets for polynomials with superat-

tracting cycles. The points xi of the superattracting cycle are in Kp since the cycle is bounded; they are in its interior since they are superattracting. (They would also be in the interior of Kp if they were only linearly attracting; see Definition 9.2.1.)

x1

x2

x2

x0

.

x1 x0

x3

Figure 10.4.1 Left: The black region is K✓ for ✓ = 1/5, with period k = 4 under angle doubling. Right: The black is K✓ for ✓ = 1/7, with k = 3. The white arrows show the orbit of x0 . (The figure at left is also K✓ for ✓ = 4/15, and that at right is also K✓ for ✓ = 2/7: 1/5 and 4/15 are companion angles, as are 1/7 and 2/7; see Definition 10.4.15 and Corollary 10.4.16.) Both sets are compact, connected, full, and locally connected (a bounded subset of C is full if its complement is connected). The sets K✓ , ✓ 2 Qodd come in a bewildering variety of shapes, and depend in a remarkably delicate way on ✓, but they are always symmetric around the critical point x0 .

The map

p

of Theorem 10.4.1 is the B¨ ottcher coordinate of the super-

attracting fixed point 1 for p. (It is the inverse of the B¨ ottcher coordinate defined in equation 9.3.8 in Appendix 9.) See Figure 10.4.2. June 25, 2015

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Polynomials with superattracting cycles

85

Theorem in 10.4.1 (The filled Julia set Kp ) Let p be a quadratic polynomial whose critical point is periodic. 1. The set Kp ⇢ C is compact, connected, full, and locally connected. 2. There exists a unique analytic isomorphism p

:C

D!C

p (z

3. The map

p

2

) = p(

Kp such that p (z)).

extends continuously to C

Definition in 10.4.2 (The Caratheodory loop) p (t)

:=

10.4.4

D.

The restriction

2⇡it ) p (e

10.4.5

of p to the boundary of C D is the Caratheodory loop of Kp (i.e., the Caratheodory loop takes the circle R/Z to the boundary Jp of Kp .) By Caratheodory’s theorem, Caratheodory loops exist if and only if Kp is locally connected (for a proof, see [79], page 183, Theorem 17.14).

! ✓

ray at angle 0

Figure 10.4.2 The mapping 1/5 takes the “checkerboard pattern” on the left to the pattern on the right; it takes the complement of the green unit disc to the complement of the filled Julia set. The two dark lines indicate external rays; see equation 10.4.7. External rays on the left and the right at the same angle are asymptotic at infinity; in particular, the ray at angle 0 on the left (dark line left) is asymptotic to its image, the dark ray at angle 0 on the right.

Proof 1. Part 1 follows from part 2 of Proposition 9.1.5 for compactness, from Theorem 9.1.6 for connectedness, from part 3 of Proposition 9.1.5 for Kp full, and from Theorem 9.4.1 for local connectivity (in the easier case, p hyperbolic). 2. Part 2 follows from Theorem 9.3.4. 3. This is again a consequence of Theorem 9.4.1. ⇤ June 25, 2015

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Theorem 10.4.3 is the main result of this section; its proof will take the remainder of the section. The theorem uses the kneading sequence S✓ to provide a completely combinatorial description of K✓ for ✓ 2 Qodd /Z. It is called the pinched disc model of the filled Julia set. Theorem in 10.4.3 (Pinched disc model of the filled Julia set K✓ ) Let ✓ be the Caratheodory loop of K✓ for ✓ 2 Qodd /Z. Then ✓ (t1 )

=

✓ (t2 )

() S✓ (t1 ) = S✓ (t2 ).

10.4.6

The kneading sequence S✓ is entirely combinatorial. Put an equivalence relation on D, where the nontrivial equivalence classes are the convex hulls of the points t that share a ✓-kneading sequence. This equivalence relation is represented for ✓ = 1/7 in Figure 10.4.3, left, where the equivalence classes are the red triangles. If we collapse the equivalence classes to points, we get a topological model of K✓ . (Actually we get much more than topology, though the geometrical information is rather subtle to extract from the combinatorics.) 2 7 1 7

2 7

1 7

4 7

4 7

Figure 10.4.3 The filled Julia set K1/7 at right (the “rabbit”) can be understood in terms of the “pinching locus” at left. The white components at left correspond to the black components at right; the convex hulls of the angles that share a 1/7th kneading sequence are triangles and points. Some are shaded red. When these triangles are “pinched” to points, they become triple points of K1/7 . The deep red triangle at left is collapsed to the point at right where three external red rays meet.

Remark Figure 10.4.3 is simpler than most: there are only countably many nontrivial equivalence classes, all inverse images of the triangle June 25, 2015

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Polynomials with superattracting cycles

87

spanned by 1/7, 2/7, 4/7 under angle doubling. In general there are uncountably many nontrivial equivalence classes, for instance, for K1/5 shown in Figure 10.4.2. 4

External and internal rays Let p be a quadratic polynomial with periodic critical point. The external ray of Kp at angle t is the set Rp (t) :=

2⇡it ), p (re

1 < r < 1.

10.4.7

(Figure 10.4.3, right shows several external rays of K1/7 .) We know that under these circumstances the Caratheodory loop p exists; we say that the external ray Rp (t) lands at p (t). Proposition in 10.4.4 Let p be a quadratic polynomial whose critical point !p is periodic, forming the cycle x0 = !p , x1 , . . . , xk = x0 . Let Vi be the component of K p containing xi , so that V0 contains !p . 1. There exists a unique homeomorphism ⌘V0 : D ! V0 , analytic in the interior, such that ⌘V0 (z 2 ) = p

k

⌘V0 (z) ,

so ⌘V0 (0) = x0 .

10.4.8

2. Let V be a connected component of Kp . Then there exists some minimal m such that p m : V ! V0 is an analytic isomorphism, so that the map ⌘V : D ! V given by ⌘V := p

m

|V

1

⌘V0

10.4.9

is a homeomorphism analytic in the interior D ⇢ D. Proof Part 1 is a special case of Theorem 9.3.2. Part 2 will be proved as Corollary 10.4.8. ⇤ The map ⌘V0 is the B¨ ottcher coordinate for the superattracting cycle containing 0. It is often called the internal parametrization of V0 , as opposed to the external parametrization p of C Kp . Definition in 10.4.5 (Internal ray, center, root) Let p be a polynomial whose critical point !p is periodic. In each component V ⇢ Kp , the arc ⌘V (re2⇡it ), 0  r  1

10.4.10

is called the internal ray of V at angle t. The point ⌘V (0) is called the center of V . The point ⌘V (1), at the end of the internal ray at angle t = 0, is called the root of V . June 25, 2015

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Figure 10.4.4 illustrates the maps ⌘V0 and ⌘V1 . In fact, the checkerboard patterns illustrate ⌘V for all components V , including all those that contain no point of the critical orbit. In particular, ⌘Vi (0) = xi , so xi is the center of Vi . We will denote by yi the root of Vi , i.e., yi := ⌘Vi (1). By Corollary 10.4.7, the roots yi form a repelling cycle (see Definition 9.2.1); the period of this cycle divides the period of !p . Unlike external rays, which exist for all polynomials p with Kp connected, internal rays exist only for polynomials for which the critical point !p is periodic, i.e., part of a superattracting cycle, since ⌘V is only defined for such polynomials. The roots, unlike the centers, are in the Julia set Jp (the boundary of Kp ): ⌘Vi takes 1, on the boundary of D, to the boundary of Vi ; by Exercise 9.1.5, the boundary of Vi is contained in Jp . ⌘V0 ⌘V1 D

" y1 = y0 V1 V0 Figure 10.4.4 The map ⌘Vi conjugates z 7! z 2 in D to p k : Vi ! Vi . The dark horizontal line at left (the internal ray of D at angle 0) is taken by ⌘V0 and ⌘V1 to the corresponding internal rays of V0 and V1 , shown as dark horizontal lines at right. The point where those two internal rays meet is both the root of V0 and the root of V1 : y0 = y1 = ⌘V0 (1) = ⌘V1 (1); see Definition 10.4.5. In this case there are two components, V0 and V1 , that contain a point of the critical orbit; there are infinitely many components V that do not contain such a point.

Let p be a quadratic polynomial whose critical point is periodic. Choose numbers 1 > r0 = rk > rk

1

> · · · > r1 > r02

10.4.11

and some number R > 1. Let DR be the disc of radius R centered at 0, let p be the map defined in equation 10.4.4, and let U be C with two sets removed: ! k [ U := C ⌘Vi (Dri ) [ p (C DR ) . 10.4.12 i=1

June 25, 2015

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Polynomials with superattracting cycles

Proposition in 10.4.6 The set U 0 := p 0 and p : U ! U is a covering map.

1

89

(U ) has compact closure in U ,

Figure 10.4.5 illustrates U and U 0 . The set U 0 is strictly inside U . In particular, the white discs at left are larger than those at right, so a point on the boundary of U 0 is strictly inside U . The black “circles” at left are the boundary of U , drawn to show that U 0 has compact closure in U . Proposition 10.4.6 is a variant of equation 9.4.3. p (C

U0

DR )

p ! U

Figure 10.4.5 Here we see U , at right, and U 0 = p U := C

k [

i=1

⌘Vi (Dri ) [

p (C

1

(U ) at left. The set ! DR ) .

is everything that is colored at right. It is bounded on the outside by an equipotential, and on the inside by the images of the circles of radius ri under ⌘Vi ; the images of those circles bound white discs, which are not in U . The set U 0 is everything that is colored at left.

Proof of Proposition 10.4.6 The set p(⌘Vi (Dri )) has compact closure in ⌘Vi+1 (Dri+1 ) and p( p (C DR )) has compact closure in p (C DR ). Thus the boundary of U maps strictly outside U , so p 1 (U ) has compact closure in U . The map p : p 1 (U ) ! U is proper, and a local homeomorphism since the critical point !p = x0 is in ⌘V0 (Dr0 ). Hence it is a covering map. ⇤ For any V ⇢ C, denote by |(z, ⇠)|V the length of the vector ⇠ 2 Tz V for the hyperbolic metric of V . Below, p0 is the derivative of p. Note the U 0 in the text and the U in the formula: the strong contraction in formula 10.4.13 is as measured in U . Corollary in 10.4.7 (Quadratic polynomial expanding on Julia set) Let p be a quadratic polynomial with periodic critical point. Then there exists C > 1 such that for every z 2 U 0 and every ⇠ 2 Tz U 0 , p(z), p0 (z)⇠

U

C|(z, ⇠)|U

June 25, 2015

10.4.13

90

Chapter 10.

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To see that the roots yi form a repelling cycle, substitute yi for z and yi+1 for p(z). Proof Since U 0 ⇢ U we have |(z, ⇠)|U 0 |(z, ⇠)|U for every z 2 U 0 and 0 every ⇠ 2 Tz U , by Proposition 3.3.4. Moreover, since U 0 has compact closure in U , the ratio |(z, ⇠)|U 0 |(z, ⇠)|U

10.4.14

tends to infinity on the boundary of U 0 , so the infimum C of the ratio is realized; of course C > 1. Since p : U 0 ! U is a covering map, it is an infinitesimal isometry for the respective hyperbolic metrics, so C|(z, ⇠)|U  |(z, ⇠)|U 0 = |(p(z), p0 (z)⇠)|U .

⇤

10.4.15

Corollary in 10.4.8 Let p be a quadratic polynomial with periodic critical point. For every component V ⇢ Kp , there exists a minimal m such that p m (V ) = V0 , and then p m : V ! V 0 is a homeomorphism analytic in the interior. Proof The main statement is that every component V of Kp is a preimage of V0 . Indeed, if this were not the case, then every component of the forward image V , V 0 := p(V ), V 00 := p(V 0 ), . . . would be a subset of U 0 , and p : V (m) ! V (m+1) would be an analytic isomorphism, since V (m) contains no critical point, and p|V (m) is the restriction of the proper analytic map p to a component of p 1 (V (m+1) ). On U 0 and in the hyperbolic metric of U , the map p expands lengths by at least C (the C of Corollary 10.4.7); since p is conformal on V (m) , it expands areas by at least C 2 . Thus Area V (m+1)

C 2 Area V (m)

C 2(m+1) Area V,

10.4.16

and this is obviously impossible since the area of U 0 is finite. Thus we can consider p

0 p

p

V ! V ! ... ! V

(m)

= V 0;

10.4.17

by the same argument as above each map is a homeomorphism analytic in the interior. ⇤

The tree of a polynomial p with a superattracting cycle Let p be a quadratic polynomial whose critical point !p is periodic. By Theorem 10.4.1, Kp is connected and locally connected, so Kp is pathwise June 25, 2015

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91

connected, and any two points of Kp can be joined by an embedded arc in Kp . We need a few properties of such arcs. Exercise 10.4.9 Suppose K ⇢ C is compact, connected, locally connected, and full. Prove the following statements: 1. If ⇢ K is an embedded arc, then for any component V ⇢ K, the intersection V \ is connected: it is a subarc of , or a single point, or empty. 2. If 0 , 00 ⇢ K are two embedded arcs joining the same points x and y, then 0 a. \ @K = 00 \ @K

b. V \ 0 = ; () V \ 00 = ; c. The sets V \ 0 and V \ 00 are either both the same single point, or both arcs that enter V at the same point and exit it at the same point. }

It follows that the only freedom in choosing an embedded arc in Kp joining two points of Kp is how to cross the interior of components. We can lift this ambiguity also. Definition in 10.4.10 (Regulated path) A regulated path in Kp is an embedded arc that intersects each component of the interior only in internal rays (hence either one ray, if the path ends at the center of the component, or two, if the path crosses the component). Given points x, y 2 Kp , each either in Jp or the center of a component

of K p , there exists a unique regulated path joining them in Kp . Regulated paths are illustrated in Figure 10.4.6; Figure 10.4.7 shows the tree T37/127 . Definition in 10.4.11 (Tree of polynomial) The tree Tp of a quadratic polynomial p with a superattracting cycle is the union of all regulated paths joining pairs of points of the critical orbit. Exercise 10.4.12 Suppose p is a quadratic polynomial whose critical point !p is periodic of period k > 1. Prove the following statements. 1. The tree Tp is a finite tree satisfying p(Tp ) = Tp . 2. All extremities of the tree are points of the critical orbit, and these extremities are the first m points of the critical orbit for some m with 1 < m  k, where k is the period of ✓ under angle doubling. (The succeeding points xm+1 , . . . , xk belong to the tree but are not extremities; see for instance x6 and x7 in Figure 10.4.7.) June 25, 2015

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3. For every x 2 Tp \ Jp , the components of Tp {x} are in distinct components of the complement of the external rays of Kp landing at x. }

z1

z2

Figure 10.4.6 The white line is the regulated path from z1 to z2 . The filled Julia set Kp (the set of points that don’t escape to infinity under iteration by p) consists of the black shapes plus dark filaments running through all the red filaments. The regulated path follows Kp , and crosses components of the interior in internal rays. Points not in Kp are colored according to their rate of escape to 1; gold points escape faster than red.

x1

x4 x1

x4

%

x6

x2

x6

x3 x0 = x7

.

x3

x2 x0 = x7

x5

x5

Figure 10.4.7 Left: The tree of p✓ for ✓ = .0100101 = 37/127. The points of the orbit of the critical point are marked x0 , x1 , x2 , . . . , x7 ; they are connected in K✓ by a regulated path. Right: The tree is a combinatorial object; here we have drawn its combinatorial structure, forgetting its complicated shape in C. All points except x6 and x0 = x7 are extremities. June 25, 2015

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Companion angles and external rays In Proposition 10.4.13, the hypothesis k > 1 implies that x0 6= x1 , i.e., p is not conjugate to the polynomial z 7! z 2 . Proposition in 10.4.13 Let p be a quadratic polynomial with a superattracting cycle x0 = !p , x1 , . . . , xk 1 of period k with k > 1. 1. There are exactly two external rays of Kp that land at y1 and whose union separates x1 from all the other xi . Each of these rays has period precisely k under angle doubling. 2. Denote the angles of these rays by ✓ and ✓0 . Then p is conjugate (by an affine map) to p✓ and to p✓0 . Corollary in 10.4.14 Every quadratic polynomial with a superattracting cycle is conjugate to p✓ for some ✓ 2 Qodd /Z. Definition in 10.4.15 (Companion angles) Two elements ✓, ✓0 of odd Q /Z are companions if p✓ and p✓0 are conjugate. Corollary in 10.4.16 Every element ✓ 2 Qodd /Z panion angle ✓0 6= ✓.

{0} has a unique com-

Proof of Proposition 10.4.13 The hypothesis k > 1 implies that Tp is not the single point {x0 }, in particular, some internal ray of V1 is part of Tp . Parts 1 and 2 of Exercise 10.4.12 imply that there is only one such ray, at angle 0, hence joining x1 to y1 . Since y1 is an endpoint of the tree, y1 is alone in its component of Tp {y1 } so part 3 says that two external rays of Kp land at y1 and separate x1 from all the other xi . Let ✓ and ✓0 be the angles of the external rays bounding the sector containing x1 (hence V1 ). By part 2 of Exercise 10.4.12, this x1 is the only point of the critical orbit in that sector, since the internal ray from x1 to y1 is one component of Tp {y1 }. It follows that ✓ and ✓0 are of period precisely k under angle doubling. This proves part 1. For part 2, by the definition of p✓ we need to construct homeomorphisms ', '0 : S 2 ! P1 that agree on Pf˜✓ , such that the diagram S2 # f˜✓ S2

'0

! '

!

P1 #p P1

10.4.18

commutes and ' is isotopic to ' rel Pf˜✓ . The same argument works for p✓0 . It will follow that p✓ is conjugate to p✓0 . 0

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e ✓,i consisting of the external ray Rp (2i 1 ✓) Consider the set of “legs” L and the internal ray at angle 0 of Vi . The union Se✓ of these legs is almost homeomorphic to the standard ✓-spider S✓ , and p does map Se✓ to itself, doubling the angles of the legs. But there is a possible problem: two legs may meet at some yi and thus not be disjoint. (This case is important; we analyze it in Propositions 10.4.17 and 10.4.18.) e ✓,1 can Because the external ray Rp (✓) is adjacent to V1 , the “leg” L be slightly deformed with endpoints {1, x1 } fixed so as to avoid y1 , and e ✓,i can be slightly deformed so as to avoid yi (see further all the “legs” L Figure 10.4.8), and these new (genuine) legs L✓,i can be taken to be disjoint. ✓10 L✓0 ,1 x1 .

L✓,1 ✓1

Figure 10.4.8 The blue petals are the Vi . If y1 (center of the flower) has lower e ✓,i period than x1 , then at least two rays L e ✓,i are not disland at y1 , so the “legs” L joint. But because Rp (2i 1 ✓) is adjacent to Vi , these pseudo-legs can be slightly e ✓,1 deformed to be embedded legs. Here L (purple and red) can be deformed to L✓,1 (gold); these legs are now disjoint. We e ✓0 ,i ; for incould do the same with the L e ✓0 ,1 (blue and red) can be destance, L formed to L✓0 ,1 (green).

Denote by S✓ the union of the legs L✓,i . The inverse image p 1 (L✓,i ) consists of two arcs meeting at 1. Denote by L0✓,i 1 the one that joins 1 to xi 1 , and call S✓0 the union of the L0✓,i . Choose a homeomorphism ' : S✓ ! S✓ , and set '0 := p 1 ' f✓ , so that '0 maps S✓ to S✓0 . Then '0 is isotopic to ' rel Pf˜✓ , and by the same Alexander trick argument as in Proposition 10.2.5, ' and '0 extend to homeomorphisms (S 2 , Pf˜✓ ) ! P1 that are isotopic rel Pf˜✓ . ⇤ When we study the structure of the Mandelbrot set, the distinction between the case where y1 has smaller period than x1 and when it has the same period will be crucial. Proposition in 10.4.17 Let p be a quadratic polynomial whose critical point is periodic of period k. Then the root y1 of the component V1 has period k0 < k if and only if at least two external rays of Kp landing at y1 belong to the same orbit under angle doubling. Proof In both directions this should be clear: if Rp (✓) lands at y1 and y1 has period k0 < k, then Rp (2k 1 ✓) also lands at y1 . Conversely, if Rp (✓) 0 and Rp (2k 1 ✓) both land at y1 , then y1 has period dividing k. ⇤ June 25, 2015

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This can be sharpened. Proposition in 10.4.18 If the angles of the external rays of K✓ landing at y1 belong to more than one orbit under angle doubling, then exactly two external rays of K✓ land at y1 , and the root y1 and the center x1 have the same period. It follows that if three or more rays land at y1 , their external angles belong to the same orbit under angle doubling, so, by Proposition 10.4.17, if y1 and x1 have the same period, then exactly two external rays land at y1 , and these rays belong to distinct orbits under angle doubling. Proof Denote by Yi the set of external rays of K✓ landing at the root yi of Vi . We will call a component of C Yi a sector at yi . For such a sector S, the subset IS ⇢ R/Z consisting of external angles of rays contained in S is an interval, whose length ↵(S) is called the opening of S. The elements of each Yi are circularly ordered; the circular order of their angles and their order as arrayed around yi coincide. We need three crucial but obvious properties of the Yi and their sectors: 1. p✓ : Yi ! Yi+1 preserves the circular order

2. If a sector S at yi does not contain the critical point, then we have 2↵(S) = ↵(p✓ (S)) 3. If S is a sector at some yi , and yj 2 S with yj 6= yi , then S contains all sectors at yj except the sector containing yi If only two external rays of K✓ land at y1 , then they are certainly adjacent to V1 . So suppose there are three or more such rays; we will first show that their external angles belong to the same orbit under angle doubling. The argument is illustrated by Figure 10.4.9. Let S be the sector at y1 containing the critical value, and let S10 be the adjacent sector at y1 with the smallest opening. Denote by S100 the other sector at y1 adjacent to S (if both adjacent sectors have the same opening, choose either one). Let S00 , S 0 1 , S 0 2 , . . . be the subsets of C defined so that S 0 i 1 is the component of p✓ 1 (S 0 i ) containing y i in its closure (remember that the yi are labeled circularly, so negative indices make sense). If the critical point does not belong to S00 [ · · · [ S 0 k , then S 0 k is a sector at y k of opening ↵(S

k)

=

1

2k+1

↵(S10 ).

