Thurston Maps Research Articles

Ground states and periodic orbits for expanding Thurston maps

Ground states and periodic orbits for expanding Thurston maps

Prime Orbit Theorems for expanding Thurston maps: Genericity of strong non-integrability condition

In the second paper [16] of this series, we obtained an analog of the prime number theorem for a class of branched covering maps on the 2-sphere S2 called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness or holomorphicity assumption. More precisely, the number of primitive periodic orbits, ordered by a weight on each point induced by a non-constant (eventually) positive real-valued Hölder continuous function on S2 satisfying the α-strong non-integrability condition, is asymptotically the same as the well-known logarithmic integral, with an exponential error bound. In this third and last paper of the series, we show that the α-strong non-integrability condition is generic in the class of α-Hölder continuous functions.

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.

Cohomology fractals, Cannon–Thurston maps, and the geodesic flow

Cohomology fractals are images naturally associated to cohomology classes in hyperbolic three-manifolds. We generate these images for cusped, incomplete, and closed hyperbolic three-manifolds in real-time by ray-tracing to a fixed visual radius. We discovered cohomology fractals while attempting to illustrate Cannon–Thurston maps without using vector graphics; we prove a correspondence between these two, when the cohomology class is dual to a fibration. This allows us to verify our implementations by comparing our images of cohomology fractals to existing pictures of Cannon–Thurston maps. In a sequence of experiments, we explore the limiting behaviour of cohomology fractals as the visual radius increases. Motivated by these experiments, we prove that the values of the cohomology fractals are normally distributed, but with diverging standard deviations. In fact, the cohomology fractals do not converge to a function in the limit. Instead, we show that the limit is a distribution on the sphere at infinity, only depending on the manifold and cohomology class.

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.

Cannon–Thurston maps for {{,textrm{CAT},}}(0) groups with isolated flats

Mahan Mitra (Mj) proved Cannon–Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group (Mitra in Topology, 37(3):527–538, 1998). We prove that Cannon–Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic {{,textrm{CAT},}}(0) groups with isolated flats with respect to the visual boundaries. We also show Cannon–Thurston maps do not exist for infinite infinite-index normal {{,textrm{CAT},}}(0) subgroups with isolated flats in non-hyperbolic {{,textrm{CAT},}}(0) groups with isolated flats. We obtain a structure theorem for the normal subgroups in these settings and show that outer automorphism groups of hyperbolic groups have no purely atoroidal mathbb {Z}^2 subgroups.

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.

Rationality is decidable for nearly Euclidean Thurston maps

Nearly Euclidean Thurston (NET) maps are described by simple diagrams which admit a natural notion of size. Given a size bound C, there are finitely many diagrams of size at most C. Given a NET map F presented by a diagram of size at most C, the problem of determining whether F is equivalent to a rational function is, in theory, a finite computation. We give bounds for the size of this computation in terms of C and one other natural geometric quantity. This result partially explains the observed effectiveness of the computer program NETmap in deciding rationality.

Quasisymmetric embeddability of weak tangents

In this paper, we study the quasisymmetric embeddability of weak tangents of metric spaces. We first show that quasisymmetric embeddability is hereditary, i.e., if $X$ can be quasisymmetrically embedded into $Y$, then every weak tangent of $X$ can be quasisymmetrically embedded into some weak tangent of $Y$, given that $X$ is proper and doubling. However, the converse is not true in general; we will illustrate this with several counterexamples. In special situations, we are able to show that the embeddability of weak tangents implies global or local embeddability of the ambient space. Finally, we apply our results to Gromov hyperbolic groups and visual spheres of expanding Thurston maps.

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.

Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra

There is a family of hyperbolic groups known as hyperbolic hydra which contain heavily distorted free subgroups. We prove the existence of Cannon–Thurston maps (that is, maps of the boundaries induced by subgroup inclusion) for these free subgroups. It is known that Cannon–Thurston maps between hyperbolic space boundaries can exist even in the presence of arbitrarily heavy (even non-recursive) distortion. The hyperbolic hydra examples show that Cannon–Thurston maps can exist even between hyperbolic group boundaries in the presence of arbitrarily heavy primitive recursive distortion.

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.

Expansion properties for finite subdivision rules I

Among Thurston maps (orientation-preserving, postcritically finite branched coverings of the 2-sphere to itself), those that arise as subdivision maps of a finite subdivision rule form a special family. For such maps, we investigate relationships between various notions of expansion—combinatorial, dynamical, algebraic, and coarse-geometric.

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.

Exponential growth of some iterated monodromy groups

Iterated monodromy groups of postcritically-finite rational maps form a rich class of self-similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have exponential growth. These groups arise from polynomials. We show exponential growth of the $\operatorname{IMG}$ of several non-polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpi\'{n}ski carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non-renormalizable polynomial with a dendrite Julia set whose $\operatorname{IMG}$ has exponential growth.

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.

Origami, Affine Maps, and Complex Dynamics

We investigate the combinatorial and dynamical properties of so-called nearly Euclidean Thurston maps, or NET maps. These maps are perturbations of many-to-one folding maps of an affine two-sphere to itself. The close relationship between NET maps and affine maps makes computation of many invariants tractable. In addition to this, NET maps are quite diverse, exhibiting many different behaviors. We discuss data, findings, and new phenomena.

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.