Weak Expansion Properties and Large Deviation Principles for Expanding Thurston Maps
In this paper, we prove that an expanding Thurston map is asymptotically -expansive if and only if it has no periodic critical points, and that no expanding Thurston map is -expansive. As a consequence, for each expanding Thurston map without periodic critical points and each real-valued continuous potential on , 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 .
Key words and phrases:Thurston map, postcritically-finite map, entropy-expansive, -expansive, asymptotically -expansive, thermodynamical formalism, equilibrium state, equidistribution, large deviation principle.
2010 Mathematics Subject Classification:Primary: 37D25; Secondary: 37D20, 37D35, 37D40, 37D50, 37B99, 37F15, 57M12
- 1 Introduction
- 2 Notation
- 3 Thurston maps
- 4 The Assumptions
- 5 Asymptotic -Expansiveness
- 6 Large deviation principles
The theory of discrete-time dynamical systems studies qualitative and quantitative properties of orbits of points in a space under iterations of a given map. Various conditions can be imposed upon the map to simplify the orbit structures, which in turn lead to results about the dynamical system under consideration. One such well-known condition is expansiveness. Roughly speaking, a map is expansive if no two distinct orbits stay close forever. Expansiveness plays an important role in the exploitation of hyperbolicity in smooth dynamical systems, and in complex dynamics in particular (see for example, [Ma87] and [PU10]).
In the context of continuous maps on compact metric spaces, there are two weaker notions of expansion, called -expansiveness and asymptotic -expansiveness, introduced by R. Bowen [Bow72] and M. Misiurewicz [Mi73], respectively. Forward-expansiveness implies -expansiveness, which in turn implies asymptotic -expansiveness [Mi76]. Both of these weak notions of expansion play important roles in the study of smooth dynamical systems (see [Bu11, DFPV12, DM09, DN05, LVY13]). Moreover, any smooth map on a compact Riemannian manifold is asymptotically -expansive [Bu97]. Recently, N.-P. Chung and G. Zhang extended these concepts to the context of a continuous action of a countable discrete sofic group on a compact metric space [CZ14].
The dynamical systems that we study in this paper are induced by expanding Thurston maps, which are a priori not differentiable. Thurston maps are branched covering maps on the sphere that generalize rational maps with finitely many postcritical points on the Riemann sphere. More precisely, a (non-homeomorphic) branched covering map is a Thurston map if it has finitely many critical points each of which is preperiodic. These maps arose in W. P. Thurston’s characterization of postcritically-finite rational maps (see [DH93]). See Section 3 for a more detailed introduction to Thurston maps.
Inspired by the analogy to Cannon’s conjecture in geometric group theory (see for example, [Bon06, Section 5 and Section 6]), M. Bonk and D. Meyer investigated extensively properties of expanding Thurston maps [BM10]. (For a precise formulation of these analogies via the so-called Sullivan’s dictionary, see [HP09, Section 1].) Such maps share many common features of rational maps. For example, for each such map , there are exactly fixed points, counted with a natural weight induced by the local degree at each point, where denotes the topological degree of the map [Li13]; there exists a unique measure of maximal entropy (see for example, [BM10]), with respect to which iterated preimages and periodic points are equidistributed in some appropriate sense (see for example, [Li13]). More generally, for each potential that is Hölder continuous with respect to some natural metric induced by , there exists a unique equilibrium state, with respect to which iterated preimages are equidistributed [Li14].
P. Haïssinsky and K. Pilgrim investigated branched covering maps in a more general context [HP09]. We will focus on expanding Thurston maps in this paper.
Let be a compact metric space, and a continuous map on . Denote, for and ,
The map is called forward expansive if there exists such that for all . By R. Bowen’s definition in [Bow72], the map is -expansive if there exists such that the topological entropy of restricted to is for all . One can also formulate asymptotic -expansiveness in a similar spirit, see for example, [Mi76, Section 2]. However, in this paper, we will adopt equivalent formulations from [Do11]. See Section 5.1 for details.