10.4.19

Since there are finitely many sectors at all the yi , the openings of sectors are bounded below, and there must be a first S 0 k 1 that contains the critical point. Then S 0 k contains the critical value. If it is a sector at a point

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S10 S0

2

S0

" 3

=S

S00

S0

1

594 Figure 10.4.9 Left: The set K✓ for ✓ = .001001010010 = . Right: A 4095 0 0 0 0 sketch of K✓ showing in gold the sectors S1 , S0 , S 1 , S 2 . The sector S is red; it is indeed S 0 3 , as the proof shows.

other than y1 , then it contains all the sectors at y1 except at most one. In particular it must contain S and S10 or S100 or both, so that ↵(S 0 k ) > ↵(S10 ). This is a contradiction, so S 0 k is a sector at y1 . Moreover, S 0 k = S: S 0 k 1 contains the critical point, so S 0 k contains the critical value. Thus p✓k maps S to the adjacent sector S10 , and as such permutes the rays of Y1 circularly. Thus the angles of all rays belonging to the Yi belong to a single cycle. ⇤ Definition in 10.4.20 (Primitive and satellite) Let ⇥ := (✓, ✓0 ) be companion angles. If ✓ and ✓0 belong to the same orbit under angle doubling, the pair is a satellite, as is p⇥ := (p✓ , p✓0 ). If ✓ and ✓0 belong to two di↵erent orbits, ⇥ and p⇥ are primitive.

Proof of Theorem 10.4.3 on the structure of the Julia set We now prove Theorem 10.4.3, which says that if ✓ is in Qodd /Z, then ✓ (t1 )

=

✓ (t2 )

() S✓ (t1 ) = S✓ (t2 ),

where ✓ is the Caratheodory loop of K✓ . For ✓ 2 Qodd /Z, the union of the external rays at angles ✓/2 and (✓ + 1)/2, together with the union of the internal rays of V0 at angles 0 and 1/2 , cuts the plane into two parts, which we will still call A✓ (the part containing the external ray at angle 0) and B✓ . We call this union of rays the dynamical dividing line. The June 25, 2015

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kneading sequence of the external ray at angle t under p✓ with respect to this decomposition coincides with the kneading sequence S✓ (t), since ✓ conjugates z 7! z 2 in C D to p✓ in K✓ . Theorem 10.4.3 is a little shaky: we don’t know how to attribute a symbol to the angles of the dividing line. The external rays with these angles form part of the dynamical dividing line; we need to say how to attribute them to A✓ or to B✓ . Let ✓ and ✓0 be companion angles, and suppose ✓ < ✓0 ; see Figure 10.4.10. Then the ray of angle ✓/2 is attributed to A✓ , and the ray of angle (✓ + 1)/2 is attributed to B✓ . This definition might make you a bit uneasy: suppose ✓ = 117/961. Should you attribute ✓/2 to A✓ or to B✓ ? To find out, you need to find the companion of ✓, and so far we haven’t seen any combinatorial way to do this. We show how to compute companion angles in Corollary 10.4.21. ✓0

V1

✓

&x

V1

x1

1

✓

✓0

✓+1 2

x0

]

]

B✓ V0

✓0 + 1 2

x0

V0 ! ✓ 2

]

A✓ ✓0 2

✓ 2

B✓

✓+1 2

A✓

Figure 10.4.10 At left, the angle ✓ is smaller than its companion ✓0 , at right it is larger. At top, the union of the external ray at angle ✓ landing at the root of V1 and the internal ray at angle 0 in V1 is shown in green. It has an inverse image (the dynamical dividing line, shown in thick green) consisting of the two external rays at angles ✓/2 and (✓ + 1)/2, together with the union of the internal rays at angles 0 and 1/2 in V0 . If ✓ < ✓0 , the angle ✓/2 (purple dot) should be attributed to A✓ , because it lands at the same point as the ray at angle ✓0 /2, which is definitely in A✓ . If ✓ > ✓0 , the angle ✓/2 should be attributed to B✓ .

Now one direction of Theorem 10.4.3 clear: if external rays of K✓ at angles t and t0 land at the same point of J✓ = @K✓ (i.e., if ✓ (t) = ✓ (t0 )), then so will all their forward images. Thus the external rays at angle 2i t and at angle 2i t0 land at the same point for all i, and the two rays are on the same side of the dynamical dividing line, so that S✓ (t) = S✓ (t0 ). For the converse, suppose that S✓ (t) = S✓ (t0 ), and that the external rays of K✓ at angles t and t0 land at points x := ✓ (t) and y := ✓ (t0 ). For June 25, 2015

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each m

Rational functions

0, join p✓m (x) to p✓m (y) by the geodesic

m

in U 0 in the homotopy

class that does not cross any of the legs Li or the dynamical dividing line p✓ 1 (L1 ). Since the legs can be approximated by disjoint legs, which do not disconnect the plane, only the dynamical dividing line does disconnect the plane, and our hypothesis that S✓ (t) = S✓ (t0 ) implies that p✓m (x) and p✓m (y) are on the same side of the dynamical dividing line. Thus this geodesic exists. The following two facts about the m are the keys to the proof: 1. One component of p✓ 1 ( 2. All

m

m)

is homotopic in U 0 to

m 1.

have length at most some constant C1 .

The first follows from the fact that m does not cross L1 , so the components of p✓ 1 ( m ) lie on opposite sides of the dynamical dividing line; hence (m 1) (m 1) the one that contains p✓ (x) also contains p✓ (y). The second follows from the compactness of J✓ . The geodesic m 1 is homotopic in U 0 to one component of p✓ 1 ( m ). By Corollary 10.4.7, this component is shorter than m by at least a factor of C, and m 1 is shorter yet since it is the geodesic in the same homotopy class. Thus 1 C1 Length( 0 )  m Length( m )  m for all m. 10.4.20 C C This implies that Length( 0 ) = 0, i.e., x = y, since 0 joins x to y. ⇤ Corollary in 10.4.21 Suppose ✓ 2 Qodd /Z has period k under angle doubling. Then its companion angle ✓0 is the unique angle periodic of period k such that ✓0 6= ✓, and S✓0 (✓) = S✓0 (✓0 ), S✓1 (✓) = S✓1 (✓0 ), . . . , S✓k

2

(✓) = S✓k

2

(✓0 ).

Note that no point in the list is an endpoint of the dividing line (the point ✓k 1 is such an endpoint), so that Corollary 10.4.21 gives a combinatorial way of calculating the companion angle. Proof Since the companion angle exists, there is such an angle. Uniqueness follows from Theorem 10.4.3. ⇤

10.5 Parameter space for quadratic polynomials Many great mathematicians of the past working on celestial mechanics shared a dream: to understand the parameter space of dynamical systems. How does changing a parameter change how the system evolves? For the June 25, 2015

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99

most part, this dream remains elusive: no one understands the parameter space even of the 3-body problem. The quadratic polynomials under iteration are probably the simplest nonlinear dynamical system, and a rare success story in this field. Proposition in 10.5.1 (A dichotomy for quadratic polynomials) Let p be a quadratic polynomial with critical point !p . Then either !p is in the filled Julia set Kp or it is not, and 1. !p 2 Kp () Kp is connected, 2. !p 2 / Kp () Kp is a Cantor set. Proof This is Theorem 9.1.6, in the case where the polynomial is quadratic, so there is only one critical point; see Remark 9.1.8. ⇤ Proposition in 10.5.2 Let p be a quadratic polynomial with critical point !p . 1. If !p is in Kp , there exists a unique analytic isomorphism p

such that

p (z

2

:C

) = p(

D!C

Kp

10.5.1

p (z)).

2. If !p is not in Kp , then p : Kp ! Kp is conjugate to the one-sided shift on two symbols. Proof Part 1 is Theorem 9.3.4. Part 2 is Theorem 9.1.6, in the case where the polynomial is quadratic. ⇤ To work in parameter space we have to choose a parameter, parametrizing the set of quadratic polynomials. We will write our quadratic polynomials as pc (z) = z 2 + c, parametrized by c, and with critical point 0. Exercise 10.5.3 Show that every quadratic polynomial is conjugate to a unique polynomial of the form z 7! z 2 + c for an appropriate c. } In particular, the “polynomial” p✓ of Section 10.4 is really a conjugacy class, and is conjugate to a unique pc✓ . To lighten notation, we write pc for the polynomial z 7! z 2 +c; we denote its filled Julia set by Kc , its Julia set by Jc , and its external ray at angle t by Rc (t). The Mandelbrot set M is the object that lives in the parameter space for quadratic polynomials. June 25, 2015

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Proposition in and Definition 10.5.4 (The Mandelbrot set) The Mandelbrot set M is M := { c 2 C | 0 2 Kc } = { c 2 C | Kc is connected } .

10.5.2

It is a compact full subset of C contained in the disc of radius 2 centered at 0. Note that the radius 2 is sharp: indeed, 2 2 M since 0 is in K 2 : the orbit of 0 is 0, 2, 2, 2, . . . . Every point c 2 M corresponds to a connected Julia set Kc : the set of z 2 C such that the sequence z, pc (z), pc 2 (z), . . . is bounded, where pc (z) = z 2 + c. Figure 10.5.1 shows six Julia sets. Those marked 2 and 6 are Cantor sets, corresponding to c 2 / M . The others are connected; in these pictures, the shading surrounding Kc represents | p |. The Julia sets 3 and 5 are dendrites; 3 is a trivial dendrite corresponding to c = 2, while 5 corresponds to c = i. Julia sets Kc for c in the boundary of the Mandelbrot set are typically dendrites. The polynomials z 7! z 2 2 and z 7! z 2 are the only quadratic polynomials with nonfractal Julia sets. These two polynomials are postcritically finite; they are the only ones with nonhyperbolic orbifolds. The complexity of the Mandelbrot set is not immediately apparent from Figure 10.5.1. Figure 10.5.2 should convince you that M really is complicated. In the strongest sense, the Mandelbrot set is not self-similar. Every neighborhood of every point in its boundary contains the address of that point, which is di↵erent from the address of all the other points. Proof of Proposition 10.5.4 Since the critical point is 0, the equivalence of the two definitions in equation 10.5.2 follows from Proposition 10.5.1. Let us check that M is in the disc {|z|  2}. The first definition says that c is in M if and only if the orbit of 0 is bounded, i.e., if and only if the sequence c1 := c, c2 := c2 + c, . . . , cn+1 := c2n + c

10.5.3

is bounded. Suppose that |c| > 2, and let us see by induction that n 7! |cn | is an increasing sequence tending to infinity: the inductive hypothesis |c1 | = |c| > 2

|cn |

10.5.4

implies |cn+1 | = |c2n + c| |cn | |c|

|cn |2

1 ,

|c|

|cn |2

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|cn | = |cn | |cn |

1

10.5.5

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Parameter space for quadratic polynomials

101

which implies that the sequence tends to infinity at least as fast as a geometric sequence with ratio |c| 1 > 1. In particular, M is closed, since it is the intersection \ { c 2 C | |cn |  2 } . 10.5.6 n

It is also bounded, so it is compact. That M is full follows from the maximum principle: if the sequence n 7! |cn | is bounded on the boundary of a bounded component of C M , then it is bounded on the interior. ⇤ 2 1 3

4 6 5

Figure 10.5.1 The black at center is the Mandelbrot set M , the object that lives in the parameter space for quadratic polynomials. Every point c 2 C corresponds to a Julia set Kc : the set of z 2 C such that the sequence z, pc (z), pc 2 (z), . . . is bounded, where pc (z) = z 2 + c. We show the Julia sets corresponding to six such points c. For 2 and 6, c is not in M ; for the others, c is in M . For 3, c = 2. June 25, 2015

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Figure 10.5.2 The Mandelbrot set M in Figure 10.5.1 looks complicated but manageable. Successive blowups reveal its immense complication. Here we show the original picture and three blowups: each time the space appears to become more complicated, and it continues this way ad infinitum. There are infinitely many small copies of M dense in the boundary of M (one is visible in the middle of the second blowup, middle left.) Each has the complication of the whole; in addition, each is surrounded by a structure of its own. June 25, 2015

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The Mandelbrot set is connected According to Adrien Douady, one “sows in the dynamical plane and reaps in the parameter space”. Proposition 10.5.5 is the sowing; Theorem 10.5.6 is the reaping. It is the first major step in understanding the structure of M ; it is due to Douady and Hubbard [32]. Let Gc : C ! R be the rate of escape function Gc (z) = lim

n!1

1 + n ln |pc (z)|, 2n

10.5.7

where ln+ (x) := sup(ln(x), 0); see Theorem 9.3.4, where Gc would be denoted Gpc . Here we are studying quadratic polynomials, so d = 2. The B¨ ottcher coordinate below is defined in equation 9.3.8 in Appendix 9.3; here we write it 'c rather than 'pc . Proposition in 10.5.5 Let Uc := { z 2 C | Gc (z) > Gc (0) }. Then the B¨ ottcher coordinate 1/dn

'p (z) := lim (p n (z)) n!1

of the superattracting fixed point at infinity for pc can be analytically continued to an analytic isomorphism n o 'c : Uc ! z 2 C |z| > eGc (0) . 10.5.8 In particular,

1. if c 2 M , then 'c extends to an isomorphism C 2. if c 2 / M , then c is in Uc , so we can define M (c)

:= 'c (c).

M

:C

Kc ! C

D.

M ! C by

10.5.9

Proposition 10.5.5 is proved in greater generality as Theorem 9.3.4. If you consider the complication of M , it is amazing that we have an “explicit formula” 10.5.9 for the conformal mapping of the complement. It is easier to compute the conformal mapping of the complement of M than it is to compute the conformal mapping of the complement of a square! Theorem in 10.5.6 The map M takes its values in C D, and as a map C M ! C D it is an analytic isomorphism. Hence the Mandelbrot set M is connected. Figure 10.5.3 compares the Mandelbrot set and its “evil twin” the tricorn. Both are connected. The Mandelbrot set is conjectured to be locally connected; the tricorn is known not to be locally connected. June 25, 2015

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Proof of Theorem 10.5.6 To prove the first statement, we will see that M is analytic, proper, and of degree 1. Indeed, M proper implies that the image is closed in C D, and all nonconstant analytic maps are open, so the image is open and closed in the codomain. Since C D is connected, M is surjective. Since an analytic map has positive degree at every point, saying that M has degree 1 is saying that every point has exactly one inverse image. So M is bijective. It is analytic because it is the composition of the analytic mappings ✓ ◆ ✓ ◆ c z c 7! and 7! 'c (z). 10.5.10 c c The composition is well defined because for c 2 C M , the image of the first lands in the domain of the second, by part 2 of Proposition 10.5.5. To see that M is proper, we will use the Green’s function 1 X 1 + n 1 + c + ln |p (z)| = ln |z| + ln 1 + . c n n (z))2 n!1 2n 2 (p c n=1 10.5.11 This is relevant, because

Gc (z) = lim

ln |

M (c)|

= Gc (c) = ln+ |c| +

1 X 1 + c ln 1 + . n n 2 (pc (c))2 n=1

10.5.12

If |c| > 2, then 1+

c 2 (pc n (c))2

10.5.13

so Gc (c) ln+ |c| is bounded, and c 7! Gc (c) is a proper map C ! [0, 1). That M : C M ! C D is proper follows immediately, since for all r1 , r2 satisfying 1 < r1 < r2 < 1, the set n o n o c 2 C r1  | M (c)|  r2 = c 2 C ln r1  |Gc (c)|  ln r2 10.5.14

is compact, and any compact subset of C D is a closed subset of such an annulus { z 2 C | r1  |z|  r2 }. Finally, since Gc (c) ln+ |c| is bounded, in particular bounded near infinity, it follows that M (c)/c is bounded near infinity, so M has a simple pole at infinity. Thus M extends as an analytic map C M ! C D, and 1 is its only inverse image, with local degree 1. So M has degree 1. This proves the first statement. The connectivity of the Mandelbrot set follows immediately. ⇤ June 25, 2015

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Figure 10.5.3 The set M and its “evil twin” the tricorn. Left: The Mandelbrot set M and two blow-ups. The set M is filled with little copies of itself; they are dense in the boundary @M . Each is connected to the main body by an “umbilical cord” that leaves from the cusp of the little Mandelbrot set. Right: The tricorn T and two blow-ups. Let gc : C ! C be given by gc (z) = z 2 +c. The filled “tricorn Julia set” KcT is the set of z 2 C such that the sequence z, gc (z), gc 2 (z), . . . is bounded; we denote by T the set of c such that KcT is connected. Like M , the set T is connected (Exercise 10.5.13): just as M is filled with little Mandelbrot sets, the set T is filled with little tricorns, connected to the main body by “umbilical cords”. But in this case the umbilical cord zig-zags towards the little tricorn, accumulating on an arc of the boundary. This implies that T is not locally connected. June 25, 2015

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Orbit portraits Definition in 10.5.7 (Orbit portrait) Let µ2 : R/Z ! R/Z be the angle doubling map µ2 (t) = 2t. An orbit portrait A is a collection of k disjoint finite subsets

such that

A := {A1 , A2 , . . . , Ak , Ak+1 = A1 }, with Ai ⇢ R/Z

1. µ2 (Ai ) = Ai+1 for all i = 1, . . . , k. 2. µ2 : Ai ! Ai+1 is bijective, and preserves circular order. 3. The Ai are pairwise unlinked : if i 6= j, the subset Ai is contained in one component of R/Z Aj . The elements of AA := [i Ai are periodic of some period m(A) that is a multiple of the cardinality k of A. Since the angle doubling map µ2 is bijective, all elements of A have the same cardinality |Ai |, called the valence v(A) of A. An orbit portrait is trivial if v(A) = 1, nontrivial if v(A) > 1. Each element Ai of a trivial orbit portrait is a singleton whose single element is a point of a single periodic orbit under angle doubling. It is less obvious, but Theorem 10.5.9 says that we also know all nontrivial orbit portraits. Although orbit portraits are purely combinatorial objects, we will describe all nontrivial orbit portraits using their relation to dynamics. Definition in 10.5.8 (Orbit portrait of cycle) If z1 , . . . , zk is a repelling or parabolic cycle of pc with c 2 M , and if we set Ai to be the set of angles of external rays landing at zi , then the ordered set A1 , A2 , . . . , Ak is an orbit portrait, called the orbit portrait of the cycle. Remark Repelling and parabolic cycles are defined in Section 9.2. Repelling cycles also often have orbit portraits when c 2 / M . More exactly, if c2 / M , external rays of the Cantor set Kc are defined near infinity, and the external ray Rc (✓) can be continued so as to land at a point of the Cantor set Kc unless 2n ✓ = arg( M (c)) for some n > 0. Thus if z1 , . . . , zk is a periodic cycle in the Cantor set Kc (necessarily repelling) and external rays do land there, then the angles Ai , i = 1, . . . , k of rays landing at zi will form an orbit portrait, and then arg( M (c)) 2 / AA . Figure 10.4.9 shows a repelling cycle; each set of three angles forms one Ai . 4

Recall (Exercise 10.5.3) that the “polynomial” p✓ of Section 10.4 is really a conjugacy class: it is conjugate to a unique pc✓ . When we know that a polynomial is of the form pc✓ for a specific ✓, we will write p✓ , K✓ , J✓ , as we do in Theorem 10.5.9. June 25, 2015

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Theorem in 10.5.9 Every nontrivial orbit portrait A is the orbit portrait

of the (repelling) cycle of roots yi of the components of K✓ containing the critical orbit of p✓ for some ✓ 2 Qodd /Z {0}.

