Thurston Map Research Articles

Eliminating Thurston obstructions and controlling dynamics on curves

AbstractEvery Thurston map $f\colon S^2\rightarrow S^2$ on a $2$ -sphere $S^2$ induces a pull-back operation on Jordan curves $\alpha \subset S^2\smallsetminus {P_f}$ , where ${P_f}$ is the postcritical set of f. Here the isotopy class $[f^{-1}(\alpha )]$ (relative to ${P_f}$ ) only depends on the isotopy class $[\alpha ]$ . We study this operation for Thurston maps with four postcritical points. In this case, a Thurston obstruction for the map f can be seen as a fixed point of the pull-back operation. We show that if a Thurston map f with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can ‘blow up’ suitable arcs in the underlying $2$ -sphere and construct a new Thurston map $\widehat f$ for which this obstruction is eliminated. We prove that no other obstruction arises and so $\widehat f$ is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem.

On elastic graph spines associated to quadratic Thurston maps

For a quadratic Thurston map having two distinct critical points and $n$ postcritical points, we count the number of possible dynamical portraits. We associate elastic graph spines to several hyperbolic quadratic Thurston rational functions. These functions have four postcritical points, real coefficients, and invariant real intervals. The elastic graph spines are constructed such that each has embedding energy less than one. These are supporting examples to Dylan Thurston's recent positive characterization of rational maps. Using the same characterization, we prove that with a combinatorial restriction on the branched covering and a cycle condition on the dynamical portrait, a quadratic Thurston map with finite postcritical set of order $n$ is combinatorially equivalent to a rational map. This is a special case of the Bernstein-Levy theorem.

Algorithmic aspects of branched coverings III/V. Erasing maps, orbispaces, and the Birman exact sequence

Let \tilde{f}\colon(S^2,\widetilde{A})\circlearrowleft be a Thurston map and let M(\tilde{f}) be its mapping class biset: isotopy classes rel \widetilde{A} of maps obtained by pre- and post-composing \tilde{f} by the mapping class group of (S^2,\widetilde{A}) . Let A\subseteq\widetilde{A} be an \tilde{f} -invariant subset, and let f\colon(S^2,A)\circlearrowleft be the induced map. We give an analogue of the Birman short exact sequence: just as the mapping class group \mathbf{Mod}(S^2,\widetilde{A}) is an iterated extension of \mathbf{Mod}(S^2,A) by fundamental groups of punctured spheres, M(\tilde{f}) is an iterated extension of M(f) by the dynamical biset of f . Thurston equivalence of Thurston maps classically reduces to a conjugacy problem in mapping class bisets. Our short exact sequence of mapping class bisets allows us to reduce in polynomial time the conjugacy problem in M(\tilde{f}) to that in M(f) . In case \tilde{f} is geometric (either expanding or doubly covered by a hyperbolic torus endomorphism) we show that the dynamical biset B(f) together with a “portrait of bisets” induced by \widetilde{A} is a complete conjugacy invariant of \tilde{f} . Along the way, we give a complete description of bisets of (2,2,2,2) -maps as a crossed product of bisets of torus endomorphisms by the cyclic group of order 2 , and we show that non-cyclic orbisphere bisets have no automorphism. We finally give explicit, efficient algorithms that solve the conjugacy and centralizer problems for bisets of expanding or torus maps.

Trees, dendrites and the Cannon–Thurston map

When 1 -> H -> G -> Q -> 1 is a short exact sequence of three infinite, word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point z in the Gromov boundary of Q an "ending lamination" on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(F_N), one can identify the resultant quotient space with a certain $\mathbb{R}$-tree in the boundary of Culler-Vogtmann's Outer space.

Algorithmic aspects of branched coverings II/V: sphere bisets and decidability of Thurston equivalence

We consider Thurston maps: branched self-coverings of the sphere with ultimately periodic critical points, and prove that the Thurston equivalence problem between them (continuous deformation of maps along with their critical orbits) is decidable. More precisely, we consider the action of mapping class groups, by pre- and post-composition, on branched coverings, and encode them algebraically as mapping class bisets. We show how the mapping class biset of maps preserving a multicurve decomposes into mapping class bisets of smaller complexity, called small mapping class bisets. We phrase the decision problem of Thurston equivalence between branched self-coverings of the sphere in terms of the conjugacy and centralizer problems in a mapping class biset. Our decomposition results on mapping class bisets reduce these decision problems to small mapping class bisets; they correspond to rational maps, homeomorphisms and maps double covered by a torus endomorphism, and their conjugacy and centralizer problems are solvable respectively in terms of complex analysis, group theory and linear algebra. Branched coverings themselves are also encoded into bisets, with actions of the fundamental groups. We characterize those bisets that arise from branched coverings between topological spheres, and extend this correspondence to maps between spheres with multicurves, whose algebraic counterparts are sphere trees of bisets. To illustrate the difference between Thurston maps and homeomorphisms, we produce a Thurston map with infinitely generated centralizer—while centralizers of homeomorphisms are always finitely generated.