Another way to formulate forward expansiveness is via distance expansion. We say that is distance-expanding (with respect to the metric ) if there exist constants , , and such that for all with , we have . If is forward expansive, then there exists a metric on such that the metrics and induce the same topology on and is distance-expanding with respect to (see for example, [PU10, Theorem 4.6.1]). Conversely, if is distance-expanding, then it is forward expansive (see for example, [PU10, Theorem 4.1.1]). So roughly speaking, if is forward expansive, then the distance between two points that are close enough grows exponentially under forward iterations of .
Since a Thurston map, by definition, has to be a branched covering map, we can always find two distinct points that are arbitrarily close to a critical point (thus arbitrarily close to each other) and that are mapped to the same point. Thus a Thurston map cannot be forward expansive. In order to impose some expansion condition, it is then natural to consider backward orbits. We say that a Thurston map is expanding if for any two points , their preimages under iterations of the map gets closer and closer. See Definition 3.4 for a precise formulation.
The expansion property of expanding Thurston maps seems to be rather strong. However, as a part of our first main theorem below, we will show that no expanding Thurston map is -expansive.
Let be an expanding Thurston map. Then is asymptotically -expansive if and only if has no periodic critical points. Moreover, is not -expansive.
When R. Bowen introduced -expansiveness in [Bow72], he mentioned that no diffeomorphism of a compact manifold was known to be not -expansive. M. Misiurewicz then produced an example of a diffeomorphism that is not asymptotically -expansive [Mi73]. M. Lyubich showed that any rational map is asymptotically -expansive [Ly83]. J. Buzzi established asymptotic -expansiveness of any -map on a compact Riemannian manifold [Bu97]. Examples of -maps that are not -expansive were given by M. J. Pacifico and J. L. Vieitez [PV08]. Our Theorem 1.1 implies that any rational expanding Thurston map (i.e., any postcritically-finite rational map whose Julia set is the whole sphere (see [BM10, Proposition 19.1])) is not -expansive.
Expanding Thurston maps may be the first example of a class of a priori non-differentiable maps that are not -expansive but may be asymptotically -expansive depending on the property of orbits of critical points.
As an immediate consequence of Theorem 1.1 and the result of J. Buzzi [Bu97] mentioned above, we get the following corollary, which partially answers a question of K. Pilgrim (see Problem 2 in [BM10, Section 21]).
An expanding Thurston map with at least one periodic critical point cannot be conjugate to a -map on the Euclidean -sphere.
Our real motivation to investigate Theorem 1.1 comes from another basic theme in the study of dynamical systems, namely, the investigation of the measure-theoretic entropy and measure-theoretic pressure, and their maximizing measures known as the measures of maximal entropy and equilibrium states, respectively.
For a continuous map on a compact metric space, we can consider the topological pressure as a weighted version of the topological entropy, with the weight induced by a real-valued continuous function, called potential. The Variational Principle identifies the topological pressure with the supremum of its measure-theoretic counterpart, the measure-theoretic pressure, over all invariant Borel probability measures [Bow75, Wa76]. Under additional regularity assumptions on the map and the potential, one gets existence and uniqueness of an invariant Borel probability measure maximizing measure-theoretic pressure, called the equilibrium state for the given map and the potential. When the potential is , the corresponding equilibrium state is known as the measure of maximal entropy. Often periodic points and iterated preimages are equidistributed in some appropriate sense with respect to such measures. See Section 6.1 for concepts mentioned here.
The existence, uniqueness, and various properties of equilibrium states have been studied in many different contexts (see for example, [Bow75, Ru89, Pr90, KH95, Zi96, MauU03, BS03, Ol03, Yu03, PU10, MayU10]).
Let be an expanding Thurston map without periodic critical points. Then the measure-theoretic entropy considered as a function of on the space of -invariant Borel probability measures is upper semi-continuous. Here is equipped with the weak topology.
Recall that if is a metric space, a function is upper semi-continuous if for all .