We then say that A is generated by ✓ and its companion angle ✓0 , and is denoted A⇥ , where ⇥ = (✓, ✓0 ). Corollary 10.5.10 follows from Theorem 10.5.9 and Proposition 10.4.18 Corollary in 10.5.10 There are two kinds of nontrivial orbit portraits: 1. either m(A) = k and v(A) = 2 (primitive portrait), or 2. m(A) = kv(A) (satellite portrait). Example 10.5.11 (Orbit portraits) 1. The simplest nontrivial orbit portrait is ⇢⇢ 1 2 {A1 } = , . 3 3

with k = 1, v = 2, and period m = 2, so it is a satellite. This is consistent with our previous use of the word “satellite” (Definition 10.4.20): 1/3 and 2/3 share the same orbit under angle doubling. 2. The next simplest, also a satellite, is generated by 1/7 and 2/7: ⇢⇢ 1 2 4 {A1 } = , , , k = 1, v = 3, m = 3. 7 7 7 3. The simplest primitive orbit portrait is generated by 3/7 and 4/7: (⇢ ) ⇢ ⇢ 1 6 2 5 4 3 , , , , , , k = 3, v = 2, m = 3. 7 7 7 7 7 7

4. The next simplest primitive orbit portrait is generated by 3/15 and 4/15: (⇢ ) ⇢ ⇢ ⇢ 3 4 6 8 12 1 9 2 , , , , , , , , 15 15 15 15 15 15 15 15 with k = 4, v = 2, m = 4.

4

Exercise 10.5.12 Find the orbit portrait corresponding to the compan820 835 ion pair 4095 , 4095 . Hint: See Figure 10.4.9. } Proof of Theorem 10.5.9 Let A = {A1 , . . . , Ak } be a nontrivial orbit portrait, and let Ii be the intervals of R/Z AA bounded by two elements of Ai . Among all the intervals of all the Ii find a shortest one I = [✓, ✓0 ]. A June 25, 2015

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priori there might be several (actually it is unique), so chose one. Renumber the Ai so that ✓, ✓0 2 A1 . Denote by I 0 and I 00 the two components of µ2 1 (I). Each has half the length of I, so one endpoint of each is in A0 = Ak and I 0 and I 00 contain no other points of AA . We obtain Figure 10.5.4. I ✓0

✓

I0

✓0 2

Figure 10.5.4. Either both red points or both purple points belong to A0 ; there are no other points of AA := [i Ai in I 0 [ I 00 . The unlinking property of the Ai says that all points of Ai are in the same component of the complement of the red or purple points, hence also in the same component of both dividing lines.

✓ 2

✓+1 2 &

I 00

-

✓0 + 1 2

By the non-linking property of an orbit portrait, all points of each Ai have the same ✓-kneading sequence and the same ✓0 -kneading sequence. So ✓ and ✓0 are companions, and A is the portrait of the orbit y1 , . . . , yk formed by the roots of V1 , . . . , Vk . ⇤

The MLC conjecture In Theorem 10.4.3 we gave a complete description of the filled Julia set K✓ for ✓ 2 Qodd /Z, based on the existence of a Caratheodory loop. The MLC conjecture (Mandelbrot set Locally Connected ) asserts that the Mandelbrot set is locally connected. By Caratheodory’s theorem, this is equivalent to saying that 1 M

:C

D!C

M

10.5.15

extends continuously to the boundary S 1 = @D, giving a Caratheodory loop M : R/Z ! @M for M . There is a lot of theoretical evidence for this conjecture (due in particular to Yoccoz, Lyubich, and Kahn; see [54] and [60]), and overwhelming computer evidence. As far as I know, no one doubts that MLC is true, but as of this writing, it remains unproved. The tricorn family shown in Figure 10.5.3 (right) illustrates the difficulties that might arise. What we have shown so far about the Mandelbrot set is true of the tricorn, but the tricorn is known not to be locally connected. Exercise 10.5.13 Prove that either 0 is in the filled “tricorn Julia set” KcT , and KcT is connected; or 0 2 / KcT and KcT is a Cantor set. Prove that T is connected. } June 25, 2015

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Some features of the Mandelbrot set A quadratic polynomial can have at most one attracting cycle since it only has one critical point. By the implicit function theorem, the set of c such that pc has an attracting cycle is open, and consists of some union of components of the interior M . In each such component, the period of the attracting cycle is constant. Definition in 10.5.14 (Hyperbolic component of M )

A hyperbolic

component of period k of M is a component consisting of c’s such that pc has an attracting cycle of period k. By an argument that we will not give, MLC implies that all components of the interior of M are hyperbolic. Figure 10.5.5 and its “cartoon version” 10.5.6 illustrate some features of the Mandelbrot set. Each hyperbolic component of period k of the interior of M has a “center” that is c✓ = c0✓ for some companion pair of angles (✓, ✓0 ) whose elements are periodic of period k under angle doubling. The points c✓ serve as “organizing centers” of M . Several are marked in Figure 10.5.6. The figure also shows points b c✓ = b c✓0 where companion pairs of external rays land. Definition in 10.5.15 (External ray of M ) ray of M at angle t is RM (t) :=

c2C

M

M (c)

For t 2 R/Z, the external

= re2⇡it , 1 < r < 1

.

10.5.16

If MLC holds, giving a Caratheodory loop M : R/Z ! @M , then the “companion equivalence relation” on angles leads to a pinched disc description of M : we have M (t1 ) = M (t2 ) if and only if the pair (t1 , t2 ) is in the closure of the companion equivalence relation. Understanding companion pairs would lead to a combinatorial description of the topology and much of the geometry of M . In the absence of MLC, we have the following. If (✓, ✓0 ) are companion angles, the external rays RM (✓) and RM (✓0 ) land at a point b c✓ = b c✓0 on the boundary of the component U✓ of M containing c✓ . The point b c✓ = b c✓0 is called the root of U✓ ; it is also called a parabolic point of M , since pbc✓ has a parabolic cycle. Rays corresponding to di↵erent companion pairs land at distinct points. This can happen in two ways: • If (✓, ✓0 ) is primitive (✓ and ✓0 belong to di↵erent orbits), then b c✓ is a

cusp on the boundary of @U✓ , and no other component of M has b c✓ in its 3 4 closure. See for instance ( 15 , 15 ) and ( 37 , 47 ) in Figure 10.5.6. June 25, 2015

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1

3/4

0

" 1/4

Figure 10.5.5 The Mandelbrot set, shown in black. The external rays of M (also shown in black) land as described in Theorem 10.5.16. If (✓, ✓0 ) are companion angles, the external rays RM (✓) and RM (✓0 ) meet at a point on the boundary of the component U✓ of M containing c✓ . This point b c✓ = b c✓0 is called the root of U✓ . For instance, RM (1/3) and RM (2/3) meet at the point 3/4; this point is the root of the component U1/3 , which turns out to be a circle of radius 1/4 centered at 1. Rays corresponding to di↵erent companion pairs land at distinct points. We have drawn all rays with angles in Qodd /Z of period  5 under angle doubling. Figure 10.5.6 is a cartoon emphasizing the combinatorics. The main cardioid above corresponds to the light blue region of the cartoon: the set of c 2 M such that pc has an attracting fixed point; its root is at the cusp 1/4. The disc centered at 1 corresponds to the green disc in Figure 10.5.6: the set of c 2 M such that pc has an attracting cycle of period 2.

• page problem If (✓, ✓0 ) is a satellite (✓ and ✓0 belong to the same orbit), b c✓ is a smooth point of @U✓ , and another component corresponding to polynomiJune 25, 2015

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Parameter space for quadratic polynomials

als

of

2 1/3

111

lower

4 3 1 2

c1/5

.

6

-b c1/5

3

1

7

c1/3 b

"

c1/3

b c0 !

0 = c0

8

14 4 9 13 6 12

2/3 5

11

Figure 10.5.6 A cartoon version of Figure 10.5.5, showing all rays with rational angles with denominators 3, 7, 15, and 31, i.e., all angles that repeat with period  5 under angle doubling. Light blue corresponds to period 1, green to period 2, purple to period 3, red to period 4, and dark blue to period 5. We have marked c✓ and b c✓ for ✓ = 0, ✓ = 1/3, and ✓ = 3/15; note from Figure 10.5.5 that c1/3 = 1 and b c1/3 = 3/4. The yellow region is the (5/7, 6/7)-wake; see Theorem and Definition 10.5.16. The components directly attached to the light blue component, corresponding to the main cardioid of M , are satellites of that component. The components shown unattached are primitive. Both satellites and primitive components have further satellites, some of which are shown. Examples of parabolic points are b c1/3 and b c1/5 ; the latter corresponds to a cusp.

period also has b c✓ as a smooth point of its boundary. See for instance ( 13 , 23 ) 5 6 and ( 7 , 7 ). June 25, 2015

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Some of the properties of the Mandelbrot set discussed above are spelled out and justified by Theorem and Definition 10.5.16. Theorem in and Definition 10.5.16 Let ⇥ := {✓, ✓0 } be a pair of companion angles in Qodd /Z, generating the orbit portrait A⇥ with period m(A⇥ ). Then 1. The external rays RM (✓) and RM (✓0 ) land at the same point b c⇥ := b c✓ = b c✓0 . 2. The polynomial pbc⇥ has a parabolic cycle z1 , . . . , zk with m(A⇥ ) 0

3. The union

(pbc⇥

) (zi ) = 1.

10.5.17

RM (✓) [ RM (✓0 ) [ {b c⇥ }

10.5.18

forms a curve in C that separates c⇥ from 0. The part containing c⇥ is called the (✓, ✓0 )-wake, also called the ⇥-wake. 4. A polynomial pc , c 2 C has a repelling cycle with orbit portrait A⇥ if and only if c belongs to the ⇥-wake. Note that this theorem tells us very little about the polynomial pbc⇥ ; in particular it does not tell us that b c⇥ is on the boundary of the component U✓ of M containing c✓ .

Proof Since only one companion pair ⇥ is under consideration, we will omit the index, and write A = A⇥ ,

A = AA⇥ ,

m = m(A⇥ ),

etc.

10.5.19

Consider the union X of the external rays RM (t), t 2 A of M , and the finite set Zm of all c such that pc has a cycle z1 , . . . , zk with (pc m )0 (zi ) = 1. Let us see that C X is open: • If c is in M , then, by Proposition 9.5.1, all external rays of Kc at rational angles land at repelling or parabolic points. In particular, for all c 2 (C X) \ M , all rays in A land at points of repelling cycles, since those c such that pc has a parabolic cycle of the appropriate period and multiplier belong to Zm . The pattern of which such rays land at the same point is constant in a neighborhood of c. • If c is in C M , set t(c) := arg( M (c)) where the argument is counted in turns. All external rays of Kc land at points of repelling cycles except those at angles t such that 2` t = t(c) for some `. The angles in A are periodic, and we have removed the rays RM (t), t 2 A. Thus for c 2 (C X) \ (C M ), all external rays of Kc at angle t with t 2 A June 25, 2015

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land at repelling cycles. Again the pattern of landing is constant in a neighborhood of c. Thus X is closed, and all rays RM (t), t 2 A land at points of Zm : they must accumulate on some subset of Zm , but their accumulation set is connected. This proves parts 1 and 2. It actually shows more: the pattern of identifications of ends of external rays RM (t), t 2 A is constant on the components of C X. Since all components of C X are unbounded, parts 3 and 4 follow from Lemma 10.5.17. Lemma 10.5.17 A polynomial pc with c 2 / M has a repelling cycle with orbit portrait A if and only if ✓ < t(c) < ✓0 . Proof of Lemma 10.5.17 By Theorem 10.5.6, the polar angle t(c) of M ) is the same as that of 'c (c) (the M (c) (i.e., the polar angle of c in C polar angle of c in C Kc ). Since c is the critical value of pc , the two rays at the half-angles t(c)/2 and (t(c) + 1)/2 meet at the critical point 0, cutting the plane into two pieces A and B, labeled so that the ray at angle 0 is in A. Recall (Proposition 10.5.1, part 3) the description of Kc for c 2 / M as the shift on two symbols: define the map c : Kc ! {A, B}N by setting i i c(t) (z) = (s0 , s1 , . . . ) where si = A if pc (z) 2 A and si = B if pc (z) 2 B. (The resulting sequence of A’s and B’s is still called the kneading sequence of z.) Then c is a homeomorphism conjugating pc |Kc to the one-sided shift on two symbols; see Figure 10.5.7. t(c) ✓0

✓ ✓0 2 &

B ✓+1 2 #

-✓ 2 ✓0

"

+1 2

A

Figure 10.5.7 Let ✓ < t(c) < ✓0 . If the orbit of an angle s under angle doubling never visits the intervals ✓ ◆ ✓ ◆ ✓ ✓0 ✓ + 1 ✓0 + 1 , or , , 2 2 2 2 then S✓ (s) = S✓0 (s) = t(c) (s), where S✓ is the ✓-kneading sequence. This is the case for the angles s 2 A, so they form the orbit portrait A.

In particular, if two angles belong to the same Ai , they have the same ✓-kneading sequence and the same ✓0 -kneading sequence, and these two sequences coincide: the orbits of the angles never visit either of the intervals (✓/2, ✓0 /2) or ((✓+1)/2, (✓0 +1)/2). Thus if t(c) is in the open interval (✓, ✓0 ), the external rays of Kc with the same angles will land on the repelling periodic point on Kc with the same symbolic sequence. If t(c) 2 / (✓, ✓0 ), then there will be two angles of A0 that lie on opposite sides of the dividing June 25, 2015

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line, so the rays at angles in A will not form the orbit portrait A. ⇤ Lemma 10.5.17 and Theorem 10.5.16 Corollary in 10.5.18 (Companion pairs are unlinked) If W1 and W2 are any two wakes, then either W1 ⇢ W2 or W2 ⇢ W1 or their interiors are disjoint. Proof Suppose the wakes correspond to the companion pairs ⇥1 = (✓1 , ✓10 ) and ⇥2 = (✓2 , ✓20 ). Choose a point c 2 W 1 \W 2 . Then (by Theorem 10.5.16, part 4) pc has two repelling cycles, one with orbit portrait A⇥1 and one with orbit portrait A⇥2 . If the pairs ⇥1 and ⇥2 are pairwise linked, then the external rays of Kc at all four angles ✓1 , ✓10 , ✓2 , ✓20 must land at the same point, contradicting the fact that they must land at points of distinct cycles. ⇤ This doesn’t quite complete our description of M . What we have proved so far would allow for the configurations shown in Figure 10.5.8.

✓2

✓10

✓2

✓20 c⇥1

c⇥1 c⇥2

✓1

c⇥2

✓1

✓10

✓20 b c⇥1 = b c⇥2

b c⇥1 = b c⇥2

Figure 10.5.8 These two configurations in parameter space satisfy Corollary 10.5.18, but b c⇥1 = b c⇥2 , contradicting Corollary 10.5.20.

Corollary 10.5.20 eliminates these possibilities. Recall that Theorem 10.5.16, part 2 guarantees that pbc⇥ has a parabolic cycle. Proposition in 10.5.19 Let ⇥ = {✓, ✓0 } be a pair of companion angles with orbit portrait A⇥ of cardinality k and period m(A⇥ ). Then 1. The parabolic cycle of pbc⇥ has orbit portrait A⇥

2. The point b c⇥ is on the boundary of a component U of M such that when c 2 U , then pc has an attracting cycle of period m(A⇥ ), and a repelling cycle with orbit portrait A⇥ on the boundary of the basin of attraction of the attracting cycle.

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Corollary in 10.5.20 If ⇥1 = {✓1 , ✓10 } and ⇥2 = {✓2 , ✓20 } are distinct companion pairs, then b c⇥1 6= b c⇥2 .

Proof of Corollary 10.5.20 This follows from the fact that a quadratic polynomial can have only one parabolic cycle. ⇤ Corollary 10.5.20 Proof of Proposition 10.5.19 To lighten notation, set pe := pbc⇥ . By Theorem 10.5.16, pe is a quadratic polynomial with a parabolic cycle. We begin by choosing families ct with special properties. The primitive and the satellite cases are a bit di↵erent, so we will discuss them separately. Let A be the orbit portrait of the parabolic cycle z1 , . . . , zk of pe of period exactly k, and suppose the angles of A have period m under angle doubling, so k divides m. By Proposition 9.5.1, 0

• In the primitive case, pe k (zi ) = 1, and m = k. 0

• In the satellite case, pe k (zi ) is a primitive (m/k)th root of unity.

The primitive case. Proposition 9.2.11 asserts that (when e2pip/q = 1 and ⌫ = 1) for any ✏ > 0 there exists a neighborhood U of z1 and an analytic map '0 : U [ pe k (U ) ! C that is an isomorphism to its image and satisfies f0 (w) := '0 pe k

'0 1 (w) = w + w2 h0 (w)

10.5.20

with h0 (0) = 1 and with |h0 (w) 1| < ✏ on '0 (U ). Figure 10.5.9, left, represents the map f0 . The attracting and repelling regions are called petals of the Fatou flower. In Figure 10.5.9 the flower has only one attracting and one repelling petal; a prettier Fatou flower is shown in Figure 10.5.11.

S⇢att !

S⇢rep

S⇢att !

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Figure 10.5.9 A primitive case. The dynamical plane for f0 and the perturbed map ft defined in equation 10.5.22. The black curves are external rays. Left: For the unperturbed map f0 , the fixed point 0 is attracting on the left and repelling on the right. When ✏ is sufficiently small, the sector S⇢att (light tan) is mapped to the dark tan region inside S⇢att : all points of S⇢att are attracted to 0 under f0 . Similarly, S⇢rep (light blue) is mapped to the dark blue region, containing S⇢rep . Right: The perturbed map ft has two fixed points, one attracting and one repelling.

There exists ⇢ > 0 such that the disc D⇢ := { w 2 C | |w| < ⇢ } is contained in '0 (U ), and by restriction we may assume that '0 (U ) = D⇢ . Using this choice of U , we see that if ⇢ < 1/(1 + ✏), then the description of the sectors below as attracting and repelling is justified. Define the regions S⇢rep = { w | |Im w| < Re w, |w| < ⇢ } S⇢att = { w | |Im w|
0,

arg at  ✏, |ht 1| < ✏ on 't 1 (D⇢0 ).

Figure 10.5.9, right, illustrates the lemma for t > 0. Proof of Lemma 10.5.21 There exists a neighborhood V of b c⇥ such that k for c 2 V , the map pc has two fixed points in U close to z1 ; further, we may assume that these fixed points are distinct for c 6= b c⇥ in V . These fixed points are not a priori functions on V : they are functions on an appropriate double cover. Consider the plane curve Ve := (c, z) 2 V ⇥ U pc k (z) = z ; this curve contains the point e c0 = (b c⇥ , z1 ). We will study Ve near this point. June 25, 2015

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Recall that z1 is the first point of the parabolic cycle for pbc✓ . The set ⇤ e V := Ve {e c0 } is a double cover of V ⇤ := V {b c⇥ }. (In fact it is ramified, but we don’t know that yet.) Then ↵ : (c, z) 7! z is a map Ve ! U that selects a fixed point of pc k ; let : Ve ! U be the map that selects the other fixed point close to z1 , i.e., ↵ composed with the deck transformation of Ve . The multiplier function m e : (c, z) 7! (pc k )0 (↵(c, z))

10.5.23

m(c e t , zt ) = 1 + t

10.5.24

is a nonconstant analytic function on Ve with m(e e c0 ) = 1, and as such there exists a continuous curve t 7! (ct , ut ) in Ve such that for t 2 [0, ] for some > 0 sufficiently small. Let t 7! (ct , vt ) be the curve obtained by applying the nontrivial deck transformation to t 7! (ct , ut ). We now find 'c0

pctk

'c01 (w) = w + (w

ut )(w

vt )kt (w)

10.5.25

for some function kt on U arbitrarily close to h0 for sufficiently small. Set ✓ ◆ ut + vt 't (w) := '0 w , 10.5.26 2

which is an isomorphism U 0 ! D⇢0 for ⇢0 < ⇢ when t is sufficiently small. Then 't pc k

't 1 (w) = w + (w2

a2t )hc (w)

10.5.27

on D⇢0 , where at = (vt ut )/2 and the function hc is arbitrarily close to kc , hence with |hc 1| < ✏ for t sufficiently small. Finally, since 1 + t = ft0 (at ) = 1 + 2at h(at ), we see that at = 2htt(at ) , and since ht is close to 1, we see that at is in a narrow wedge around the positive real axis. ⇤ Lemma 10.5.21 Lemma 10.5.22 Let ft be as in equation 10.5.22. For t sufficiently small, the sector S⇢rep := { w | |Im w| < Re w, |w| < ⇢0 } 0

10.5.28

is repelling for ft , and the sector S⇢att := { w | |Im w| < 0

Re w, |w| < ⇢0 }

10.5.29

is attracting. Proof of Lemma 10.5.22 We will show that the boundary of S⇢rep is 0 mapped outside S⇢rep 0 . For w in the part of the line Re w = Im w where June 25, 2015

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Im w 0 and |w|  ⇢0 , of course w2 is purely imaginary, so since at is nearly real and positive, the vector w2 a2t makes an angle with the line that is only slightly smaller than ⇡/4, say bounded below by ⇡/6, and since h(w) ⇡ 1, the increment (w2 a2t )h(w) points strictly outside of S⇢rep 0 . att The argument for the part of the boundary of S⇢0 where Re w = Im w, Im w  0 is identical. So far we have only used the fact that at is nearly real and positive; now we need that it is small. On the part of @S⇢rep where |w| = ⇢0 ei✓ the 0 2 0 2 2i✓ 2 0 2 term w = (⇢ ) e points out, and since |at |  (⇢ ) /2 for t sufficiently small, w2 a2t points out also. We leave to the reader to check by a similar argument that S⇢att is mapped inside itself. ⇤ Lemma 10.5.22 0 The argument so far justifies that Figure 10.5.9 describing the primitive case is correct. The satellite case. We now prove Proposition 10.5.19 in the satellite case. Again to lighten notation, set pe := pbc⇥ . In the satellite case, pe has a 0 parabolic cycle z1 , . . . , zk of period exactly k such that 0 := pe k (zi ) is a primitive qth root of unity with q > 1. Proposition 9.2.11 asserts that there exists a neighborhood U of z1 and an analytic map '0 : U [ pe k (U ) ! C that is an isomorphism to its image, with '0 (U ) = D⇢ such that f0 (w) := '0 pe k

'0 1 (w) =

0 w(1

+ wq + o(wq )).