Quotients of Torus Endomorphisms and Latt\xe8s-Type Maps

We show that if an expanding Thurston map is the quotient of a torus endomorphism, then it has a parabolic orbifold and is a Lattès-type map.

Expansion properties for finite subdivision rules II

We prove that every sufficiently large iterate of a Thurston map which is not doubly covered by a torus endomorphism and which does not have a Levy cycle is isotopic to the subdivision map of a finite subdivision rule. We determine which Thurston maps doubly covered by a torus endomorphism have iterates that are isotopic to subdivision maps of finite subdivision rules. We give conditions under which no iterate of a given Thurston map is isotopic to the subdivision map of a finite subdivision rule.

On NET maps: Examples and nonexistence results

A Thurston map is called nearly Euclidean if its local degree at each critical point is 2 and it has exactly four postcritical points. Nearly Euclidean Thurston (NET) maps are simple generalizations of rational Lattès maps. We investigate when such a map has the property that the associated pullback map on Teichmüller space is constant. We also show that no Thurston map of degree 2 has constant pullback map.

On the structure of isentropes of real polynomials

In this paper we will modify the Milnor–Thurston map, which maps a one dimensional mapping to a piece-wise linear of the same entropy, and study its properties. This will allow us to give a simple proof of monotonicity of topological entropy for real polynomials and better understand when a one dimensional map can and cannot be approximated by hyperbolic maps of the same entropy. In particular, we will find maps of particular combinatorics which cannot be approximated by hyperbolic maps of the same entropy.

Presentations of NET maps

A branched covering $f: S^2 \to S^2$ is a nearly Euclidean Thurston (NET) map if each critical point is simple and its postcritical set has exactly four points. We show that up to equivalence, each NET map admits a normal form in terms of simple affine data. This data can then be used as input for algorithms developed for the computation of fundamental invariants, now systematically tabulated in a large census.

Modular groups, Hurwitz classes and dynamic portraits of NET maps

An orientation-preserving branched covering f: S^2 \to S^2 is a nearly Euclidean Thurston (NET) map if each critical point is simple and its postcritical set has exactly four points. Inspired by classical, non-dynamical notions such as Hurwitz equivalence of branched covers of surfaces, we develop invariants for such maps. We then apply these notions to the classification and enumeration of NET maps. As an application, we obtain a complete classification of the dynamic critical orbit portraits of NET maps.

Algorithmic aspects of branched coverings IV/V. Expanding maps

Thurston maps are branched self-coverings of the sphere whose critical points have finite forward orbits. We give combinatorial and algebraic characterizations of Thurston maps that are isotopic to expanding maps as maps and as maps with contracting biset. We prove that every Thurston map decomposes along a unique minimal multicurve into Levy-free and finite-order pieces, and this decomposition is algorithmically computable. Each of these pieces admits a geometric structure. We apply these results to matings of post-critically finite polynomials, extending a criterion by Mary Rees and Tan Lei: they are expanding if and only if they do not admit a cycle of periodic rays.

Algorithmic aspects of branched coverings I/V. Van Kampen’s theorem for bisets

We develop a general theory of bisets : sets with two commuting group actions. They naturally encode topological correspondences. Just as van Kampen’s theorem decomposes into a graph of groups the fundamental group of a space given with a cover, we prove analogously that the biset of a correspondence decomposes into a graph of bisets : a graph with bisets at its vertices, given with some natural maps. The fundamental biset of the graph of bisets recovers the original biset. We apply these results to decompose the biset of a Thurston map (a branched selfcovering of the sphere whose critical points have finite orbits) into a graph of bisets. This graph closely parallels the theory of Hubbard trees. This is the first part of a series of five articles, whose main goal is to prove algorithmic decidability of combinatorial equivalence of Thurston maps.

