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Boundary Values of the Thurston Pullback Map

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Boundary Values of the Thurston Pullback Map Boundary Values of the Thurston Pullback Map Russell Lodge Indiana University Bloomington, Indiana [email protected] July 3, 2012 Abstract The Thurston characterization and rigidity theorem gives a beauti- ful description of when a postcritically finite topological branched cover mapping the 2-sphere to itself is Thurston equivalent to a rational func- tion. Thurston equivalence remains mysterious, but great strides were taken when Bartholdi and Nekrashevych used iterated monodromy groups to solve the \Twisted Rabbit Problem" which sought to identify the Thurston class of the rabbit polynomial composed with an arbi- trary Dehn twist. Selinger showed that the Thurston pullback map on Teichm¨ullerspace can be extended to the Weil-Petersson boundary. We compute these boundary values for the pullback map associated to 3z2 f(z) = 2z3+1 and demonstrate how the dynamical properties of this map on the boundary can be used as an invariant to solve the twisting problem for f. Contents 1 Introduction 2 1.1 Thurston's Theorem . 4 1.2 Boundary Values of the Pullback Map . 5 1.3 The Twisting Problem . 6 1.4 Outline . 7 2 Thurston Maps with n Postcritical Points 8 2.1 Notation, Definitions, and Examples . 8 2.2 Two definitions of Teichm¨ullerspace; Virtual Endomorphisms 12 2.3 Schreier Graphs and the Reidemeister-Schreier Algorithm . 22 2.4 Iterated Monodromy Groups and Wreath Recursions . 23 1 3 General facts in the case jPf j = 4 29 3z2 4 Analysis of a specific example: f(z) = 2z3+1 37 4.1 The Dynamical Plane of f .................... 37 4.2 The Correspondence on Moduli Space . 41 4.3 The Virtual Endomorphism and Wreath Recursion on Moduli Space . 43 4.4 The Wreath Recursion on the Dynamical Plane; Covering and Hurwitz Equivalence . 49 5 Boundary Values of σf 52 5.1 The Boundary Maps to the Boundary . 52 5.2 Dynamical behavior of φf .................... 53 6 Properties of σf : Q ! Q 61 7 Slopes of Curves in Cb n P when jP j = 4 66 7.1 Slopes in Cb n P when jP j =4 .................. 66 7.2 Slopes in Cb n P when P = f0; 1; !; !¯g . 69 8 The Twisting Problem for f 71 8.1 Limiting behavior of the Extended Virtual Endomorphism . 71 8.2 Solution to the Twisting Problem . 78 9 Future Work 83 1 Introduction Broadly speaking, this dissertation deals with the interplay between geome- try, topology, algebra and complex dynamics; the primary focus of this work is Teichm¨ullerspace and Thurston's characterization theorem for rational maps. In the early 1980's, John Hubbard and Adrien Douady highlighted a close relationship between combinatorics and complex dynamics when they created combinatorial models for the Julia sets of critically finite polyno- mials using Hubbard trees [7]. Critically finite quadratic polynomials are highly significant because they could be used to prove the famous conjec- ture that hyperbolicity is dense|this conjecture was shown by McMullen to be equivalent to the statement that certain quadratics can be approxi- mated by a sequence of critically finite quadratics obtained by a procedure 2 called renormalization [20]. Douady and Hubbard also present an algorithm that in principle finds the formula for a polynomial realizing any admissible Hubbard tree. These results are very illuminating, and there are a number of other combinatorial invariants that can faithfully encode polynomials, al- lowing much to be said about the relationship between combinatorics and Julia sets in the case of polynomials. Complex rational functions are another matter, however, and it is not known how to extend results for polynomials to the rational case. This sets the stage for Thurston's theorem. Complex rational func- tions are a very rigid class of functions (e.g. they are conformal outside of a finite set), and finding explicit formulas for one that realizes partic- ular combinatorial data can be challenging. Thus, we focus our attention on a more flexible class of functions. A Thurston map is a critically fi- nite orientation-preserving branched cover from the two-sphere to itself. To pursue the agenda of relating combinatorics to dynamics, one can create a combinatorial model of a dynamical system using Thurston maps, and then Thurston's theorem characterizes when the Thurston map is equivalent to a rational function. Furthermore, the theorem asserts a rigidity result|if the Thurston map is equivalent to a rational function, this rational function is es- sentially unique (ignoring a family of well-understood Euclidean examples). Thurston's theorem for rational functions is proven in [12] using iteration of an analytic map on Teichm¨ullerspace called Thurston's pullback map. The proof parallels the proof of Thurston's theorem on the hyperbolization of 3-manifolds, which studies dynamics of the skinning map on Teichm¨uller