10.5.30

Unlike the primitive case, in the satellite case the fixed point z1 of pe k does not bifurcate: the entire cycle z1 , . . . , zk moves holomorphically with c. But when this same point z1 is viewed as a fixed point of pe kq , it does bifurcate; see Figure 10.5.10. By the implicit function theorem, there is an analytic function ↵ : V ! C defined on a neighborhood V of b c⇥ such that ↵(c) is a fixed point of pc k and ↵(b c⇥ ) = z1 . Set (c) := (pc k )0 (↵(c));

10.5.31

then is a nonconstant analytic function defined on V , and (b c⇥ ) = 0 . Write t = (1 + t) 0 . There exists a parametrized curve t 7! ct in V such that (ct ) = t . Temporarily we will assume that t 0 and small; the case t  0 is also interesting. We want to make a change of variables analogous to that of Lemma 10.5.21, but here it seems reasonable to center the new variable at ↵(ct ). Lemma 10.5.23 There exists a local coordinate 't such that ⇣ ⌘ ft (w) := 't pctk 't 1 (w) = t w 1 + wq + o(wq ) . June 25, 2015

10.5.32

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119

Proof of Lemma 10.5.23 Set t (z)

Then we have t

⇣ := '0 z

pctk

t

1

(w) =

⌘ ↵(ct ) + z1 . tw

where gt (w) = a1 w + · · · + aq

1w

q 1

⇣ ⌘ 1 + gt (w)

+ aq wq + o(wq )

10.5.33

10.5.34

10.5.35

with |a1 |, . . . , |aq 1 |, |aq 1| all arbitrarily small for t sufficiently small. The same formal power series argument as in Proposition 9.2.11 shows that by a further succession of q changes of variables, all arbitrarily close to the identity, we can eliminate a1 , . . . , aq 1 , and make aq = 1. Set 't to be the resulting change of variables: we have achieved ⇣ ⌘ 't pctk 't 1 (w) = t w 1 + wq + o(wq ) , 10.5.36

⇤ Lemma 10.5.23

as required.

z1

z2

z3

z4

Figure 10.5.10 In the satellite case the fixed point z1 of pe k does not bifurcate. However, when z1 is viewed as a fixed point of pe kq , it does bifurcate, into a cycle of period k, and a cycle of period kq, consisting of q points in a small neighborhood of each of z1 , . . . , zk . Here k = 4 and q = 3; the blue points z1 , . . . , z4 , form a cycle of period 4, represented by the blue arrows. The red arrows represent a cycle of period kq = 12. This is a cartoon of Figure 10.4.9; each disc represents one of the four blow-ups of that figure.

We next need an analogue of Lemma 10.5.22 adapted to the satellite case, illustrated by Figure 10.5.11. Each of the two flowers in Figure 10.5.11 represents what one sees in one disc in Figure 10.5.10: on the left, the case where the fourth iterate has a parabolic fixed point; on the right, the case where it has a repelling fixed point with real multiplier. June 25, 2015

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Figure 10.5.11 A satellite case. Left: The attracting and repelling petals of the Fatou flower in the case 0 = e2⇡i/3 . The three external rays enter through the three repelling sectors. The map w 7! e2⇡i/3 w(1 + w3 + o(w3 )) mainly rotates by a third of a turn, but it also maps each light blue petal to the dark blue petal outside the light blue petal a third of a turn around, and maps each light tan petal to the dark tan petal inside the light tan petal a third of a turn around. Right: In the perturbed case (Lemma 10.5.24) the petals are almost the same as on the left, but the dynamics are quite di↵erent. There is now an attracting cycle in the dark tan petals, which attracts all of the pale tan petals (and more besides). Under the inverse map, the blue petals (and a neighborhood of the origin) are attracted to the center.

Lemma 10.5.24 For all ✏ > 0 and t

0 sufficiently small, the sectors

S✏rep := { w 2 C | Re 1/wq > 1/✏ }

10.5.37

are repelling for ft , and the sectors S✏att := { w 2 C | Re 1/wq
1; they are unions of curves corresponding to periods dividing n. It is true but not obvious that for each such divisor j, the closure of the set of (c, z) such that z has period exactly j under pc is an irreducible component Yj of Xn [16, 94]. In many settings, it is a good idea to study the irreducible curves Yj ; in the present setting it just complicates things. June 25, 2015

10.5

Parameter space for quadratic polynomials

123

0

is 0 if and only if Qn and Qn have a root in common. But di↵erentiating Qn (c) = Q2n 1 (c) + c and reducing mod 2 we find 0

0 1 (c)Qn 1 (c)

Qn (c) = 2Qn

+ 1 = 1,

10.5.41

since 2 = 0 in Z/2. In particular this derivative has no roots, so none in common with Qi . Thus the discriminant of Qi is an odd integer, hence nonzero. ⇤ With this under our belt, we can proceed to use more standard methods. Recall that U is a component of M such that for some c 2 U , the polynomial pc has an attracting cycle of period n, and that the multiplier map mU is defined in Theorem 10.5.25. Proposition in 10.5.27 Let U ⇤ be the complement of the polynomials with superattracting points, and set D⇤ := D {0}. Then the restriction mU : U ⇤ ! D⇤ is a finite covering map. Proof Recall that a map is a finite covering map if and only if it is proper and a local homeomorphism. The map Xn ! C given by (c, z) 7! (Pc n )0 (z) is a proper map; its restriction to U is the restriction to one component of the inverse image of the unit disc. As such mU : U ⇤ ! D⇤ is proper. We will show that mU is a local homeomorphism by writing a local inverse; this is fairly elaborate, involving Beltrami forms on C invariant under pc . This is an important tool in complex dynamics, central in particular in Sullivan’s proof of the no wandering domain theorem (see Appendix C7.1). Choose (c, z) 2 U ; then z is a linearly attracting fixed point under pc n , and there exists a linearizing coordinate, i.e., there exist ⇢ > 0, a neighborhood W of z in C, and an isomorphism ' : V ! D⇢ such that mU '(w) = '(pc n (w)). Begin by finding Beltrami forms on D⇢ invariant under multiplication by := mU (c). To do this, we invite the reader to check that the diagram H⇢t ? ? y⇣7!⇣+ 1

H⇢µ

⇣7!⇣+t⇣

!

1 |t|2

ln

t

!

⇣7!⇣+t⇣

H⇢ ? ? y⇣

⇣7!e⇣

!

!⇣+ln

H⇢

!

⇣7!e⇣

D⇢ ? ? yz7!

z

10.5.42

D⇢

commutes, where H⇢ is the halfplane Re ⇣ < ln |⇢|, and H⇢t is the halfplane Re (⇣ + t⇣) < ln |⇢|. Of course, the top line is not analytic from left to right: it is quasiconformal, and can be made conformal if we change the conformal structure either in the domain (obviously) or in the codomain (less obviously: that June 25, 2015

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requires knowing that the diagram commutes). We define µt to be the Beltrami form on D⇢ such that ⇣ 7! e⇣+t⇣ is analytic, going from H⇢t with the standard complex structure to D⇢ with the Beltrami form µt . As a map (D⇢ )µt ! (D⇢ )µt , the map z 7! z has derivative ✓ ◆1 |t|2 . 10.5.43 t

Now we pull this Beltrami form µt back to W by ': set ⌫t = '⇤ µt . It is important to realize that ⌫t can now be pulled back to all of the basin of (n 1) attraction of the cycle z, pc (z), . . . , pc (z); we will still call the resulting Beltrami form ⌫t , and we will keep that name even after extending ⌫t by 0 to all of C by setting it to be 0 at points not in the basin. Let t : C ! C be a quasiconformal homeomorphism integrating this Beltrami form; this map t exists by the mapping theorem (Theorem 4.6.1). Normalize t so that 0 is fixed and t is tangent to the identity at 1 (this is possible because t is analytic in the complement of Kc , hence in a neighborhood of 1; see Theorem 4.7.4). The map ft :=

t

pc

t

1

:C!C

10.5.44

is a quadratic polynomial; in fact our normalizations force ft (z) = z 2 + ct for some analytic function t 7! ct . The inverse image of the attracting cycle 2 t by t is an attracting cycle for ft , but its multiplier is now ( / )1 |t| . t 1 |t|2 The map t 7! ( / ) is an analytic isomorphism of a neighborhood of 0 in the t-plane to a neighborhood of in the multiplier plane; let be a local inverse, defined in some neighborhood V of . Then the map ⌘ 7! (c (⌘), (⌘)) is a local inverse of the multiplier map mU , showing that mU is a local homeomorphism. ⇤ Proposition 10.5.27 We can now prove part 2 of Theorem 10.5.25: that mU : U ! D is an analytic isomorphism. Proposition 10.5.27 shows that mU : U ! D is a ramified cover, ramified only above 0. So if we can show that m0U (c0 ) 6= 0 when (c0 , 0) 2 Xn and 0 has period exactly n under pc0 , we will be done. By the chain rule and p0c (z) = 2z, we see that mU (c) = 2n z0 (c) · · · zn

1 (c).

10.5.45

where the zi are labeled so that z0 (c0 ) = 0. Di↵erentiating gives ⇣ ⌘ m0U (c0 ) = 2n z00 (c0 )z1 (c0 ) · · · zn 1 (c0 ) + z0 (c0 )z10 (c0 ) · · · zn 1 (c0 ) + · · ·

and since z0 (c0 ) = 0, only the first term on the right can be nonzero, and moreover all the factors z1 (c0 ), . . . , zn 1 (c0 ) are nonzero. So if we can show that z00 (c0 ) 6= 0, we will be done. June 25, 2015

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Parameter space for quadratic polynomials

125

Di↵erentiate the equation Pn (c, z0 (c)) = z0 (c): @Pn @Pn (c, z0 (c)) + (c, z0 (c))z00 (c) = z00 (c). 10.5.46 @c @z Evaluate this at (c0 , 0). To get the first term, di↵erentiate Pn (c, 0) = Qn (c) and evaluate the derivative at c0 ; this gives Q0n (c0 ), which we know to be nonzero by Proposition 10.5.26. The factor @Pn (c0 , 0) 10.5.47 @z of the second term vanishes: it is the multiplier of the superattracting cycle. Putting these together, we see that z00 (c0 ) = Q0n (c0 ) 6= 0. ⇤ Theorem 10.5.25 We now know that mU is an isomorphism. Since U is bounded by a real-algebraic curve, the map mU 1 : D ! U extends to the closure, giving a map S 1 ! @U , which is a homeomorphism since M is full. The root of U is the image of 1, whereas its center, i.e., the only c0 2 U such that pc0 has a superattracting point, must be a polynomial of the form pc⇥ . Thus every hyperbolic component of M is of the form U⇥ , with center c⇥ for some companion pair ⇥ := {✓, ✓0 }. Moreover, Theorem 10.5.16 and Proposition 10.5.19 say that the external rays RM (✓) and RM (✓0 ) meet at the root b c⇥ of U⇥ . To complete our description of M we want one more result. Proposition in 10.5.28 If ⇥ is a primitive companion pair, b c⇥ is a cusp of the boundary of U⇥ . If ⇥ is a satellite with orbit portrait A⇥ = {A1 , . . . , Ak } with each |Ai | = q 2 so that c⇥ has period kq, then b c⇥ is both a smooth point of @U⇥ and a smooth point of the boundary of a component corresponding to attracting cycles of period k.

Proof Let us summarize what we already know. For each companion pair ⇥, let U⇥ be the component of M with center c⇥ . This component U⇥ consists of all the polynomials with an attracting cycle, and with a repelling cycle with orbit portrait P⇥ on the boundary of the basin of attraction. Moreover, for each ⇥ the multiplier map mU⇥ : U⇥ ! D of the attracting cycle is an isomorphism. We will discuss the primitive case and the satellite case separately. The primitive case. We defined the proof of Lemma 10.5.21 the double cover Ve :=

(c, z) 2 V ⇥ U

pc k (z) = z June 25, 2015

with e c0 = (b c⇥ , z1 ).

10.5.48

126

Chapter 10.

Rational functions

This double cover has a single branch, i.e., it is a double cover of V ramified above b c⇥ . (If it consisted of two branches, then we could construct pct in each branch, which would give two curves leading into two distinct copies of U⇥ . But U⇥ does not have two components.) The map mU⇥1 : D ! U⇥ lifts uniquely to a map m e U⇥1 : D ! Ve , picking out the point of the attracting cycle in the component containing the critical value; see equation 10.5.23. Since m e is defined on Ve , with values in a neighborhood of 1, and since the inverse image of the unit disc is connected, we see that m e U⇥1 must be an isomorphism to its image. In particular the inverse image of the boundary of the disc is a smooth curve through the ramification point. Finally, Proposition 10.5.28 follows in the primitive case from the following generality: if C is a smooth curve through the origin in C and the map z 7! z 2 is injective on C, then the image of C has a cusp. The satellite case. This is easier. For c near a point b c⇥ where A⇥ is a satellite orbit portrait of period m(A⇥ ), there is a single cycle of period m(P⇥ ) near the parabolic cycle. The multiplier of this cycle is well defined in a neighborhood of b c⇥ . The region where |m| < 1 is again connected, since it is U⇥ , so the multiplier map is a local isomorphism, and the boundary @U⇥ is smooth at b c⇥ . ⇤

10.6 The Thurston pullback mapping σf

Sections 10.2–10.5 were devoted to seeing what Theorem 10.1.14 on the topological characterization of rational functions says about quadratic polynomials. Now we prove the theorem. Recall from the outline given in Section 10.1 that the proof depends on defining a map f from an appropriate Teichm¨ uller space to itself, and looking for a fixed point of that mapping. Let f : S 2 ! S 2 be a Thurston mapping of degree d, i.e., a postcritically finite ramified covering map of degree d. Recall that Crit f denotes the set of critical points, and Pf the set of postcritical points. Set Tf := T(S 2 Pf ) . We often find it easier to think of the points of Pf as “pinned” rather than removed, and write (S 2 , Pf ) rather than S 2 Pf ; see Figure 10.6.1. The Teichm¨ uller space Tf is the Teichm¨ uller space of a Riemann surface of genus 0, with Pf removed, and so by Proposition 7.1.1 has dimension |Pf | 3. Teichm¨ uller spaces of genus 0 are easier to understand than general Teichm¨ uller spaces, but this will not help as much as we might hope. The Thurston pullback mapping f : Tf ! Tf is defined as follows. June 25, 2015

10.6

The Thurston -mapping

127

Definition in 10.6.1 (Thurston pullback map f ) Let f be a Thurston map of degree d; suppose that ⌧ 2 Tf is represented by ' : (S 2 , Pf ) ! P1 . Then the restrictions of ' f to open subsets of S 2 on which they are injective define a complex structure on S 2 , except at critical points of f . At a critical point ! 2 Crit f , we can choose a branch of (' f )1/deg! f that is a local homeomorphism. With this complex structure, S 2 is a Riemann surface homeomorphic to a sphere, hence (by Theorem 1.1.3) isomorphic to P1 . Let '0 : (S 2 , Pf ) ! P1 be such an isomorphism. Then we define f (⌧ ) 2 Tf to be the element represented by '0 : we have f (['])

Set f' := ' f

('0 )

1

= ['0 ].

10.6.1

. The commutative diagram (S 2 , Pf ) f# (S 2 , Pf )

'0

!

'

!

P1 # f'

10.6.2

P1

encodes f , in the sense that f ([']) = ['0 ]. Note that f' is analytic by construction, hence a rational map of degree d. Some examples of f are given in Appendix C5.

Figure 10.6.1 We think of the points of Pf as “pinned” rather than removed, so we often write (S 2 , Pf ) rather than S 2 Pf . Curves drawn on S 2 Pf are caught on the pins, and in the course of a homotopy cannot go through them. We think of the Xm in Theorem 10.3.3 as such pins, those of the same color sharing a kneading sequence.

Figure 10.6.2 illustrates the bottom line of diagram 10.6.2. The figure illustrating the top line of the diagram would be similar, but the geometry of P1 '0 (Pf ) is not identical to that of P1 '(Pf ); see Figure 10.8.1. We will be interested in how that geometry changes as we iterate f , i.e., as we compute first f ([']) = ['0 ], then f (['0 ]) = ['00 ], then f (['00 ]) = ['000 ], and so on. June 25, 2015

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Chapter 10.

a

Rational functions

'(a)

. '

P1 .

(S , Pf ) 2

'( )

Figure 10.6.2 When we think of a point ⌧ 2 Tf represented by ' : (S 2 , Pf ) ! P1 , we imagine a picture like this. At left is S 2 with Pf marked as green dots. It has “topology”, but no “geometry”. In contrast, P1 has the hyperbolic metric of P1 '(Pf ); for every a 2 Pf , the puncture '(a) is at infinity and has a “trumpetlike” neighborhood. Every curve on P1 '(Pf ) has a geodesic in its homotopy class with a well-defined length. The homotopy classes of such curves correspond under ' to the homotopy classes on the model surface S 2 Pf . If is a curve on S 2 Pf , then '( ) is really only defined up to homotopy; the length of '( ) is meaningless, but the length of the geodesic in its homotopy class is meaningful.

Exercise 10.6.2 Check that the Thurston pullback map f : Tf ! Tf is well defined by the construction given in Definition 10.6.1, and that it depends only on the Thurston equivalence class of f . }

We have already seen (Exercise 10.1.5) that Pf doesn’t change under iteration. The map f also behaves well under iteration. Exercise 10.6.3

Show that if f is a Thurston map, then f

m

=(

f)

m

}

By definition f' = ' f ('0 ) 1 , and by construction f' is analytic, hence a rational function. Diagram 10.6.2 looks like diagram 10.1.2 in Definition 10.1.6 of Thurston equivalence, but it does not say that f is Thurston equivalent to f' ; for that to be the case, we would need to know (part 1 of the definition) that we can choose '0 so that ' and '0 coincide on Pf and are isotopic rel Pf . Di↵erent choices of '0 di↵er by an automorphism of P1 ; only when ['0 ] = ['] will it be possible to choose '0 so that f' is Thurston equivalent to f . Theorem in 10.6.4 A Thurston map f is Thurston equivalent to a rational function if and only if the Thurston pullback map f has a fixed point. Remark We will prove Thurston’s theorem 10.1.14 by showing that when f admits no Thurston obstruction, then f has a fixed point. We will use the June 25, 2015

10.6

The Thurston -mapping

129

Banach fixed point theorem, which requires that mappings be contracting with a fixed Lipschitz constant k < 1. All analytic mappings Tf ! Tf are weakly contracting for the Teichm¨ uller metric (see Corollary 6.10.7). We will not need this result, since we will compute the derivative of f and verify it directly. Still, Corollary 6.10.7 justifies the feeling that something has been accomplished when a question has been reduced to whether a map Tf ! Tf has a fixed point: f cannot be very wild, so we have some hope of determining under what circumstances it has a fixed point. 4 Proof of Theorem 10.6.4 If f is Thurston equivalent to a rational map g, then there are homeomorphisms ', '0 : (S 2 , Pf ) ! (P1 , Pg ) that coincide on Pf , are isotopic rel Pf , and such that the diagram '0

P1 #g ' ! P1

S2 f# S2

!

10.6.3

commutes (Definition 10.1.6). In the context of diagram 10.6.2, this means that g = f' , and that if ⌧ is represented by ', then f (⌧ ) = ⌧ 0 is represented by '0 . But since ' and id '0 coincide on Pf and are isotopic rel Pf , this means that ⌧ = ⌧ 0 . For the converse, represent ⌧ by ' : (S 2 , Pf ) ! P1 and suppose that f (⌧ ) = ⌧ . Consider the commutative diagram 10.6.2. The condition 0 2 1 f (⌧ ) = ⌧ means that ' : (S , Pf ) ! P also represents ⌧ , so there exists 1 1 an isomorphism ↵ : P ! P such that the diagram '0

(S 2 , Pf ) id # (S 2 , Pf )

P1 #↵ ' ! P1 !

10.6.4

commutes on Pf and commutes up to isotopy rel Pf . Now the diagram (S 2 , Pf ) f# (S 2 , Pf )

↵ '0

!

'

!