In [Li14], we established the existence and uniqueness of the equilibrium state for an expanding Thurston map and a given real-valued Hölder continuous potential. Here the sphere is equipped with a natural metric induced by , called a visual metric. (See Theorem 6.1.) The tools we used in [Li14] are from the thermodynamical formalism. Neither Theorem 1.1 nor Corollary 1.3 was used there. Note that Corollary 1.3 implies a partially stronger existence result than the one obtained in [Li14].
Let be an expanding Thurston map without periodic critical points and be a real-valued continuous function on (with respect to the standard topology). Then there exists at least one equilibrium state for the map and the potential .
See Section 6.1 for a quick proof after necessary definitions are given precisely.
Once we know the existence and uniqueness of the equilibrium states, one natural question to ask is how periodic points and iterated preimages are distributed with respect to such measures. We know that for an expanding Thurston map, iterated preimages and preperiodic points (and in particular, periodic points) are equidistributated with respect to the unique measure of maximal entropy (see [Li13] and [HP09]).
Some versions of equidistribution of iterated preimages with respect to the unique equilibrium state for an expanding Thurston map and a Hölder continuous potential were obtained in [Li14]. We record them in Proposition 6.6. However, similar results for periodic points were inaccessible by the methods used in [Li14] due to technical difficulties arising from the existence of critical points.
In this paper, thanks to Theorem 1.1 and Corollary 1.3, rather than trying to establish the equidistribution of periodic points directly, we derive some stronger results using a general framework devised by Y. Kifer [Ki90]. More precisely, we obtain level-2 large deviation principles for periodic points with respect to equilibrium states in the context of expanding Thurston maps without periodic critical points and Hölder continuous potentials. We use a variant of Y. Kifer’s result formulated by H. Comman and J. Rivera-Letelier [CRL11], which is recorded in Theorem 6.2 for the convenience of the reader. For related results on large deviation principles in the context of rational maps on the Riemann sphere under additional assumptions, see [PSh96, PSr07, XF07, PRL11, Com09, CRL11].
Denote the space of Borel probability measures on a compact metric space equipped with the weak topology by . A sequence of Borel probability measures on is said to satisfy a level-2 large deviation principle with rate function if for each closed subset of and each open subset of we have
We refer the reader to [CRL11, Section 2.5] and the references therein for a more systematic introduction to the theory of large deviation principles.
In order to apply Theorem 6.2, we just need to verify three conditions:
The existence and uniqueness of the equilibrium state.
The upper semi-continuity of the measure-theoretic entropy.
The first condition is established in [Li14]. The second condition is weaker than the equidistribution results, and is within reach. The last condition is known for expanding Thurston maps without periodic critical points by Corollary 1.3. Thus we get the following level-2 large deviation principles.
Let be an expanding Thurston map with no periodic critical points, and a visual metric on for . Let denote the space of Borel probability measures on equipped with the weak topology. Let be a real-valued Hölder continuous function on , and be the unique equilibrium state for the map and the potential .
For each , let be the continuous function defined by
and denote for . Fix an arbitrary sequence of functions satisfying for each and each . We consider the following sequences of Borel probability measures on :
Iterated preimages: Given a sequence of points in , for each , put
Periodic points: For each , put
Then each of the sequences and converges to in the weak topology, and satisfies a large deviation principle with rate function given by
Furthermore, for each convex open subset of containing some invariant measure, we have
and (1.2) remains true with replaced by its closure .
As an immediate consequence, we get the following corollary. See Section 6.6 for the proof.
Let be an expanding Thurston map with no periodic critical points, and a visual metric on for . Let be a real-valued Hölder continuous function on , and be the unique equilibrium state for the map and the potential . Given a sequence of points in . Fix an arbitrary sequence of functions satisfying for each and each .
Then for each , and each convex local basis of at , we have
Here and are as defined in Theorem 1.5.
As mentioned above, equidistribution results follow from corresponding level-2 large deviation principles.
Let be an expanding Thurston map with no periodic critical points, and a visual metric on for . Let be a real-valued Hölder continuous function on , and be the unique equilibrium state for the map and the potential . Fix an arbitrary sequence of functions satisfying for each and each .