Equilibrium States for Expanding Thurston Maps

In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $${\mu_\phi}$$ for each expanding Thurston map $${f : S^2\rightarrow S^2}$$ together with a real-valued Holder continuous potential $${\phi}$$ . Here the sphere S2 is equipped with a natural metric induced by f, called a visual metric. We also prove that identical equilibrium states correspond to potentials that are co-homologous up to a constant, and that the measure-preserving transformation f of the probability space $${(S^2,\mu_\phi)}$$ is exact, and in particular, mixing and ergodic. Moreover, we establish versions of equidistribution of preimages under iterates of f, and a version of equidistribution of a random backward orbit, with respect to the equilibrium state. As a consequence, we recover various results in the literature for a postcritically-finite rational map with no periodic critical points on the Riemann sphere equipped with the chordal metric.

Algorithmic aspects of branched coverings

This is the announcement, and the long summary, of a series of articles on the algorithmic study of Thurston maps. We describe branched coverings of the sphere in terms of group-theoretical objects called bisets, and develop a theory of decompositions of bisets. We introduce a canonical Levy decomposition of an arbitrary Thurston map into homeomorphisms, metrically-expanding maps and maps doubly covered by torus endomorphisms. The homeomorphisms decompose themselves into finite-order and pseudo-Anosov maps, and the expanding maps decompose themselves into rational maps. As an outcome, we prove that it is decidable when two Thurston maps are equivalent. We also show that the decompositions above are computable, both in theory and in practice.

A characterization of Sierpiński carpet rational maps

In this paper we prove that a postcritically finite rational map with non-empty Fatou set is Thurston equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpiński carpet.

Cannon–Thurston maps for hyperbolic free group extensions

This paper gives a detailed analysis of the Cannon{Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free groups of nite rank at least 3 and consider a convex cocompact subgroup Out(F), i.e. one for which the orbit map from into the free factor complex of F is a quasi-isometric embedding. The subgroup determines an extension E of F, and the main theorem of Dowdall{Taylor (DT1) states that in this situation E is hyperbolic if and only if is purely atoroidal. Here, we give an explicit geometric description of the Cannon{Thurston maps @F! @E for these hyperbolic free group extensions, the existence of which follows from a general result of Mitra. In particular, we obtain a uniform bound on the multiplicity of the Cannon{Thurston map, showing that this map has multiplicity at most 2 rank(F). This theorem generalizes the main result of Kapovich and Lustig (KL5) which treats the special case where is innite cyclic. We also answer a question of Mahan Mitra by producing an explicit example of a hyperbolic free group extension for which the natural map from the boundary of to the space of laminations of the free group (with the Chabauty topology) is not continuous.

Periodic points and the measure of maximal entropy of an expanding Thurston map

In this paper, we show that each expanding Thurston map f : S 2 → S 2 f\colon S^2\!\rightarrow S^2 has 1 + deg ⁡ f 1+\deg f fixed points, counted with appropriate weight, where deg ⁡ f \deg f denotes the topological degree of the map f f . We then prove the equidistribution of preimages and of (pre)periodic points with respect to the unique measure of maximal entropy μ f \mu _f for f f . We also show that ( S 2 , f , μ f ) (S^2,f,\mu _f) is a factor of the left shift on the set of one-sided infinite sequences with its measure of maximal entropy, in the category of measure-preserving dynamical systems. Finally, we prove that μ f \mu _f is almost surely the weak ∗ ^* limit of atomic probability measures supported on a random backward orbit of an arbitrary point.

Constructive Geometrization of Thurston Maps and Decidability of Thurston Equivalence

The key result in the present paper is a direct analogue of the celebrated Thurston’s Theorem Douady and Hubbard (Acta Math 171:263–297, 1993) for marked Thurston maps with parabolic orbifolds. Combining this result with previously developed techniques, we prove that every Thurston map can be constructively geometrized in a canonical fashion. As a consequence, we give a partial resolution of the general problem of decidability of Thurston equivalence of two postcritically finite branched covers of $$S^2$$ (cf. Bonnot et al. Moscow Math J 12:747–763, 2012).

Weak expansion properties and large deviation principles for expanding Thurston maps

In this paper, we prove that an expanding Thurston map f:S2→S2 is asymptotically h-expansive if and only if it has no periodic critical points, and that no expanding Thurston map is h-expansive. As a consequence, for each expanding Thurston map without periodic critical points and each real-valued continuous potential on S2, there exists at least one equilibrium state. For such maps, we also establish large deviation principles for iterated preimages and periodic points. It follows that iterated preimages and periodic points are equidistributed with respect to the unique equilibrium state for an expanding Thurston map without periodic critical points and a potential that is Hölder continuous with respect to a visual metric on S2.