P1 # f' ↵

1

10.6.5

P1

has exactly the right properties to be a Thurston equivalence between f and the rational function f' ↵ 1 : because diagram 10.6.4 commutes, ↵ '0 coincides with ' on Pf . In addition, diagram 10.6.5 commutes, and ↵ '0 is isotopic rel Pf to '. ⇤ Theorem 10.6.4 Remark The above proof produces a bijection of the set of fixed points of the pullback map f onto the set of conjugacy classes under Aut P1 of rational functions equivalent to f . 4 June 25, 2015

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Corollary in 10.6.5 (The case where |Pf |  3) If the postcritical set Pf has at most three elements, then f is equivalent to a rational map, unique up to conjugacy by Aut P1 . Proof If |Pf | = 3, the space Tf has one point, which is therefore fixed by uller space (if there were, f . If |Pf | = 2, then there really isn’t a Teichm¨ it would have dimension 1); f is a ramified cover S 2 ! S 2 with exactly two critical values, and it is then easy to see that it is Thurston equivalent to z 7! z k for some k with |k| 2. Negative values of k correspond to the case where f exchanges the critical values, positive values to the case where the critical values are fixed. The case |Pf | = 1 does not occur, since the complement of one point in S 2 is simply connected. ⇤

The Thurston pullback map σf in terms of Beltrami forms There is another way to describe f , using the description of Teichm¨ uller space Tf in terms of Beltrami forms (Proposition 6.4.11). This requires that f be quasiconformal, so that pullbacks of Beltrami forms by f are well defined. Exercise 10.6.6 shows that this does not restrict the class of maps to which Theorem 10.1.14 applies: we can always choose C 1 elements of the Thurston equivalence class of f . Remark Here it is essential that Pf be finite. There are quasiconformal branched mappings f1 , f2 with infinite postcritical sets that are topologically Thurston equivalent such that there is no quasiconformal homeomorphism (S 2 , Pf1 ) ! (S 2 , Pf2 ). 4 Exercise 10.6.6 Let f be a Thurston map. Show that there exists a C 1 Thurston map g that is Thurston equivalent to f . Hint: For each x 2 Pf , choose a smooth path t 7! x (t), t 2 [0, 1] in P1 joining '0 (x) to '(x); show that these paths can be chosen disjoint. Choose disjoint neighborhoods Ux of x [0, 1] and smooth isotopies x (t) : Ux ! Ux that are the identity o↵ some fixed compact subset of Ux , and which map x (0) to x (t). Let (t) be the map that is the identity on P1 [x Ux and coincides with x (t) on each Ux . Finally, set g := f' (1). }

Without loss of generality, we will assume in the rest of the chapter that f is C 1 . We can then consider the map µ 7! f ⇤ µ from Bel(S 2 , Pf ) to itself, where Bel(X) is the space of Beltrami forms on a Riemann surface X.6 6

Following a suggestion by Xavier Bu↵, we have changed our notation for the space of Beltrami forms. In the first printing of volume 1, we used the standard notation M, which we have found to be obscure and inauspicious; it doesn’t adapt well to the space of infinitesimal Beltrami forms, which will now be denoted bel, by analogy to the notation for Lie groups and Lie algebras. June 25, 2015

10.7

The derivative of

f

131

If ' integrates µ 2 Bel(S 2 , Pf ) and '0 integrates f ⇤ µ, and if we set fµ := ' f ('0 ) 1 , so that the diagram (S 2 , Pf ) f# (S 2 , Pf )

'0

!

'

!

P1 # fµ

10.6.6

P1

commutes, it is easy to see that fµ is analytic: ('0 )⇤ pulls the standard complex structure of P1 back to the structure f ⇤ µ, the map f takes f ⇤ µ to µ, and ' takes µ back to the standard structure. Hence fµ is a rational function, and we find the following characterization. Proposition in 10.6.7 The map Bel(P1 ) ! Bel(P1 ) given by µ 7! f ⇤ µ

induces

f

: Tf ! Tf on Tf = Bel S 2 /QC0 (S 2

Pf ).

Proof This exactly our original description of f , restricted to complex structures on S 2 that can be represented by Beltrami forms. But all points of Tf are represented by such complex structures. ⇤

10.7 The derivative and coderivative of σf The proof of Theorem 10.1.14 will, after a few detours, come down to applying the Banach fixed point theorem to the Thurston pullback f . As such, we are crucially interested in computing the derivative of f and showing that it has norm < 1. Actually, it is easier to deal with the coderivative. > We begin by recalling the cotangent space T['] Tf to Teichm¨ uller space. Let ' : (S 2 , Pf ) ! P1 represent a point of Tf . To lighten notation, denote by Q' := Q1 (P1 '(Pf )) the space of holomorphic quadratic di↵erentials on P1 '(Pf ) that equivalently (see Exercise 5.3.10) R • satisfy P1 '(Pf ) |q| < 1 (i.e., are integrable) • are meromorphic on P1 with at most simple poles on '(Pf ).

> It follows from Proposition 6.6.2 that the cotangent space T['] Tf is canonically isomorphic to Q' : since Tf is finite dimensional, there is no complication associated to pre-duals and post-duals . More specifically, the isomorphism between the cotangent space and Q' is induced by the pairing Z h⌫, qi = ⌫q 10.7.1 P1

of infinitesimal Beltrami forms and quadratic di↵erentials. June 25, 2015

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When f : Y ! X is a covering map of Riemann surfaces, we defined in Definition 5.4.15 the direct image operator f⇤ : Q1 (Y ) ! Q1 (X), where Q1 (X) and Q1 (Y ) are the spaces of integrable holomorphic quadratic differentials on X and Y respectively. In order to compute the derivative of f , we need to generalize this to the case where f is allowed to have critical points and the quadratic di↵erentials are allowed to have simple poles. Proposition in 10.7.1 (The direct image operator for proper maps of Riemann surfaces) Let X, Y be Riemann surfaces, and let f : Y ! X be a proper analytic mapping. Let P ⇢ X be a discrete set including the critical values of f , and let P 0 ⇢ f 1 (P ) be a subset, necessarily discrete. Then the integrable holomorphic quadratic di↵erentials in Q(Y P 0 ) are meromorphic on Y , with at worst simple poles on P 0 ; those in Q(X P ) are meromorphic on X, with at worst simple poles on P . Moreover, the map f : Y f and the direct image operator f⇤ : Q(Y

f

1

(P ) ! X

1

P is a finite covering map,

(P )) ! Q(X

P)

10.7.2

restricts to a map f⇤ : Q1 (Y

P 0 ) ! Q1 (X

P)

satisfying kf⇤ k  1.

10.7.3

Proof The map f : Y f 1 (P ) ! X P is proper. It is also a local homeomorphism: P contains all the critical values of f , so f 1 (P ) contains all the critical points. Thus this restriction of f is a covering map, and Proposition 5.4.16 applies: f⇤ : Q1 (Y P 0 ) ! Q1 (X P ) is well defined and satisfies kf⇤ k  1. It remains to see that elements of Q(Y P 0 ) and of Q(X P ) are integrable if and only if they have at worst simple poles on P 0 or P respectively. This follows from the computation ◆ Z Z 2⇡ ✓Z 1 dz 2 r = dr d✓ = 2⇡, 10.7.4 z D 0 0 r showing that simple poles are integrable. An obvious modification shows that more serious singularities are not. ⇤

Remark Proposition 10.7.1 is mainly repetition, but we should emphasize one point: there is no map f⇤ : Q1 (Y ) ! Q1 (X). Even if q is analytic on Y , the quadratic di↵erential f⇤ q may acquire poles on X at the critical values of f . But if q is integrable on Y , then f⇤ q is integrable on X, and in particular it has at worst simple poles. June 25, 2015

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The derivative of

f

133

To see how this may occur, let f be given by f (z) = z k , and let w be the variable in the codomain. If 1 1 X X a(n+2)k 2 n 2 q= am z m dz 2 , then f⇤ q = w dw . 10.7.5 k2 m= 1 n= 1 In particular, if q has at worst a simple pole at 0, then so does f⇤ q, and f⇤ q may well have a simple pole at 0 even if q does not: the coefficient ak 2 contributes the pole, and k 2 0. 4 Proposition in 10.7.2 (Derivative of the Thurston pullback map 0 0 2 1 f ) Let ⌧ 2 Tf and ⌧ := f (⌧ ) be represented by ', ' : (S , Pf ) ! P ; let f' be the rational map defined by diagram 10.6.2. Then the transpose [D f (⌧ )]> : Q'0 ! Q' of the derivative [D f (⌧ )] is given by the formula [D

> f (⌧ )] q

= (f' )⇤ q.

10.7.6

Proof We will use the description of f in terms of the pullback of complex structures given in Proposition 10.6.7. The hint for Exercise 10.6.6 shows that we can choose ' to represent the base complex structure on S 2 , which corresponds to µ = 0. The map f is induced by the map Bel(S 2

Pf ) ! Bel(S 2

Pf ) given by µ 7! f ⇤ µ.

10.7.7

Denote by bel(X) the space of infinitesimal Beltrami forms. Clearly, the derivative bel(P1 Pf ) ! bel(P1 Pf ) at µ = 0 of this map is ⌫ 7! f'⇤ ⌫. The proposition then follows from the observation that f'⇤ is the transpose of (f' )⇤ : Z ⇣ ⌘ Z ⇣ ⌘ ⌫ (f' )⇤ q = (f' )⇤ ⌫ q. 10.7.8 S2

S2

This equality would be obvious if we replaced the domain of integration on the left by a simply connected open subset U ⇢ P1 '(Pf ), and that on F the right by f' 1 (U ) = Ui . Clearly considering ⌫ with support in such open sets is enough. ⇤

Recall from equation 5.4.3 that the space Q1' carries the L1 norm kqk1 (which we will write kqk since it is the only norm involved in this discussion): Z kqk = |q|. 10.7.9 P1

Recall from Theorem 6.6.5 that the Teichm¨ uller metric is twice the metric induced by the infinitesimal metric given by the dual of this norm on each tangent space. June 25, 2015

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Proposition in 10.7.3 (Contraction by second iterate of be a Thurston map of degree d. Then 1.

k(f' )⇤ k  1,

so

> f (⌧ )] k

k[D

2. If the orbifold Of is hyperbolic, then contracting, i.e., k[D f 2 (⌧ )]k < 1.

f

2

f)

Let f

 1.

is infinitesimally strictly

Proof Part 1 is a generality about direct images; see Proposition 5.4.16. It also follows from knowing that the Teichm¨ uller metric is the Kobayashi metric; see Proposition 6.10.2. The underlying principle for part 2 is that contraction occurs when there are postcritical points whose inverse images are not postcritical. Usually there are lots of such points, since points have d inverse images by f , so the tree of inverse images grows exponentially, whereas Pf is finite. But every now and then the postcritical set manages to so organize itself that all the postcritical points have inverse images that are postcritical or critical. Understanding just how this can happen requires a messy combinatorial argument, using the following two lemmas. Lemma 10.7.4 Let F : P1 ! P1 be a rational map of degree d 2 and let q be a meromorphic quadratic form with at most simple poles on P1 . Let Y be the set of poles of q. If kF⇤ qk = kqk, then 1. q = d1 F ⇤ F⇤ q. 2. F

1

(F (Y )) ⇢ Y [ Crit F .

Proof of Lemma 10.7.4 1. On a neighborhood of a noncritical value, the terms in F⇤ q coming from the di↵erent sheets of the covering must have the same argument. Thus F ⇤ F⇤ q is a multiple of q by a function that is meromorphic and real, hence constant, and that constant must be d. 2. Since F ⇤ does not introduce poles, every point of F (Y ) is a pole of F⇤ q by part 1. Thus every point of F 1 (F (Y )) is either a pole of q or a critical point of F . ⇤ Lemma 10.7.4 Lemma 10.7.5 Let f : S 2 ! S 2 be a Thurston map of degree d, and suppose that Z ⇢ Pf satisfies f 1 (Z) ⇢ Pf [ Crit f . Then 1. |Z|  4.

2. If |Z| = 4, then f has local degree 2 at all critical points, Z contains the set of critical values, and Z \ Crit f = ;. Set Z 0 := f

1

(Z)

Crit f.

June 25, 2015

10.7

The derivative of

f

135

Then either a. f

1

(Z 0 ) 6⇢ Pf [ Crit f , or

b. f 1 (Z 0 ) ⇢ Pf [ Crit f , in which case Z = Pf and the orbifold Of is not hyperbolic. Figure 10.7.2, left, illustrates parts 2a and 2b. Proof of Lemma 10.7.5 1. Since f 1 (Z) is a subset of Pf [ Crit f , for every point of f 1 (Z) we can find a sequence of inverse images that leads back to a critical point (for instance, the point itself might be critical). It may be that there is such a sequence of inverse images that contains no points of Z, or it may happen that all such sequences contain points of Z. We write f 1 (Z) as a disjoint union X1 [ X2 according to this criterion. More precisely, define subsets Y1 ⇢ Crit f,

Y2 ⇢ Z,

X1 ⇢ f

1

(Z),

X2 ⇢ f

1

(Z)

10.7.10

as follows: • Y1 : The critical point ! 2 Crit f belongs to Y1 if and only if there exists k! 0, necessarily unique, such that f

k!

(!) 2 f

1

(Z) and {!, . . . , f

k!

!} \ Z = ;.

• X1 := {f k! (!) | ! 2 Y1 }. Clearly ! 7! f map Y1 ! X1 , so |X1 |  |Y1 |  | Crit f |.

k!

• X2 := f kz (z) | z 2 Y2 . Clearly z 7! f mapping Y2 ! X2 , so |X2 |  |Y2 |  |Z|.

kz

10.7.11

(!) defines a surjective

• Y2 : z 2 Z belongs to Y2 if there is a first element f kz (z) of the sequence z, f (z), . . . that belongs to f 1 (Z), and that element does not belong to X1 . (z) defines a surjective

I found it impossible to understand this labeling without going through some examples. Example 10.7.6 and Exercise 10.7.7 should help. Example 10.7.6 Consider the sets Crit f , Pf , and Z shown in Figure 10.7.1: the critical points are a, b, c; the postcritical points are b, d, e; and Z ⇢ Pf consists of b and e, so Z satisfies f 1 (Z) ⇢ Pj [ Crit f . We have a 2 Y1 , because it satisfies equation 10.7.11 with ka = 0, hence a 2 X1 also. Next, b 2 / Y1 since b 2 Z. Next, c 2 Y1 with kc = 1, so d = f (c) 2 X1 . Thus Y1 = {a, c} and X1 = {a, d}. Now label the elements of Z: we have b 2 Y2 with kb = 0, so b 2 X2 also, and e 2 / Y2 since no element of its orbit belongs to f 1 (Z). So Y2 = X2 = {b}. June 25, 2015

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3

a

b

Rational functions Figure 10.7.1 A possible set Crit f [ Pf for a branched map f of degree 3. The critical points a, b, c are red and purple; the number over the exiting arrow is the local degree. The postcritical points are blue and purple. The set Z consists of the two circled points b and e. Small empty circles are undistinguished points, not in Pj [ Crit f . 4

2

e

2

c

d

Exercise 10.7.7 Figure 10.7.2 represents three examples of Crit f [ Pf : two of degree 2 on the left and one of degree 3 on the right. The red points are critical (all of local degree 2); the blue points are postcritical. The set Z for each consists of circled points. Find the labeling.

Figure 10.7.2 Three examples of Crit f [ Pf . In each, the circled points have all their inverse images in Pf [ Crit f . The top picture at left illustrates part 2a of Lemma 10.7.5; the bottom picture at left illustrates part 2b. }

Proof of Lemma 10.7.5, continued By construction, X1 \ X2 = ;. Moreover X1 [ X2 = f 1 (Z). We see this as follows. Choose x 2 f 1 (Z). Since f 1 (Z) ⇢ Pf [ Crit f , there certainly are sequences x k , x k+1 , . . . , x of inverse images of x with x k 2 Crit f . If there is any such sequence with no x i 2 Z, then x 2 X1 . Otherwise, every such sequence contains elements of Z. The last such point belongs to Y2 , and x belongs to X2 . Putting these decompositions together, we find |f

1

(Z)| = |X1 | + |X2 |  | Crit f | + |Z|  2d

2 + |Z|.

10.7.12

On the other hand, f 1 (Z) has d|Z| elements in its inverse, counted with multiplicity, where the multiplicity at an inverse image is the local degree there. By the Riemann-Hurwitz formula (Theorem A3.4), X (degx f 1) = 2d 2, 10.7.13 x2Crit f

June 25, 2015

10.7

where d is the degree of f , so X d|Z| = degx f = |f 1

(Z)| + 2d

X

(Z)| +

1 (Z)

x2f

 |f

1

x2f

2.

The derivative of

(degx f

f

137

1)

1 (Z)

10.7.14

Putting equations 10.7.12 and 10.7.14 together, we find (d

1)|Z|  4(d

1);

10.7.15

since d > 1, this proves |Z|  4.

2. If |Z| = 4, then all the inequalities above must be equalities. In particular, | Crit f | = 2d 2, so all the critical points are ordinary, i.e., degx f = 2 when x 2 Crit f . If a point of Z is critical, the first inequality in equation 10.7.12 cannot be an equality. Moreover, in order for the inequality in equation 10.7.14 to be an equality, all the critical points must be in f 1 (Z), so that Z contains the critical values. This proves the first half of part 2. For the second half, if |Z| = 4, then by equation 10.7.14 we have 4d = |f (recall that Z 0 = f W1 = Z

1

Z 0,

(Z)

1

(Z)| + 2d

2 = |Z 0 | + 4d

4

10.7.16

Crit f ), hence |Z 0 | = 4. Set W2 = Z 0

W20 = f

Z,

1

(W2 ).

10.7.17

We have |W1 | = |W2 |, and since W2 contains no critical value, we have |W20 | = d|W2 |. Suppose now that f 1 (Z 0 ) ⇢ Pf [ Crit f . Then W20 ⇢ Pf . For each w 2 W20 , we can choose x 2 f (Crit f ) and k 0 such that f k (x) = w. j Take the last j 2 {0, . . . , k} such that f (x) 2 Z, and set w0 := f j (x) and i := k j. Then w0 2 W1 ,

f i (w0 ) 2 W20 ,

and f

i0

(w0 ) 2 / W20 for i0 < i.

10.7.18

It follows that the map w 7! w0 is injective, and |W1 |

|W20 | = d|W2 |.

10.7.19

This implies |W1 | = |W2 | = 0 and Z = Z 0 = Pf , and thus Of is not hyperbolic. ⇤ Lemma 10.7.5 With these two lemmas, we are ready to complete the proof of Proposition 10.7.3. Proof of Proposition 10.7.3, part 2, continued June 25, 2015

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Define '0 , '00 , f' , f'0 so that the left diagram below commutes: '00

(S 2 , Pf ) #f

'0

(S 2 , Pf ) #f 2 (S , Pf )

q 00 2 Q1 (P1 '00 (Pf )) # (f'0 )⇤

P1 # f'0

!

q 0 2 Q1 (P1 '0 (Pf )) # (f' )⇤ 1 1 q 2 Q (P '(Pf ))

P1 # f' P1

!

'

!

10.7.20

By Proposition 10.7.3, kqk  kq 0 k  kq 00 k. We will show that if there exists q 00 2 Q'00 with q 00 6= 0 such that k[D

f

2

(⌧ )]> q 00 k = k(f' f'0 )⇤ q 00 k = kq 00 k,

10.7.21

so that k[D f (⌧ )] k = 1, then the orbifold Of is not hyperbolic. Set q 0 := (f'0 )⇤ q 00 and q := (f' )⇤ q 0 = (f' f'0 )⇤ q 00 ; it follows from equation 10.7.21 that kqk = kq 0 k = kq 00 k. Let Y be the set of poles of q; Y 0 the set of poles of q 0 ; and Y 00 the set of poles of q 00 ; finally set Z := ' 1 (Y ) and Z 0 := ('0 ) 1 (Y 0 ). Note that each of these sets has cardinality at least 4, since all quadratic di↵erentials on P1 have at least four poles counted with multiplicity, and in this case the poles are simple. From kq 0 k = k(f' )⇤ q 0 k we derive q 0 = d1 (f' )⇤ (f' )⇤ q 0 , hence >

2

(f' )

1

(Y ) ⇢ '0 (Pf [ Crit f ), hence f

1

Z ⇢ Pf [ Crit f.

Thus Lemma 10.7.5 applies, so |Z| = 4, the map f has local degree 2 at all its critical points, and Z contains all critical values and no critical points. In particular, Y 0 is the set of non-critical points of f' 1 (Y ). The argument can be repeated for Y 0 , showing that f 1 (Z 0 ) ⇢ Pf [ Crit f . Thus we are in part 2b of Lemma 10.7.5, and Of is not hyperbolic. ⇤ Corollary in 10.7.8 (Thurston rigidity) If f is a Thurston mapping and the orbifold Of is hyperbolic, then 1. The map

f

2

is strictly contracting: for all ⌧, ⌧ 0 2 Tf with ⌧ 6= ⌧ 0 , d

2 f (⌧ ),

2 0 f (⌧ )

< d(⌧, ⌧ 0 ).