We consider the following sequences of Borel probability measures on :
Iterated preimages: Given a sequence of points in , for each , put
Periodic points: For each , put
Then as ,
Here is defined as in Theorem 1.5.
Since for if , we get
for . In particular, when ,
when , since for if , we have
See Section 6.6 for the proof of Corollary 1.7. Note that the part of Corollary 1.7 on iterated preimages generalizes (6.15) and (6.16) in Proposition 6.6 in the context of expanding Thurston maps without periodic critical points. We also remark that our results Corollary 1.3 through Corollary 1.7 are only known in this context. In particular, the following questions for expanding Thurston maps with at least one periodic critical point are still open.
Is the measure-theoretic entropy upper semi-continuous?
Are iterated preimages and periodic points equidistributed with respect to the unique equilibrium state for a Hölder continuous potential?
Note that regarding Question 2, we know that iterated preimages, counted with local degree, are equidistributed with respect to the equilibrium state by (6.15) in Proposition 6.6. If Question 1 can be answered positively, then the mechanism of Theorem 6.2 works and we get that the equidistribution of periodic points from the corresponding large deviation principle. However, for iterated preimages without counting local degree, (i.e., when in Corollary 1.7, and in particular, when ,) the verification of Condition (2) mentioned earlier for Theorem 6.2 to apply still remains unknown. Compare (6.17) and (6.18) in Proposition 6.7.
We will now give a brief description of the structure of this paper.
After fixing some notation in Section 2, we give a quick review of Thurston maps in Section 3. We direct the reader to [Li14, Section 3] for a more detailed introduction to such maps and the terminology that we use in this paper. However, we do record explicitly most of the results from [BM10, Li13, Li14] that will be used in this paper.
In Section 4, we state the assumptions on some of the objects in this paper, which we are going to repeatedly refer to later as the Assumptions. Note that these assumptions are the same as those in [Li14, Section 4].
We first introduce basic concepts in Section 5.1. We review the notion of topological conditional entropy of a continuous map (on a compact metric space ) given an open cover of , and the notion of topological tail entropy of . The latter was first introduced by M. Misiurewicz under the name “topological conditional entropy” [Mi73, Mi76]. We adopt the terminology and formulations by T. Downarowicz in [Do11]. We then define -expansiveness and asymptotic -expansiveness using these notions.
In Section 5.2, we prove four lemmas that will be used in the proof of the asymptotic -expansiveness of expanding Thurston maps without periodic critical points. Lemma 5.5 states that any expanding Thurston map is uniformly locally injective away from the critical points, in the sense that if one fixes such a map and a visual metric on for , then for each sufficiently small and each , the map is injective on the -ball centered at as long as is not in a -ball of any critical point of , where can be made arbitrarily small if one lets go to . In Lemma 5.6 we prove a few properties of flowers in the cell decompositions of induced by an expanding Thurston map and some special -invariant Jordan curve. Lemma 5.7 gives a covering lemma to cover sets of the form by -flowers, where , , and each is an -flower. Finally, we review some basic concepts in graph theory, and provide a simple upper bound of number of leaves of certain trees in Lemma 5.8. Note that we will not use any nontrivial facts from graph theory in this paper.