space. Thurston's pullback map is very mysterious, and one goal of my research has been to understand its dynamics on the Weil-Petersson completion of Teichm¨ullerspace using the fact of Selinger that the pullback map extends to the boundary [30]. I did this for the Thurston pullback map associated to 3z2 the rational function f(z) = 2z3+1 because f was investigated in [8]. This paper showed that the Thurston pullback map associated to f is surjective, and up to some nondynamical equivalence it is essentially the only pullback function where this property has been observed. A second major focus of my work has been to better understand the notion of equivalence used in Thurston's theorem. Pilgrim described the lack of solution to the Twisted Rabbit Problem a "humbling reminder" of the lack of suitable invariants for Thurston equivalence [27], and though the problem has been solved, the need for invariants remains. The following study of the Thurston pullback map and Thurston equivalence exploit some rich connections between the 3 Thurston pullback map, combinatorics, Weil-Petersson geometry, hyperbolic geometry, group theory, and arithmetic dynamics. 1.1 Thurston's Theorem Let F be an orientation-preserving branched cover which maps the two- sphere to itself by degree greater than one. Denote by PF the postcritical set as defined in Section 2.1. If PF is finite, F is called a Thurston map. There are several valuable sources for producing examples of Thurston maps: an obvious example would be any post-critically finite rational maps with degree greater than two. Finite subdivision rules of the two-sphere are a con- venient way to produce combinatorial models of Thurston maps, and they have been studied extensively in the recent past [9]. A third way to produce examples of Thurston maps is by mating two critically finite polynomials of the same degree [23]. Having shown that a wealth of examples of Thurston maps exist, we define Thurston equivalence and then state Thurston's the- orem: Definition: Let F and G be Thurston maps with postcritical sets PF and PG respectively. Then F is Thurston equivalent to G if there are orien- 2 2 tation preserving homeomorphisms h0; h1 :(S ;PF ) −! (S ;PG) with h0 homotopic to h1 rel PF , so that the following commutes: 2 h1 2 (S ;PF ) / (S ;PG) F G 2 h0 2 (S ;PF ) / (S ;PG) Thurston's Theorem: Let F be a Thurston map which is not a Latt`es example. Then F is Thurston equivalent to a rational map if and only if the Thurston pullback map has a fixed point. Furthermore, if F is equivalent to a rational map, this rational map is unique up to M¨obiusconjugation. Thurston maps are called obstructed if they are not equivalent to a ra- tional function. A different formulation of Thurston's theorem from the one above gives a better sense for why certain Thurston maps are or are not obstructed; having a fixed point of the pullback map is equivalent to the Thurston map not having a special family of simple closed curves in the two- sphere that behave a certain way under preimage. The mapping properties of curves under preimage of a Thurston map have deep implications for the pullback map on Teichm¨ullerspace, and we will pursue this theme in both 4 of the following sections. It is also worth noting that mapping properties of simple closed curves play an important role in the three other groundbreak- ing theorems proved by Thurston relating geometry and topology, as noted by John Hubbard [15]. Recently there has also been interest in non-dynamical problems involv- ing branched covers of the sphere by higher genus surfaces. See for example the discussion of the Birman-Hilden property in [5, 13]. As was the case in the dynamical setting just described, the preimages of essential curves seem to be significant actors. 1.2 Boundary Values of the Pullback Map Thurston's pullback map is very complicated|it is transcendental, and in all known examples it is infinite-to-one. It may be surprising, then, that 3z2 the pullback map associated to f(z) = 2z3+1 has boundary values that can be computed by transforming the even continued fraction expansions of rational numbers by a prescribed algorithm (see [8] for an image of the Thurston pullback map studied here). We call a simple closed curve γ ⊂ Cb n PF essential if both components bounded by γ contain at least two post-critical points. The collection of homotopy classes of essential curves in Cb n PF can be put in correspondence with the rational numbers when there are four post-critical points. Further- more, when jPF j = 4, it is a standard fact that Teichm¨ullerspace can be identified with the upper half-plane in such a way that the Weil-Petersson p completion is obtained by adding the rational points q + 0i along with the 1 point 0 to the boundary of the upper half-plane where each point corre- sponds to collapsing the sphere with four marked points along the curve p corresponding to q .
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