10.7.22

2. If g1 and g2 are rational functions that are Thurston equivalent to f , then g1 and g2 are conjugate by an automorphism of P1 . Proof Part 1 follows immediately from part 2 of Proposition 10.7.3: a di↵erentiable map on a connected Finsler manifold that is infinitesimally strictly contracting is strictly contracting; recall from Theorem 6.6.5 that Teichm¨ uller space is a complete Finsler manifold. The second part follows, since a strictly contracting map can have at most one fixed point. ⇤ June 25, 2015

10.7 The derivative of

f

139

Remarks 1. Even though Teichm¨ uller space is complete, part 1 does not imply the existence of a fixed point: there is no constant k < 1 on the right side of inequality 10.7.22. 2. The nonhyperbolic case, where |Pf | = 4 and ⌫f (x) = 2 if x 2 Pf , does occur; we will examine it in detail in Appendix C6. In that case Tf is 1-dimensional, and f is an isometry, which may be elliptic, parabolic, hyperbolic, or the identity; see Lemma C6.6. 4

10.8 The necessity of the eigenvalue criterion Here we show that the eigenvalue criterion in Theorem 10.1.14 is necessary. We will need the following exercise, which underlies the inverses of degrees in Definition 10.1.12 of the Thurston linear transformation f . Exercise 10.8.1 Show that if f : A ! B is a covering map of annuli of degree d, then Mod A = d1 Mod B. } Theorem in 10.8.2 Let f : (S 2 , Pf ) ! (S 2 , Pf ) be a Thurston map that is Thurston equivalent to a rational function. Let = { 1 , . . . , n } be an f-stable multicurve on (S 2 , Pf ), and let f be the associated Thurston linear transformation, with leading eigenvalue . Then  1, and if the orbifold Of is hyperbolic, then < 1. Proof Since f is Thurston equivalent to a rational function, the map f has a fixed point ⌧0 by Theorem 10.6.4. Consider first the case where Of is hyperbolic. Then (by Corollary 10.7.8) all points of Tf must be attracted to ⌧0 under iteration of f . We will show that if 1, there are points of Tf (those of the set XC ( ) of Lemma 10.8.3) that are not attracted to ⌧0 , so we must have < 1. Represent the fixed point ⌧0 by '0 : (S 2 , Pf ) ! P1 . Choose a nonzero eigenvector m0 of f with eigenvalue and all entries nonnegative, and let c be the largest number such that there exist disjoint annuli A , 2 on P1 '0 (Pf ), with A homotopic to '0 ( ), and such that the vector a 2 R with a = Mod A satisfies a cm0 . (By a cm0 we mean that every entry of a is at least equal to the corresponding entry of cm0 .) Set m := cm0 ; denote the entries of m by m , 2 . Consider the region XC ( ) ⇢ Tf formed of those ' : (S 2 , Pf ) ! P1 such that there exists a family of disjoint annuli (B ) 2 in P1 '(Pf ), with each B homotopic to '( ), and Mod B Cm . (Thus XC ( ) is the region in Tf made up of those Riemann surfaces in which the curves of have annuli of large modulus around them.) These spaces are nonempty for all C: take any point ['] 2 Tf , choose annuli in the homotopy classes June 25, 2015

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of the curves of '( ), 2 , and choose a Beltrami form on these annuli to give them a large modulus. The fixed point ⌧0 is not in the closure of XC ( ) when C > 1. In Figure 10.8.1, the picture downstairs is a point of XC ( ), and the picture upstairs is its image under f . The figure also illustrates the Thurston linear transformation f ; here 0 plays the role of [ ] in Definition 10.1.12. Lemma 10.8.3 If

is a Thurston obstruction, i.e., if f (XC (

1, then

)) ⇢ XC ( ).

10.8.1

Thus no point of XC ( ) can be attracted to ⌧0 when C > 1, proving Theorem 10.8.2 when Of is hyperbolic. Proof of Lemma 10.8.3 Use the notation of diagram 10.6.2. Take a point ⌧ 2 XC ( ), represented by ' : (S 2 , Pf ) ! P1 . Paring down the annuli B if necessary, we may assume that Mod B = Cm . Those components of f' 1 ([ 2 B ) that are homotopic in S 2 '0 (Pf ) to form disjoint annuli, the sum of whose moduli is (f (Cm)) = C

B0 1 # B0

'0 = f (') ! (S 2 , Pf )

'0 (

'0 (x) ! x 2 Pf

10.8.2

Cm .

'0 (Pf )

P1

point of f' 1 ('(Pf )) not in '0 (Pf ) !

#f

m

'0 (

3)

!

2

"

'0 (

0)

"

B0 3 '0 (

1)

# f' d2

' ! '(

1)

"

!

'(

-B

P

1

1

B

d3

d1 (S 2 , Pf )

2)

'(x), x 2 Pf

June 25, 2015

0)

'(Pf )

B

2

3

10.9

Convergence in moduli space

141

Figure 10.8.1 A Thurston map f with an f -invariant multicurve = 0 , 1 , . . . ; suppose that 1 , 2 , 3 are the only elements of whose inverse images under f have a component ⌘i homotopic to 0 , which covers i with degree di . At bottom we have P1 '(Pf ), with ' representing some element of Tf . We have chosen disjoint annuli B in the homotopy class of each '( ), 2 . Saying that ' represents a point of XC ( ) is saying that the (red) annuli B 0 , . . . , B 3 can be chosen tall and skinny (large modulus). The inverse images under f' of the cusps '(Pf ) downstairs are the cusps '0 (Pf ) upstairs and various other points. In orange (at top) we see the components B 0 i of f' 1 (B i ) homotopic in P1 '0 (Pf ) to '0 ( 0 ). Denote by b 2 R the vector of moduli Mod B i . By Exercise 10.8.1, Mod B 0 i =

1 Mod B i ; di

it follows from Theorem 3.2.6 that the smallest annulus in P1 the three orange annuli B 0 1 , B 0 2 , B 0 3 has modulus at least 1 Mod B d1 the first (0th) entry of the vector f (b). Mod B 0 1 + Mod B 0 2 + Mod B 0 3 =

1

+

1 Mod B d2

'0 (Pf ) containing

2

+

1 Mod B 3 , d3

Among these annuli, there are two outer ones, which together bound an annulus B 0 to which Theorem 3.2.6 applies, so Mod B 0 Cm . ⇤ Lemma 10.8.3 When Of is not hyperbolic, it is not necessarily true that all points of Tf are attracted to ⌧0 under iteration of f : the map f might be (and in fact is; see Lemma C6.6) an isometry. But it is true that the distance to ⌧0 is non-increasing, since f is weakly contracting. If is an f -invariant multicurve with > 1, then equation 10.8.2 shows that C > C, this shows that there are points f (XC ( )) ⇢ X C ( ). Since of Tf that will be moved further and further away from ⌧0 under iteration; this is a contradiction. ⇤ Theorem 10.8.2

10.9 Convergence in moduli space implies ¨ller space convergence in Teichmu By Theorem 10.6.4, we are looking for a fixed point of the Thurston pullback map f . The obvious way to do this is to iterate f , i.e., to choose any ⌧0 2 Tf and to define a sequence n 7! ⌧n by ⌧n+1 = f (⌧n ). If the sequence converges, we have found a fixed point. If it does not converge, we need to find a Thurston obstruction. One crucial step in the proof will be to prove that if the orbifold Of is hyperbolic, either the sequence n 7! ⌧n converges in Teichm¨ uller space, or the projections of the ⌧n diverge in the moduli space Modulif . By the Mumford compactness theorem 7.3.3, divergence in moduli space implies June 25, 2015

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that very short curves develop. These short curves will form the Thurston obstruction. The moduli space defined below is really labeled moduli space of genus 0 . This moduli space is much easier to understand than moduli space of higher genus or with unlabeled points. Definition in 10.9.1 (Moduli space) The moduli space Modulif is the space of equivalence classes of injections ⌘ : Pf ! P1 , where ⌘1 and ⌘2 are equivalent if there exists a M¨ obius transformation ↵ such that ⌘2 = ↵ ⌘1 . Denote by p : Tf ! Modulif the natural projection. Let a, b, c be three distinct points of Pf . For any injection ⌘ as above, we can find a unique M¨ obius transformation ↵ 2 Aut P1 such that ↵ ⌘(a) = 0,

↵ ⌘(b) = 1,

↵ ⌘(c) = 1.

10.9.1

Then the complex numbers ↵ ⌘(x), for x 2 Pf {a, b, c}, determine the point of Modulif represented by ⌘. This identifies Modulif with the open subset of (P1 )|Pf | 3 where all coordinates are distinct and di↵erent from 0, 1, 1. In particular, Modulif is a complex manifold, and the natural projection is a universal covering map. Let n 7! ⌧n be a sequence in Teichm¨ uller space. In general, knowing that the sequence n 7! p(⌧n ) converges in moduli space tells us nothing about whether the sequence n 7! ⌧n converges. But for the sequence n 7! ⌧n defined by ⌧n+1 = f (⌧n ), this is not true. Theorem in 10.9.2 Pick ⌧0 2 Tf . If the orbifold Of is hyperbolic, then the sequence ⌧0 , ⌧1 , . . . converges in Tf if and only if the closure of the sequence p(⌧0 ), p(⌧1 ), . . . in Modulif is compact. In that case, lim ⌧n is the unique fixed point of

f.

Proof In one direction this is obvious: if the sequence n 7! ⌧n converges, so does the sequence n 7! p(⌧n ). The other direction follows from the fact that the amount by which f contracts at ⌧ depends only on p(⌧ ) and a finite amount of extra information. p1

p2

Lemma 10.9.3 There exist a tower Tf ! Moduli0f ! Modulif of covering maps, with p2 finite and p2 p1 = p, and a map f : Moduli0f ! Modulif June 25, 2015

10.9

Convergence in moduli space

143

such that the following diagram commutes: f

Tf # p1

!

Moduli0f # p2 Modulif

&

Tf

? ? yp

f

10.9.2

Modulif .

Proof of Lemma 10.9.3 Let W be the set W :=

(⌘, ⌘ 0 , g) ⌘, ⌘ 0 : Pf ! P1 , g : P1 ! P1 rational of degree d

,

such that there exist homeomorphisms ', '0 : (S 2 , Pf ) ! P1 with ⌘ = '|Pf , ⌘ 0 = '0 |Pf , and such that the following diagram commutes: (S 2 , Pf ) f# 2 (S , Pf )

'0

P1 #g ' ! P1 . !

10.9.3

Let Moduli0f be the quotient W/ ⇠, where (⌘1 , ⌘10 , g1 ) ⇠ (⌘2 , ⌘20 , g2 ) if there exist M¨ obius transformations ↵, such that ⌘2 = ↵ ⌘1 , ⌘20 = ⌘10 , 1 and g2 = ↵ g1 . Denote by [(⌘, ⌘ 0 , g)] the class of (⌘, ⌘ 0 , g) modulo ⇠. The map p1 is defined by p1 ['] = [('|Pf , '0 |Pf , f' )];

see diagram 10.6.2. The maps p2 and

p2 [(⌘, ⌘ , g)] = [⌘] and 0

f

10.9.4

are defined by f [(⌘, ⌘

0

, g)] = [⌘ 0 ].

10.9.5

It should be clear that the construction is well defined: p2 [(⌘, ⌘ 0 , g)] defines ⌘ up to post-composition with an element ↵ 2 Aut P1 , hence ⌘ is a well-defined element of Modulif ; similarly, ⌘ 0 is defined up to postcomposition with an element 2 Aut P1 . If ' : (S 2 , Pf ) ! P1 represents an element ⌧ 2 Tf , then '|Pf is also defined up to such post-composition, 1 as is f (⌧ )|Pf . Then f' is defined up to the equivalence f' ⇠ ↵ f' . Next, we need to see that p2 is a covering map; then p1 being a covering map is a generality on covering spaces. Let ⌘0 : Pf ! P1 represent the point [⌘0 ] 2 Modulif , and choose disjoint discs Dx of ⌘0 (x) for each x 2 Pf . Let H be the set of injections ⌘ : Pf ! P1 such that ⌘(x) 2 Dx for each Q x 2 Pf ; clearly H is isomorphic to the polydisc x2Pf Dx . For every point (⌘0 , ⌘00 , g0 ) 2 p2 1 (⌘0 ) we will construct a map s : H ! W with s(⌘0 ) = (⌘0 , ⌘00 , g0 ). Choose a family of C 1 di↵eomorphisms h⌘ : P1 ! P1 parametrized by ⌘ 2 H and such that June 25, 2015

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1. h⌘ is the identity on P1 [x Dx , and h⌘0 is the identity map P1 ! P1 2. h⌘ depends smoothly on ⌘ 3. h⌘ (⌘0 (x)) = ⌘(x) Set µ⌘ = @h⌘ /@h⌘ , so that h⌘ solves the Beltrami equation @h⌘ = µ⌘ @h⌘ . For any [(⌘0 , ⌘00 , g)] 2 p2 1 [⌘0 ] and ⌘ 2 H, define µ0⌘ = g ⇤ µ⌘ , and let h0⌘ be the solution of the Beltrami equation @h0⌘ = µ0⌘ @h0⌘ , normalized to fix three points of ⌘ 0 (Pf ) so that h0⌘ depends analytically on ⌘. We can now define our section s : H ! W by the formula ⇣ ⌘ s(⌘) = h⌘ ⌘0 , h0⌘ ⌘00 , h⌘ g (h0⌘ ) 1 . 10.9.6

We leave it to the reader to check that s does define a section of p2 after quotienting by ⇠. Finally we need to check that p2 is a finite covering map. Given a point in Modulif represented by an inclusion ⌘ : Pf ! P1 , there exist only finitely many isomorphism classes of covering maps g : P1 ! P1 of degree d ramified only over ⌘(Pf ) up to precomposition by an element of Aut P1 . Indeed, pick x 2 P1 ⌘(Pf ); such a class is determined by an action of the fundamental group G := ⇡1 P1 ⌘(Pf ), x on a set with d elements, up to isomorphism of G-spaces. Since G is finitely generated, there are only finitely many such G-spaces, a fortiori finitely many isomorphism classes. For each such map g, there are finitely many injections ⌘ 0 : Pf ! g 1 (⌘(Pf )). The fiber of p2 above [⌘] is the (obviously finite) subset of this finite set for which there exist homeomorphisms ', '0 : (S 2 , Pf ) ! P1 such that S2 #f S2

'0

P1 #g ' ! P1 !

10.9.7

commutes and '|Pf = ⌘ and '0 |Pf = ⌘ 0 .

⇤ Lemma 10.9.3

Proof of Theorem 10.9.2, continued If p1 (⌧ ) = [(⌘, ⌘ 0 , g)], then by equation 10.9.4 and Proposition 10.7.2, [D

> f]

= g⇤ : Q⌘0 ! Q⌘ .

10.9.8

So the norm kD f (⌧ )k depends only on p1 (⌧ ). Let 0 be a C 1 curve from ⌧0 to ⌧1 , with length l0 ; set n := f n ( 0 ) and := [n 1 n . Let p, p1 , and p2 be as in diagram 10.9.2. The map p2 is finite, hence proper, so { p(⌧n ) | n

{ p1 (⌧n ) | n

0 } has compact closure in Modulif () 0 } has compact closure in Moduli0f ()

p( ) has compact closure in Modulif . June 25, 2015

10.10

Asymptotic geometry of Riemann surfaces

145

Set K := sup⌧ 2 kD f2 (⌧ )k. By Proposition 10.7.3 and the equivalences above, K < 1 if p(⌧n ) has compact closure in Modulif , and since Length( n )  KLength(

n 2 ),

10.9.9

the sequence n 7! ⌧n is Cauchy if the set {p(⌧n ), n 0} has compact closure in Modulif . Since Teichm¨ uller space is complete, the sequence n 7! ⌧n then converges, and clearly ⌧ := limn!1 ⌧n is a fixed point of f , necessarily unique by Corollary 10.7.8 and Theorem 10.6.4. ⇤ Theorem 10.9.2

10.10 Asymptotic geometry of Riemann surfaces Theorem 3.2.6 states for appropriate subannuli Ai of an annulus A, X Mod A Mod Aj . 10.10.1 j

The extent to which this inequality fails to be an equality is called the Gr¨ otzsch defect. Theorem 3.2.6 gives a stringent condition for when the Gr¨ otzsch defect is 0. But Theorem 10.10.3 shows that under fairly general circumstances, the Gr¨ otzsch defect is not too big. This will require the following consequence of the collaring theorem (Theorem 3.8.3). The “standard collar” is the collar of thickness ⌘(l( )) around described in that theorem. Proposition in 10.10.1 Let X be a Riemann surface, a simple closed geodesic on X of length l, and A the standard collar around . Then ⇡ Mod A 1. 10.10.2 l

e with the band model B Proof Identify the universal covering space X of the hyperbolic plane, so that the real axis R ⇢ B is a component of the inverse image of . The quotient B/lZ is an annulus of modulus ⇡/l, and e of X on which is the only simple is identified with the covering space X closed geodesic. e . This subannulus A is The standard collar A is a subannulus of X e /lZ, where A e is best visualized in the band model B as the quotient A

the set of points in B at distance at most ⌘(l) from the real axis in the Poincar´e metric of B, as shown in Figure 10.10.1. Let ↵ be the geodesic in B joining a to b. To prove Proposition 10.10.1, we need to see that ↵ lies inside the semicircle with diameter [a, b]. June 25, 2015

146

Chapter 10.

Rational functions

⌘(l) hyperbolic

l !

e A

⇡ Euclidean

a b Consider the harmonic function g on B with boundary values 1 on [a, b] and 0 elsewhere on the boundary @B (in particular on the “top” of B, the line Im z = ⇡/2). Exercise 10.10.2 Show that the geodesic ↵ is the set where g(z) = 1/2. } Now consider the harmonic function h on the halfplane Im z > ⇡/2, with boundary values 1 on [a, b] and 0 elsewhere; this “elsewhere” is the rest of the line Im z = ⇡/2. The function h is 1/2 on the geodesic in the halfplane joining a to b, i.e., on the semicircle . On the boundary of B we have h g, and since h(z) > 0 on the “top”, we have h > g everywhere in B. It follows that g(z) = 1/2 only where h > 1/2, i.e., inside the semicircle. ⇤ Let X be a hyperbolic Riemann surface, P ⇢ X a finite subset, and set X 0 := X P . Let 0 be a closed geodesic on X 0 , and let be the geodesic on X homotopic to 0 . Recall from Section 7.6 that l (Y ) denotes the hyperbolic length in Y of the geodesic homotopic to the closed curve . Since the inclusion X 0 ! X is contracting, and the geodesic is lengthminimizing in its homotopy class, we have l 0 (X 0 )

l 0 (X)

l (X).

10.10.3

Theorem 10.10.3 sharpens this inequality. Theorem in 10.10.3 (Asymptotic geometry of Riemann surfaces) Let X be a Riemann surface and let P ⇢ X be a finite set, p with cardinality |P | > 0. Set X 0 := X P , and choose M < ln(3 + 2 2). Let be a simple closed geodesic on X, and suppose there are s distinct closed geodesics 10 , . . . , s0 on X 0 homotopic to in X and of length < M . Then 1. s  |P | + 1.

2. For all i, we have l i0 (X 0 ) > l (X). s X 1 1 |P | + 1 1 1 |P | + 1 3. < < + . 0 0 l (X) ⇡ M l i (X ) l (X) ⇡ i=1 June 25, 2015

10.10

The inequality l 0 (X 0 )

Asymptotic geometry of Riemann surfaces

147

l (X) says that for each l i0 (X 0 ) we have 1 1  . l i0 (X 0 ) l (X)

10.10.4

Because of the sum in the middle, the right side of part 3 of Theorem 10.10.3 improves on this inequality; more significantly, the left side shows that the improvement is close to optimal. p Proof 1. By Corollary 3.8.7, the i0 are disjoint since M < ln(3 + 2 2). At least s 1 of the components of X [i i0 are annuli, and at least one point of P must belong to each, so |P | s 1. 2. The inclusion X 0 ! X is analytic, hence length decreasing.

3. First let us verify the inequality on the right. According to Theorem 3.8.3 and Proposition 10.10.1, there exist disjoint cylinders Ci0 ⇢ X 0 with equators i0 and moduli satisfying ⇡ Mod(Ci0 ) > 1. 10.10.5 l i0 (X 0 ) e homotopic to the equator These cylinders lift to disjoint cylinders in X . By Theorem 3.2.6, whenever an annulus A contains disjoint annuli Ai P homotopic to the equator of A, we have Mod A i Mod Ai . Therefore, ⇡ l (X)

Theorem 3.2.6

proof Prop. 3.2.1

z}|{ =

>

X i

z}|{

e Mod X

⇡ l i0 (X 0 )

!

1

>

X i

X

Mod Ci0

i

⇡ l i0 (X 0 )

!

10.10.6 (|P | + 1).

Now consider the second inequality. According to Theorem 3.8.3 and Proposition 10.10.1, there is a cylinder C ⇢ X with equator and with ⇡ Mod C > 1. 10.10.7 l (X) The parallels of C (curves at constant distance from the equator) passing through the points of C \ P cut C into s0 annuli Cj , j 2 J := {1, . . . , s0 }, with s0  |P | + 1; see Figure 10.10.2. Figure 10.10.2 The parallels of C through the (red) points of P cut C into annuli, some large and some small. The crucial point of Theorem 10.10.3 is that whenever the modulus of C is large, the large subannuli contribute almost all its modulus.