Section 5.3 consists of the proof of Theorem 1.1 in the form of three separate theorems. Namely, we show in Theorem 5.9 the asymptotic -expansiveness of expanding Thurston maps without periodic critical points. The proof relies on a quantitative upper bound of the frequency for an orbit under such a map to get close to the set of critical points. Lemma 5.8 and terminology from graph theory is used here to make the statements in the proof precise. We then prove in Theorem 5.10 and Theorem 5.12 the lack of asymptotic -expansiveness of expanding Thurston maps with periodic critical points and the lack of -expansiveness of expanding Thurston maps without periodic critical points, respectively, by explicit constructions of periodic sequences of -vertices for which one can give lower bounds for the numbers of open sets in the open cover needed to cover the set , for sufficiently large. Here denotes the -flower of (see (3.3)), and is the set of all -flowers (see (3.4)). These lower bounds lead to the conclusion that the topological tail entropy and topological conditional entropy, respectively, are strictly positive, proving the corresponding theorems (compare with Defintion 5.3 and Definition 5.4). The periodic sequence of -vertices in the proof of Theorem 5.10 shadows a certain infinite backward pseudo-orbit in such a way that each period of begins with a backward orbit starting at a critical point which is a fixed point of , and approaching as the index increases, and then ends with a constant sequence staying at . The fact that the constant part of each period of can be made arbitrarily long is essential here and is not true if has no periodic critical points. The periodic sequence of -vertices in the proof of Theorem 5.12 shadows a certain infinite backward pseudo-orbit in such a way that each period of begins with a backward orbit starting at and , and approaching as the index increases, and then ends with . In this case is a critical point whose image is a fixed point. In both constructions, we may need to consider an iterate of for the existence of with the required properties. Combining Theorems 5.9, 5.10, and 5.12, we get Theorem 1.1.
Section 6 is devoted to the study of large deviation principles and equidistribution results for periodic points and iterated preimages of expanding Thurston maps without periodic critical points. The idea is to apply a general framework devised by Y. Kifer [Ki90] to obtain level-2 large deviation principles, and to derive the equidistribution results as consequences.
In Section 6.1, we review briefly the theory of thermodynamical formalism and recall relevant concepts and results in this theory from [Li14] in the context of expanding Thurston maps and Hölder continuous potentials. After the necessary concepts are introduced, we provide a quick proof of Theorem 1.4, which asserts the existence of equilibrium states for expanding Thurston maps without periodic critical points and given continuous potentials.
In Section 6.2, we give a brief review of level-2 large deviation principles in our context. We record the theorem of Y. Kifer [Ki90], reformulated by H. Comman and J. Rivera-Letelier [CRL11], on level-2 large deviation principles. This result, stated in Theorem 6.2, will be applied later to our context.
After proving and recording several technical lemmas in Section 6.3, we generalize some characterization of topological pressure in Section 6.4 in our context. More precisely, we use equidistribution results for iterated preimages from [Li14] recorded in Proposition 6.6 to show in Proposition 6.7 and Proposition 6.8 that
where the sum is taken over preimages under in Proposition 6.7, and over periodic points in Proposition 6.8, the potential is Hölder continuous with respect to a visual metric , and the weight for and . We note that for periodic points, the equation (1.4) is established in Proposition 6.8 for all expanding Thurston maps, but for iterated preimages, we only obtain (1.4) for expanding Thurston maps without periodic critical points in Proposition 6.7.
In Section 6.5, by applying Theorem 6.2 to give a proof of Theorem 1.5, we finally establish level-2 large deviation principles in the context of expanding Thurston maps without periodic critical points and given Hölder continuous potentials.
Section 6.6 consists of the proofs of Corollary 1.6 and Corollary 1.7. We first obtain characterizations of the measure-theoretic pressure in terms of the infimum of certain limits involving periodic points and iterated preimages (Corollary 1.6). Such characterizations are then used in the proof of the equidistribution results (Corollary 1.7).
Acknowledgments. The author wants to express his gratitude to the Institut Henri Poincaré for the kind hospitality during his stay in Paris from January to March 2014, when a major part of this work was carried out. The author also would like to thank N.-P. Chung for explaining his work on weak expansiveness for actions of sofic groups. Last but not least, the author wants to express his deepest gratitude to M. Bonk for his patient teaching and guidance as the advisor of the author.
Let be the complex plane and be the Riemann sphere. We use the convention that and . As usual, the symbol denotes the logarithm to the base .
The cardinality of a set is denoted by . For , we define as the greatest integer , and the smallest integer .
Let be a function between two sets and . We denote the restriction of to a subset of by .
Let be a metric space. For subsets , we set , and for . For each subset , we denote the diameter of by , the interior of by , the closure of by , and the characteristic function of by , which maps each to . For each , we define to be the open -neighborhood of , and the closed -neighborhood of . For , we denote the open ball of radius centered at by .