June 25, 2015

148

Chapter 10.

Rational functions

For each j, let j be the equator of Cj , and let ⌘j be the geodesic of X 0 homotopic to j . We write J as the union J t J + , where J

:=

j2J

l⌘j (X 0 ) < M

,

J + :=

j2J

l⌘j (X 0 )

, 10.10.8 so that the curves {⌘j , j 2 J } are among the curves i0 (there might be other curves i0 not homotopic to any j ). In all cases, we have l j (Cj ) > l j (X 0 ) > l⌘j (X 0 ),

M

10.10.9

so for j 2 J we have l j (Cj ) > l⌘j (X 0 ), and for j 2 J + we have l j (Cj ) M . Moreover, for j 2 J we have l⌘j (X 0 ) = l i0 (X 0 ) for an appropriate i. Thus we find X X ⇡ ⇡ 1 < Mod C = Mod Cj = 10.10.10 l (X) l (Cj ) j2J j2J j X X ⇡ X ⇡ ⇡ ⇡ < +  + (|P | + 1) . ⇤ 0 0 l⌘j (X ) M l i0 (X ) M + i j2J

j2J

10.11 Sufficiency of the eigenvalue criterion In this section we complete the proof of Theorem 10.1.14, by showing that the eigenvalue criterion is sufficient. Proposition in 10.11.1 Let f : S 2 ! S 2 be a Thurston mapping with hyperbolic orbifold. If f is not equivalent to a rational function, then there is a Thurston obstruction: there exists an f -stable multicurve on S 2 Pf with 1. Proof Choose ⌧0 2 Tf and set ⌧n+1 := f (⌧n ). We have proved that the sequence n 7! ⌧n converges if and only if f is equivalent to a rational function. If the sequence does not converge, then the sequence n 7! p(⌧n ) leaves every compact subset of Modulif , and by the Mumford compactness theorem, the hyperbolic surface P1 'n (Pf ) must develop arbitrarily short geodesics for sufficiently large n. We will see that with an appropriate interpretation of “sufficiently large n” and “sufficiently short curve”, these short curves do form a Thurston obstruction. We will again use the gap principle, which we first used in Section 8.4. Let be a closed curve on S 2 Pf , and let ⌧ 2 Tf be represented by ' : (S 2 , Pf ) ! P1 . As in Section 8.4 we denote by l (⌧ ) the hyperbolic length of the geodesic on P1 '(Pf ) homotopic to '( ), and by w (⌧ ) the log of that length, called the log-length of : w (⌧ ) := ln l (⌧ ). Recall (Theorem 7.6.4) that for all , the function w is Lipschitz with Lipschitz ratio 1. June 25, 2015

10.11

Sufficiency of the eigenvalue criterion

149

Let L⌧ be the logarithmic spectrum L⌧ :=

w (⌧ )

a simple closed curve on S 2

Pf

.

10.11.1

Figure 10.11.1 represents what such a logarithmic spectrum can look like. gap [a, b] " W (⌧ )

" a

"0 b

1 W0 ⇠ .567 . . .

Figure 10.11.1 The red dots and blue dots represent a possible set L⌧ ⇢ R. A red dot represents an element w (⌧ ) that is < W0 , a blue dot an element W0 . The set of dots is of course infinite, since there are infinitely many homotopy classes of simple closed curves on S 2 Pf , but there are at most |Pf | 3 red dots, since these are log-lengths of disjoint geodesics on P1 '(Pf ). In particular, there is a smallest element W (⌧ ) of L⌧ , and when this infimum W (⌧ ) is very negative, L⌧ must develop very large gaps. Proposition 10.11.3 says that if a gap [a, b] is sufficiently large, the curves such that the corresponding log-lengths w (⌧ ) are to the left of the gap form an f -stable multicurve. Note that if a red dot is to the left of b, it is automatically to the left of a.

Exercise 10.11.2 Show that on a sphere with a finite set P removed, a maximal multicurve has at most |P | 3 elements. }

The logarithmic spectrum L⌧ has infinitely many points since there are infinitely many non-peripheral simple closed curves on S 2 Pf (unless |Pf | = 3). But by Exercise 10.11.2 and Corollary p 3.8.7, at most |Pf | 3 elements of L⌧ are smaller than W0 := ln ln(3 + 2 2) ⇠ .567 . . . . In particular, the function W (⌧ ) := inf L⌧ on Teichm¨ uller space is well defined. The map W measures how “degenerate” the Riemann surface P1 '(Pf ) is: how close it is to falling apart. It is sometimes called the systole, but I find the word ugly. See Figure 10.11.2. (S , Pf ) 2

.

a

' '(a) P1 '( ) !

⌧

Figure 10.11.2 Suppose ⌧ 2 Tf is represented by ' : (S 2 , Pf ) ! P1 . In its hyperbolic metric, P1 '(Pf ) looks something like the bottom picture. The points of Pf become punctures, with “trumpet-like” neighborhoods; see a and '(a). The image '( ) of the closed curve is really just a homotopy class in P1 '(Pf ); this homotopy class contains a unique geodesic ⌧ , of length l (⌧ ). When l (⌧ ) is very small, the Riemann surface P1 '(Pf ) is close to falling apart: it is being “strangled” around the green “neck” in the homotopy class of '( ).

June 25, 2015

150

Chapter 10.

Rational functions

By the Mumford compactness theorem (see the argument at the beginning of this section), if the sequence n 7! ⌧n does not converge, then lim inf W (⌧n ) = n!1

1.

10.11.2

Set D := d(⌧0 , ⌧1 ). Note that since f is contracting, d(⌧n , ⌧n+1 )  D for all n (the inequality is strict if the orbifold Of is hyperbolic). Set J := D+ln d. This quantity is interesting because of Lemma 10.11.3. Lemma 10.11.3 If for some n there is an interval [a, b] ⇢ R L⌧n with W (⌧n ) < a < b < W0 and b a > J, then the multicurve on S 2 Pf defined by := {

| w (⌧n )  a }

is an f -stable multicurve.

Proof The condition W (⌧n ) < a guarantees that 6= ;. As usual, use notation suggested by diagram 10.6.2: represent ⌧n by 'n , and so on. Then for any 2 , the geodesic homotopic to any component ⌘ of the inverse image f'n1 ('( )) has length at most dea in P1 f'n ('n (Pf )), and if ⌘ is nonperipheral on P1 'n+1 (Pf ), the geodesic 0 of P1 'n+1 in its homotopy class is shorter yet, so w 0 (⌧n+1 ) < a + ln d. But for all simple closed curves on S 2 Pf not in we have w (⌧n+1 ) > b D, since d(⌧n , ⌧n+1 ) < D and the function w is Lipschitz of ratio 1. Since b a > ln d + D = J, the geodesic 0 cannot be such a , and so must belong to . We have proved that for every 2 , every non-peripheral component of f 1 ( ) is homotopic in S 2 Pf to an element of , as required. ⇤ Lemma 10.11.3 It follows that if W (⌧n ) < (|Pf | 2)J, there exists a gap [a, b] ⇢ R L(⌧n ) with b a > J and a (|Pf | 2)J, leading to an f -stable multicurve made up of short curves. We need to show that under appropriate circumstances, 1. We will need Lemma 10.11.4, which provides some crucial uniformity. We will use the sup norm on R and the corresponding operator norm for f . Lemma 10.11.4 For every d 2 and p 3, there exists an integer m 1 such that if f : (S 2 , Pf ) ! (S 2 , Pf ) is a Thurston mapping of degree d with |Pf | = p, and is an f -stable multicurve on S 2 Pf such that f : R ! R has leading eigenvalue < 1, then kf

m

k

1 . 2

10.11.3

Proof For given d and p, there are only finitely many possible Thurston matrices. Indeed, if f : (S 2 , Pf ) ! (S 2 , Pf ) is a Thurston mapping of degree d with |Pf | = p, and is an f -stable multicurve on S 2 Pf , the June 25, 2015

10.11

Sufficiency of the eigenvalue criterion

151

matrix corresponding to the linear transformation f : R ! R is at most of size (p 3) ⇥ (p 3), so has at most (p 3)2 entries. Each such entry is of the form X 1 , 10.11.4 di,j,↵ ↵

where ↵ runs through the nonperipheral components of f 1 ( j ) homotopic to i . So there are at most d terms in the sum, each of the form 1/d↵ with d↵ an integer between 1 and d. Let be a multicurve with < 1. Then kf n k ! 0 as n ! 1, and since there are only finitely many possible matrices for f , we can choose m, depending only on d and p, such that kf

m

This proves Lemma 10.11.4. ⇤

k
2C.

10.11.13

10.11.14

This contradicts the following lemma. Lemma 10.11.8 If for some constant B > 0 a sequence of numbers n 7! xn satisfies |xn+1 |  12 |xn | + B, and if |x0 |  2B, then |xn |  2B for all n. June 25, 2015

10.11

Sufficiency of the eigenvalue criterion

153

Proof The map x 7! 12 x + B maps the interval [ 2B, 2B] to itself. ⇤ Thus 1, and is a Thurston obstruction. This ends the proof of Proposition 10.11.1, hence that of Theorem 10.1.14.

Postscript: Curves belonging to Thurston obstructions Many people have studied (and still study) the behavior of lengths of curves under iteration of the Thurston pullback mapping f . When the sequence n 7! ⌧n does not converge, the formula lim inf W (⌧n ) = 1 says that there must be ⌧n such that the corresponding P1 'n (Pf ) has arbitrarily short geodesics. When Douady and I wrote the proof in [33], we thought that all curves forming Thurston obstructions would have lengths tending to 0, but we couldn’t prove it (just as well, since I later found a counterexample). Another picture is that some curves are short on some P1 'n (Pf ), then as we continue iterating they become long again, then later yet some other curves get short, and so forth. This is wrong again, as is shown Lemma 10.11.9. But it still leaves open the possibility that no particular curve has length tending to 0; that as the iteration progresses, the short curves (which always exist) keep changing. Pilgrim in [87] showed that this isn’t true either. He showed that there exists a canonical Thurston obstruction and that the curves of this Thurston obstruction have lengths tending to 0. Here we will show the weaker Lemma 10.11.9. Lemma 10.11.9 If the sequence n 7! ⌧n does not converge, then lim W (⌧n ) =

n!1

1.

10.11.15

Proof The distances dn := d(⌧n , ⌧n+1 ) are monotone decreasing. Let K, K 0 be compact subsets of Modulif , with K ⇢ K 0 . If p(⌧n ) 2 K for infinitely many n, the steplength dn decreases by a fixed factor at each visit, so there will be n such that ⌧n 2 K with dn arbitrarily small. But dn also decreases by a (larger) fixed factor when ⌧n 2 K 0 , and eventually after a visit to K, the steplength dn will be too small for the sequence to ever leave K 0 . ⇤

June 25, 2015

C1 The Perron-Frobenius theorem Square real matrices with all entries nonnegative are important in many fields; examples include adjacency matrices of graphs (combinatorics) and transition matrices of Markov processes (probability). Let A be a real matrix (perhaps a column matrix, i.e., a vector). We will write • A 0 if all entries ai,j satisfy ai,j 0, and • A > 0 if all entries ai,j satisfy ai,j > 0.

Note that if A > 0 and 0  v  w with v 6= w, then Av < Aw. Pn We will use the L1 norm kvk = i=1 |vi | on Rn .

Theorem in C1.1 (Perron-Frobenius theorem) If A is a real n ⇥ n matrix such that A > 0, there exists a unique real eigenvector v > 0 with kvk = 1. This eigenvector has a simple real eigenvalue > 0, and any other eigenvalue µ 2 C of A satisfies |µ| < . The eigenvector v is called the leading or dominant eigenvector; its eigenvalue is the leading (or dominant) eigenvalue. Proof Let Q ⇢ Rn be the “quadrant” w 0, and let be the standard P simplex = Q \ P , where P ⇢ Rn is the affine hyperplane xi = 1. If w 2 , then w 0 and w 6= 0, and then Aw > 0, so that Aw 2 Q. Consider the function g : ! R given by g(w) := sup { s | sw  Aw } :

C1.1

since the set of such s is closed, we have g(w)w  Aw for all w 2 . Since g is a continuous positive-valued function on the compact set , it achieves its maximum at some v 2 . Let us see that v is an eigenvector of A with eigenvalue := g(v). By contradiction, suppose g(v)v 6= Av, so that g(v)v < Av, and apply A to both sides of this inequality: g(v)Av = A(g(v)v) < A(Av),

so

g(Av) > g(v),

C1.2

contradicting the hypothesis that v is a maximum. Thus Av = g(v)v. We have found an eigenvector v 2

, with eigenvalue

:= g(v). Pre-

cisely the same argument shows that A has an eigenvector v1 2 eigenvalue 1 > 0, so that v1> A = (A> v)> = 1 v1> . >

June 25, 2015

with

Appendix C1.

Now suppose v0 2

The Perron-Frobenius theorem

is an eigenvector of A with eigenvalue

> 0 1 v1 v

= (v> A)v0 = v> (Av0 ) =

0

155

, Then

0 > v1 v,

and since v1> v 6= 0 we have 0 = 1 ; this is true for all eigenvectors of A in , in particular for v, so 0 = 1 = . We can now see that v is unique: if there were another v0 linearly independent from v, then the restriction of A to the subspace spanned by v and v0 would be id, contradicting A( ) ⇢ Q. If w 6= v, then g(w)w  Aw and g(w)w 6= Aw, so g(w)Aw = A(g(w))w < A(Aw)),

so g(Aw) > g(w).

C1.3

Thus for any w 2 with w 6= v, the sequence k 7! g(Ak w) is strictly increasing, and so the sequence k 7! Ak w/kAk wk converges to v. If there are any eigenvectors ui linearly independent from v and with eigenvalues µi satisfying |µi | , then the subspace of Cn spanned by v and these eigenvectors intersects Rn in a subspace of dimension > 1, since the non-real eigenvectors come in conjugate pairs. Thus there are elements w 6= v in that can be written X X w = av + ai ui , which gives Ak w = a k v + ai µki ui , C1.4 i

i

and the sequence k 7! A w/kA wk cannot converge to v. This shows that all the other eigenvalues µ 6= of A satisfy |µ| < . To see that is a simple eigenvalue, choose w 2 \ ker(A I)2 ; then Aw = w + av for some a 2 R. Then k

Ak w =

k

k

w+k

k 1

av =

k 1

( w + kav),

so w + kav > 0 for all k, so a 0. It then follows that g(w) g(v), so g(w) = g(v), and w also achieves the maximum of g, hence w = v. So ker(A so

I)2 = ker(A

I),

C1.5

is a simple eigenvalue. some commented out material ⇤

Corollary in C1.2 (The case A v satisfying v 0.

0) If A

0, then A has an eigenvector

Proof Let n 7! An be a sequence of matrices tending to A, with An > 0. Then by Theorem C1.1, An has an eigenvector vn 2 . Moreover the eigenvalues n are bounded, for instance by the sum of all the elements of An . Since is compact and the n are bounded, by passing to a subsequence we may assume that the sequence n 7! vn converges to some v 2 , and that the sequence n 7! n converges to 0. Passing to the limit in the equation An vn = n vn , we see that Av = v. June 25, 2015

156

Appendix C1.

The Perron-Frobenius theorem

Corollary in C1.3 (Improved Perron-Frobenius theorem) If A 0 n and A > 0 for some n 1, there exists an eigenvector v > 0 with simple eigenvalue > 0, and any other eigenvalue µ 2 C of A satisfies |µ| < . Proof The matrix A has an eigenvector v 0 with eigenvalue 0; the vector v is still an eigenvector for An , this time with eigenvalue n . Thus n is larger than |µ|n for all the other eigenvalues µ of A, we have v > 0 and v is the unique eigenvector in . ⇤

June 25, 2015

C2 The Alexander trick The proof we give of Proposition C2.1 is due to the American topologist James Alexander (1888–1971); it is called the Alexander trick. Proposition in C2.1 1. Every orientation-preserving homeomorphism f : S n extends to a homeomorphism fe: Dn ! Dn . 2. Any two such extensions fe1 , fe2 are isotopic.

Proof 1. Let f : S n extension by

1

fe(x) :=

1

! Sn

1

! Sn

1

be a homeomorphism. Define the radial

(

⇣

x |x|

|x| f 0

⌘

if x 2 Dn

{0}

if x = 0.

C2.1

Clearly this map is a homeomorphism Dn ! Dn extending f . 2. Let ge : Dn ! Dn be another homeomorphism extending f . Consider the one-parameter family of homeomorphisms Ft defined for 0 < t  1 by ⇢ e f (x) if |x| t Ft (x) := C2.2 te g (x/t) if |x|  t (If |x| = t, then te g (x/t) = tf (x/t) = |x|f (x/t) = fe(x). ) Then Ft converges to fe uniformly on Dn as t ! 0. This constructs an isotopy between ge and fe, showing that any extension is isotopic to the radial extension, hence all extensions are isotopic to each other. ⇤ Remarks

1. Even if f is a di↵eomorphism, the radial extension is emphatically not di↵erentiable at 0. 2. The map f 7! fe is a group homomorphism Homeo(S n

1

) ! Homeo(B n )

C2.3

that is a section of the restriction homomorphism Homeo(B n ) ! Homeo(S n

1

).

3. If we replace “homeomorphism” by “di↵eomorphism”, the result remains true for n = 1 and n = 2, but it is false in higher dimensions. For n = 2 the result is very hard [25]. 4 June 25, 2015

C3 Homotopy implies isotopy In topology, we are constantly up against an irritating distinction between homotopy and isotopy. The object of Theorem C3.1 is to lay this ghost for once and for all. We will use the results and notation of Section 3.6. Theorem in C3.1 Let S be a second-countable connected orientable surface. If two homeomorphisms f1 , f2 : S ! S are homotopic, then they are isotopic. Proof We will prove this when S carries a complete hyperbolic structure and admits a decomposition by a multicurve into geodesic trousers and half-annuli; the special cases where S is a sphere, a torus, a disc, or an annulus are a bit di↵erent. Exercise C3.3 sketches how to modify the proof to make it apply to those cases. We are also assuming that S has no boundary. Exercise C3.4 sketches how to adapt the proof to the case where S has a boundary (consisting of circles). We will further make the (less innocent) assumption that the set Z of points of curves 2 is closed. It follows from the discussion in the second paragraph after Figure 3.6.3 that Z being closed is equivalent to requiring that all components of the ideal boundary I(S) be circles; none are homeomorphic to R. This hypothesis on S is satisfied by all surfaces S such that H1 (S, Z) has finite rank, and by many more. Exercise C3.5 sketches how to adapt the proof if Z is not closed. Theorem C3.1 is clearly equivalent to proving that if f : S ! S is a homeomorphism homotopic to the identity, then it is isotopic to the identity. By Proposition 6.4.9, this is true when S has a conformal structure for which f is quasiconformal, so here the emphasis will be on the case where we have no smoothness hypotheses whatsoever on f . Further, in Lemma A2.3 (in the appendix of the first volume), we proved a result about smooth curves in S; we will revisit this result here without smoothness conditions. We are really doing topology, and can choose our hyperbolic surface S within its topological class: it will be convenient to choose all the FenchelNielsen twists to be 0, so that geodesic segments joining curves of within trousers and in neighboring trousers continue each other. With these preliminaries out of the way, let us embark on the proof of Theorem C3.1, which consists of five parts. We start with a homeomorphism f0 := f homotopic to the identity, and construct f1 , . . . , f5 , each fi isotopic to fi 1 , each fi a little nicer than fi 1 , and with f5 = id. June 25, 2015

Appendix C3.

Homotopy implies isotopy

Part 1: f isotopic to f1 with f1 ( ) real analytic for

159

2

On S, the standard annuli A around the 2 form a system of disjoint annuli. Each of these annuli is a union of two sub-annuli A ,1 and A ,2 on either side of . Since f is a homeomorphism, f (A ,1 ) and f (A ,2 ) are also disjoint annuli on S, this time with probably horribly jagged edges. Each f (A ,i ) is an annulus in its own right and contains a unique simple closed geodesic of some length li , for its own metric. By Proposition 3.2.1 there are conformal maps '

,i

: B/li Z ! f (A

,i ),

C3.1

and the foliation of f (A ,i ) by images of vertical lines in B extends to the boundary. Recall that B = { z 2 C | Im z > ⇡/2 } has height ⇡, so the leaves of the foliation on the two sides of f ( ) can be naturally parametrized by [ ⇡, 0] and [0, ⇡], giving a homeomorphism (not a conformal map) ' : ⇥ [ ⇡, ⇡] ! f (A ) extending f : ! f ( ). Let ⌘ be the image by ' ,1 of the closed geodesic on B/l1 Z, i.e., ⌘ = ' ( ⇥ {⇡/2}) (we have privileged A ,1 over A ,2 arbitrarily). We can then construct a homeomorphism h : S ! S isotopic to the identity such that S 1. h is the identity outside 2 f (A ) 2. h(f ( )) = ⌘ for all 3. h maps each leaf of the foliation of f (A ) to itself, and is piecewise linear using the parametrization of the leaf by [ ⇡, ⇡] (see Figure C3.1). Set f1 := h f , so that f1 ( ) = ⌘ for all

2 . ⇡

⌘ 0 % f( ) ⇡

0

⇡/2 ⇡

Figure C3.1 The pale pink region bounded by the green curves is f (A ). The map h takes each black line (leaf of the foliation) to itself, so as to take the red dot to the orange dot. This is sketched at right: each leaf of the foliation of f (A ) is parametrized by [ ⇡, ⇡]. We can isotope the identity of A to a homeomorphism h : A ! A ; this isotopy is defined leaf by leaf as shown on the right. The isotopy turns f ( ) (red, jagged) to the (smooth orange) curve ⌘ := h f . June 25, 2015

160

Appendix C3.

Homotopy implies isotopy

Part 2: f1 is isotopic to f2 , which is the identity on the multicurve This is more delicate than part 1; there we were working in the codomain of f exclusively. Here we must deal with the possibility that although the ⌘ , 2 are disjoint, ⌘ may intersect any element of (including itself) in complicated ways. In moving ⌘ back to by an isotopy, we need to worry about moving other 0 2 in unaccountable ways. It doesn’t seem possible to do all these motions at once, so we will do them one by one, being careful at each step not to undo what we did earlier. Choose an order 1 , 2 , . . . for the curves of . We will construct a sequence h0 = id, h1 , h2 , . . . of di↵eomorphisms S ! S, each isotopic to the preceding one, such that hi f1 is the identity on 1 , . . . , i . Further, on each trouser and each half-annulus, all hi f1 coincide for i large enough, so that f2 := lim (hi f1 ) : S ! S i!1

C3.2

exists and is a homeomorphism isotopic to f . Suppose by induction that hi 1 has been constructed for some i 1 (true for i = 1). For each i, the smooth curves hi 1 (⌘ i ) = hi 1 (f1 ( i )) and i are homotopic on S; further hi 1 (⌘ i ) does not intersect any of the j = hi 1 (f1 ( j )) for j < i, since hi 1 f1 is a homeomorphism. It then follows from Lemma A2.3 that we can smoothly isotope hi 1 (⌘ i ) to be disjoint from i , and then to be the identity on i . (This last involves a choice; see Figure C3.2. In part 3 we will show how to correct our choice if we make the wrong one.) To see that these isotopies do not a↵ect j for j < i, note that the isotopies above take place in arbitrarily small neighborhoods of regions bounded by arcs of i and hi 1 (⌘ i ); but j does not intersect the boundary of any such region, and all such regions are simply connected, so they cannot completely contain any j . Part 3: Homotopies fixing endpoints Recall that Z denotes the set of points of curves 2 . For each 2 let A be the standard annulus around . Each trouser T of S Z is bounded by curves 0 , 00 , 000 2 and contains three half-annuli A0T , A00T , 0 00 000 A000 T ; let BT , BT , BT be the other half-annuli (usually in di↵erent trousers, and sometimes in half-annuli of S Z ). Let 0 , 00 , 000 be the minimizing geodesics joining 00 to 000 , 000 to 0 , and 0 to 00 respectively. We have shown that f can be isotoped to f2 , which is the identity on Z , in particular on @T . The problem that this part 3 addresses is that although f2 ( 0 ) is an arc in T joining the endpoints of 0 , it may not be homotopic in T , with endpoints fixed , to 0 , and similarly for all such geodesics in all trousers (see Figure C3.3). June 25, 2015

Appendix C3.

hi

i

Homotopy implies isotopy

1 (⌘ i )

161

x

"0 ⌘ i

⌘0 i

Figure C3.2 Left: The orange curve hi 1 (⌘ i ) can be moved by an isotopy of S to be a curve ⌘ 0 i (red and orange) disjoint from i (blue). The curve ⌘ 0 i and i together bound an annulus, which is naturally isomorphic to a cylinder. Right: A map from i to ⌘ 0 i . The curve ⌘ 0 i can then be isotoped to i , by drawing geodesics in the annulus (for its own metric) joining each x 2 i to its image. This isotopy involves a choice; a power of a Dehn twist in the annulus will take one set of geodesics to a di↵erent one. At the moment we choose arbitrarily; part 3 will tell us how to correct the choice if needed; see Figure C3.5. Figure C3.3 The green curve is , the red is f2 ( ). If you keep the endpoints of f2 ( ) fixed and pull taut, the resulting geodesic will loop around: it will not coincide with . We would need to apply +1 Dehn twists in an annulus on the left, and 2 Dehn twists in an annulus on the right, to get a curve homotopic to with endpoints fixed.

Lemma C3.2 says that we can isotope f2 to a homeomorphism f3 so that the curves f3 ( (i) ) are all homotopic in T , with endpoints fixed, to (i) . (We use (i) to denote the general case of prime, double prime, triple prime.) Denote by DA(i) the positive geometric Dehn twist in the half-annulus T

(i)

(i)

AT ; note that this Dehn twist is the identity on the boundary of AT , in particular on Z . Lemma C3.2 (i)

(i)

1. For each half-annulus AT there is an integer n(AT ) such that if (i) n(AT ) (i) AT

f3 is f2 composed with all the Dehn twists D trouser T and every j, the curve f3 ( with endpoints fixed. June 25, 2015

(j) T )

, then for each

is homotopic in T to

(j)

162

Appendix C3.

Homotopy implies isotopy

(i)

2. If BT is contained in a trouser (not a half-annulus of S Z ) so (i) (i) (i) that n(BT ) is defined by part 1, we have n(AT ) = n(BT ). 3. The homeomorphism f3 is isotopic to f2 , hence to f . Proof 1. All simple arcs joining one boundary component of a trouser to another are in the same homotopy class. Thus there is a homotopy in T that deforms 0 to f2 ( 0 ), keeping the endpoints on 00 and 000 (but not fixed on those curves). During this homotopy the endpoint on 00 travels around 00 some integral number n(A00T ) of times since it ends at the same point where it starts: the endpoint of 0 on 00 . This homotopy can be realized by n(A00T ) Dehn twists in A00T . 2. Remember we are assuming that BT00 is contained in a trouser, so that n(BT00 ) is defined. If we perform n(A00T ) Dehn twists in A00T and n(BT00 ) Dehn twists in BT00 , the result is to do n(A00T ) + n(BT00 ) Dehn twists in A 0 . Let be an arc consisting of the geodesic joining 0 to 00 , continued on the other side of 0 by a similar geodesic leading to the curve ⌘ 2 in the boundary of the neighboring trouser. (This is possible because we have assumed that the Fenchel-Nielsen twists of our trouser decomposition are all 0.) One (pleasant) possibility is that is a closed curve. This can happen in two ways, shown in Figure C3.4, left and middle. The simple closed curve f2 ( ) is then homotopic to since f2 is homotopic to the identity, but it is also homotopic to the image of by n(A00T ) + n(BT00 ) twists around 0 , which is not even in the same homology class as unless n(A00T ) + n(BT00 ) = 0. 00

"0

=

00

=⌘

=⌘

0

.

.

⌘

00 0

Figure C3.4 Three possible (green). At left and middle, is a closed curve; at left, it is contained in a single trouser, at middle in two. At right, is a line segment. In purple we show how to construct the curve ↵ in that case.

In the (more common) case where is not a closed curve, let ↵ be the closed curve obtained by going around ⌘, then following , then going around 00 , then following backwards; see the purple curve in Figure C3.4, right. The curve f2 (↵) is homotopic to the curve obtained from ↵ by twisting n(A00T ) times in AT 00 and n(BT00 ) times in BT00 ; if n(A00T ) + n(BT00 ) 6= 0 we see that ↵ and f2 (↵) must intersect in 2(n(A00T ) + n(BT00 )) points and cannot be homotopic. Thus n(A00T ) + n(BT00 ) = 0. June 25, 2015

C3.3

↵

Appendix C3.

Homotopy implies isotopy

163

Figure C3.5 If in the half-annulus on one side of 2 you perform a power of a Dehn twist, and in the adjoining half-annulus you perform the opposite power, you construct a homeomorphism that is isotopic to the identity within the annulus

. T |{z} |{z} A00T BT00

3. Figure C3.5 shows that the homeomorphism g consisting of doing a certain number of Dehn twists in A00T and undoing them in BT00 is isotopic to the identity. Thus the homeomorphism f3 := g f2 is isotopic to f2 , hence to f . ⇤ Lemma C3.2 Part 4: f is isotopic to a map f4 , which is the identity on the We will now show that f is isotopic to a map f4 that is the identity on the , while keeping it equal to the identity on Z (i.e., on the multicurve); this is similar to the proof of part 1. Cut a trouser T by the three curves f3 ( 0 ), f3 ( 00 ), f3 ( 000 ), to obtain two hexagons, H1 and H2 . Choose conformal maps '1 : D ! H1 and '2 : D ! H2 . Each disc has six distinguished points on its boundary, corresponding to the corners of the hexagons. As shown in Figure C3.6, '1 maps the segments between these points alternately to (blue) arcs of (i) , which are smooth, and (homeomorphically) to (red) arcs f3 ( (i) ), which may be very jagged. The map '2 does the same for the hexagon H2 . f3 (

.

foliation .

(i)

)

D H1 '1 (i)

⇢Z

Figure C3.6 The conformal map '1 extends as a homeomorphism from the closed disc on the left to the “front hexagon” H1 on the right, whose boundary is alternately a segment of Z (blue) and the jagged (red) curves f3 ( (i) ). A similar map '2 extends as a homeomorphism from a closed disc to the “back hexagon”. (The front and back hexagons make up a trouser on S.) The map '1 takes the (purple) geodesics at left to (purple) real-analytic curves on S; it takes the (orange) chords at left to (orange) real-analytic curves on S. The orthogonals (black) to the geodesics provide a foliation which allows us to push the jagged red curves to the smooth orange curves. June 25, 2015

164

Appendix C3.

Homotopy implies isotopy

Using the Poincar´e metric of the disc, draw the geodesics joining the endpoints of the arcs f3 ( (i) ). For the copy of the disc corresponding to H1 , draw also the chords joining the same points. The region inside the Poincar´e geodesic is foliated by the orthogonals to the geodesic; carry over the foliation to T by '1 and '2 . This provides each f3 ( (i) ) with a “tubular neighborhood” in which we can construct a homeomorphism g : S ! S isotopic to the identity such that (g f3 )( (i) ) is the real analytic curve that is the image of the corresponding chord. A further isotopy like the one illustrated by Figure C3.2, left, can then make (g f3 )( (i) ) disjoint from (i) (this is where part 3 is needed, to know that regions between (g f3 )( (i) ) and (i) are homeomorphic to discs). A further isotopy will create a map f4 that is isotopic to f3 , and is the identity on all curves and . Part 5: f is isotopic to the identity on S For each trouser component T of S Z , apply the Alexander trick (Appendix C2) to each hexagon bounded by arcs of ’s and ’s; this provides an isotopy between f4 and the identity. For each half-annulus component of S Z , we can similarly isotope f4 , use the 1-point compactification to get a disc, and again apply the Alexander trick to get the identity on bigger and bigger annuli, pushing any complication nearer and nearer the added point until it disappears. This shows that “f homotopic to the identity” implies “f isotopic to the identity”. ⇤ ⇤ Theorem C3.1 We proved Theorem C3.1 under various hypotheses; the exercises below show they aren’t necessary. Exercise C3.3 1. Show that an orientation-preserving homeomorphism f of the sphere is isotopic to the identity. Hint: Using the techniques of parts 1 and 2, isotope f to be the identity on the equator, then use the Alexander trick as in part 5. 2. Show that any homeomorphism f of the torus that induces the identity on the homology is isotopic to the identity. Hint: Draw two simple closed curves ↵ and that generate the homology. Show that you can isotope f to be the identity first on ↵, then on . Then use the Alexander trick. 3. Show that every orientation-preserving homeomorphism f of the (open) disc is isotopic to the identity. Hint: Use the 1-point compactification to reduce this to part 1. 4. Show that every orientation-preserving homeomorphism f of an annulus A is isotopic to the identity. Hint: Use the 2-point (endpoint) compactification to reduce this to part 1. } June 25, 2015

Appendix C3.

Homotopy implies isotopy

165

Exercise C3.4 Suppose S is a surface with boundary, and that the components of the boundary are circles. Show that a homeomorphism f : S ! S homotopic to the identity is isotopic to the identity. This is really two results: 1. Suppose that f simply maps the boundary to the boundary and the isotopy is required to be among homeomorphisms that map the boundary to the boundary. In that case, begin by removing the boundary to get a surface S 0 , and go through the proof of Theorem C3.1. We now have half-annuli in S 0 whose closure in S is compact, and the map constructed is the identity on the interior component. It can then be adjusted to be the identity on S by an isotopy that maps boundary to boundary. 2. The other case is where f is the identity on the boundary and is assumed to be homotopic to the identity with boundary fixed; we then need to show that f is isotopic to the identity also with boundary fixed. In this case you need to prove Lemma C3.2 for the geodesics that are homotopic to boundary components; the same argument goes through. (But in the proof of part 2 is never closed; you must consider ↵.) } Exercise C3.5 Suppose Z is not closed. Then some components of the ideal boundary of S are intervals, so that the corresponding components of S Z are halfplanes bounded by an infinite geodesic. Show that Theorem C3.1 is still true. Hint: First isotope f to a homeomorphism that is the identity on geodesics bounding the halfplanes of S Z . Then isotope f to be the identity on the curves i converging to . You must show that these isotopies move points arbitrarily little as i ! 1, in particular, the integers n(A0 i ) are all 0 for i sufficiently large. }

June 25, 2015

C4 The mapping class group and outer automorphisms The mapping class group is a central actor in this book; it is interesting to see that it has an entirely algebraic interpretation, in terms of outer automorphisms. Recall Definition C9.1.2 [this is from amalgamated sums appendix, not in this volume of a pointed map. A pointed map f : (X, x) ! (X, x) induces a homomorphism f⇤ : ⇡1 (X, x) ! ⇡1 (X, x) given by [ ] 7! [f

].

C4.1

If f is not pointed, there is no such well-defined map; outer automorphisms are a weak substitute for f⇤ . For any group G, we denote by Inn G ⇢ Aut G the subgroup of inner automorphisms, i.e., conjugation by elements of G. Clearly Inn G is a normal subgroup of Aut G and Inn G = G/Z(G), where Z(G) denotes the center of G, i.e., the set of elements of G that commute with all elements. Definition in C4.1 (Outer automorphisms) For any group G, the group of outer automorphisms of G is Out G := Aut G/ Inn G. Note that elements of Out G are not automorphisms of G, unless G is commutative. Example C4.2 shows that Out G can be tremendously complicated. Example C4.2 The additive group Zn is just about the simplest group there is, and Inn Zn is trivial, since Zn is commutative. But Out Zn is GLn (Z), which is already complicated for n = 2 and very mysterious for n 3. 4 Let (X, x) be a pointed CCC space, and let f : X ! X be a continuous map, not necessarily pointed. Any choice of path : [0, 1] ! X with (0) = x and (1) = f (x) induces a map [ ⇤f ⇤

1

] : ⇡1 (X, x) ! ⇡1 (X, x),

If 1 and 2 are two such paths, then [ conjugated by [ 2 ⇤ 1 1 ]. If f is a homeomorphism, [ ⇤f ⇤

1

[ ] 7! [ ⇤ (f 1

⇤f ⇤

1

1

] and [

] : ⇡1 (X, x) ! ⇡1 (X, x) June 25, 2015

)⇤ 2

1

].

⇤f ⇤

C4.2 2

1

] are

C4.3

Appendix C4.

The mapping class group and outer automorphisms

167

is an automorphism for any choice of , and since any two paths induce conjugate automorphisms, we see that f induces a homomorphism f⇤ from the group of homotopy classes of homeomorphisms to itself.

The topologist’s mapping class group Let X be a path-connected space with base point x. The topologist’s mapping class group of X, denoted tMCG(X), is the group of isotopy classes of homeomorphisms X ! X. We will mainly be concerned with the case where X is a surface S. The definition of tMCG(S) and Definition 6.4.13 of the Teichm¨ uller modular group MCG(S) agree if and only if S is of finite type, i.e., compact with finitely many punctures. If S is of infinite type, then tMCG(S) is not the Teichm¨ uller modular group, and does not act on Teichm¨ uller space. A homeomorphism f : X ! X defines an isomorphism f⇤ : ⇡1 (X, x) ! ⇡1 X, f (x) .

C4.4

A path joining x to f (x) defines an isomorphism f⇤ : ⇡1 (X, x) ! ⇡1 (X, x) given by f⇤ :

7! ⇤ f ( ) ⇤

1

,

C4.5

where ⇤ denotes concatenation (see Notation C9.1.5). If 1 is another path joining x to f (x), then f⇤ and f⇤ 1 di↵er by an inner automorphism (conjugacy by ⇤ 1 1 ). Thus f 7! f⇤ : ⇡1 (X, x) ! ⇡1 (X, x) defines a map : tMCG(X) ! Out ⇡1 (X, x) written map is obviously a group homomorphism.

C4.6 (f ) = f⇤out . This

Theorem in C4.3 Let S be a connected orientable surface with base point s. 1. The map

: tMCG(S) ! Out ⇡1 (S, s) is injective.

2. If S is compact and without boundary,

is an isomorphism.

Proof 1. This is trivial if S is a sphere, and we leave the case where S is a torus to the reader, so we will assume that S is hyperbolic. We will call a loop joining s to itself a “loop at s”. Of course, is injective if ker = {1}. To say that f⇤out is the identity element of Out ⇡1 (S, s) is to say that f⇤ is the identity automorphism of ⇡1 (S, s) for some path from s to f (s). Untangling the definitions, we need to show that June 25, 2015

168

Appendix C4.

The mapping class group and outer automorphisms

– if there exists a path : [0, 1] ! S with (0) = s and (1) = f (s) and such that for every loop at s, the loop ⇤ f ( ) ⇤ 1 is homotopic to among loops at s, – then f is isotopic to the identity. If the hypothesis is true, then for any x 2 S, there is a well-defined homotopy class of curves joining x to f (x). Indeed, let ⌘x be any curve joining x to s. Then the path ⌘x ⇤ ⇤f (⌘x 1 ) is, up to homotopy, independent of the choice of ⌘x . Indeed, any other such path ⌘x0 is homotopic to ⌘x ⇤ for some loop at s. See Figure C4.1. f( ) f (s) f (⌘x )

s

% ⌘x

f (x)

x

Figure C4.1 The path (blue) from s to f (s) has the property that for any loop (red) at s, the path ⇤ is homotopic with endpoints fixed to ⇤ f ( ). It follows that there is a privileged path (green) from any x to f (x).

The corresponding path from x to f (x) is ⌘x ⇤ ⇤ ⇤ f ( 1 ) ⇤ f (⌘x 1 ). But our hypothesis says that ⇤ f ( 1 ) is homotopic with endpoints fixed 1 to ⇤ , so ⌘x ⇤

⇤ ⇤ f(

1

) ⇤ f (⌘x 1 ) is homotopic to ⌘x ⇤

⇤

1

⇤ ⇤ f (⌘x 1 ),

which is homotopic to ⌘x ⇤ ⇤ f (⌘x 1 ), as required. Recall that our surface S is hyperbolic. For every x 2 S let ⇢x : [0, 1] ! S be the geodesic joining x to f (x) in the homotopy class of ⌘x , in unit time. Then the map F : S ⇥ I ! S given by F (x, t) := ⇢x (t) is a homotopy between f and the identity. By Theorem C3.1 f is then isotopic to the identity. 2. This uses the fact that S is an Eilenberg-Maclane space K ⇡1 (S, s), 1 . Thus if ' : ⇡1 (S, s) ! ⇡1 (S, s) is an automorphism, there is a homotopy equivalence f : (S, s) ! (S, s) such that fs = '. (See for instance Lemma 4.31 in [49].) We must show that f is homotopic to a homeomorphism. e s˜) ! (S, e s˜) of the universal covering space. Lift f to a map fe : (S, Lemma C4.4 is where we use the hypothesis “S compact”. We will use the e s˜), with s˜ as the center. Let be the Fuchsian group disc model D of (S, of covering transformations. e s˜) ! (S, e s˜) extends to S 1 := @D and the Lemma C4.4 The map fe : (S, 1 1 extension f : S ! S is an orientation-preserving homeomorphism. June 25, 2015

Appendix C4.

The mapping class group and outer automorphisms

169

Proof of Lemma C4.4 We will show that fe is a quasi-isomorphism; the result then follows from Theorem 12.1.12. Since f is a homotopy equivalence, there exists g : S ! S such that f g and g f are homotopic to the identity. These homotopies select a homotopy class of paths x joining x to g f (x) for each x 2 S. Denote by C the maximum length of a geodesic joining x to g f (x) in the homotopy class of x ; C exists because S is compact. Lifting the homotopy leads to a specific lift ge of g such that ge fe is homotopic to the identity; we have d x, ge fe(x)  C

for all x 2 Se = D.

C4.7

This immediately shows that fe satisfies the “quasi-surjective” condition to be a quasi-isometry (see Definition 12.1.1). Again since S is compact, both f and g are uniformly continuous, so there exists > 0 such that d(x, y)