We set to be the space of continuous functions from to , by the set of finite signed Borel measures, and the set of Borel probability measures on . For , we use to denote the total variation norm of , the support of , and
for each . For a point , we define as the Dirac measure supported on . For we set to be the set of -invariant Borel probability measures on . If we do not specify otherwise, we equip with the uniform norm , and equip both and with the weak topology.
The space of real-valued Hölder continuous functions with an exponent on a compact metric space is denoted as . For given and , we define
for and . Note that when , by definition we always have , and by convention .
3. Thurston maps
This section serves as a minimal review for expanding Thurston maps. Most of the definitions and results here were discussed in [Li14, Section 3]. The reader is encouraged to read Section 3 in [Li14] for a quick introduction to expanding Thurston maps and the terminology that we use in this paper. For a more thorough treatment of the subject, we refer to [BM10].
Let denote an oriented topological -sphere. A continuous map is called a branched covering map on if for each point , there exists a positive integer , open neighborhoods of and of , open neighborhoods and of in , and orientation-preserving homeomorphisms and such that , , and
for each . The positive integer above is called the local degree of at and is denoted by . The degree of is
for and is independent of . If and are two branched covering maps on , then so is , and
A point is a critical point of if . The set of critical points of is denoted by . A point is a postcritical point of if for some and . The set of postcritical points of is denoted by . Note that for all .
Definition 3.1 (Thurston maps).
A Thurston map is a branched covering map on with and .
Let be a Thurston map, and be a Jordan curve containing . Then the pair and induces natural cell decompositions (see [Li14, Definition 3.2]) of , for , such that
consisting of -cells, where the set consists of -tiles, the set consists of -edges, and where the set consists of -vertices. The interior of an -cell is denoted by (see the discussion preceding Definition 3.2 in [Li14]). The -skeleton, for , of is the union of all -cells of dimension in this cell decomposition.
We record Proposition 6.1 of [BM10] here in order to summarize properties of the cell decompositions defined above.
Proposition 3.2 (M. Bonk & D. Meyer, 2010).
Let , let be a Thurston map, be a Jordan curve with , and .
The map is cellular for . In particular, if is any -cell, then is an -cell, and is a homeomorphism of onto .
Let be an -cell. Then is equal to the union of all -cells with .
The -skeleton of is equal to . The -skeleton of is the set , and we have .
, , and .
The -edges are precisely the closures of the connected components of . The -tiles are precisely the closures of the connected components of .
Every -tile is an -gon, i.e., the number of -edges and the number of -vertices contained in its boundary are equal to .
From now on, if the map and the Jordan curve are clear from the context, we will sometimes omit in the notation above.
If we fix the cell decomposition , , we can define for each the -flower of as
Note that flowers are open (in the standard topology on ). Let be the closure of . We define the set of all -flowers by
For and , we have
where , and are all the -tiles that contains as a vertex (see [BM10, Lemma 7.2]). Moreover, each flower is mapped under to another flower in such a way that is similar to the map on the complex plane. More precisely, for and , there exists orientation preserving homeomorphisms and such that is the unit disk on , , , and
for all , where . Let and , where are all the -tiles that contains as a vertex, listed counterclockwise, and are all the -tiles that contains as a vertex, listed counterclockwise, and . Then , and if , where . (See also Case 3 of the proof of Lemma 5.2 in [BM10] for more details.)
Definition 3.4 (Expansion).
A Thurston map is called expanding if there exist a metric on that induces the standard topology on and a Jordan curve containing such that
It is clear that if is an expanding Thurston map, then so is , for .
For an expanding Thurston map , we can fix a particular metric on called a visual metric for . For the existence and properties of such metrics, see [BM10, Chapter 8]. For a fixed expanding Thurston map, each visual metric corresponds to a unique expansion factor . One major advantage of a visual metric is that in we have good quantitative control over the sizes of the cells in the cell decompositions discussed above (see [BM10, Lemma 8.10]).
Lemma 3.5 (M. Bonk & D. Meyer, 2010).
Let be an expanding Thurston map, and be a Jordan curve containing . Let be a visual metric on for with expansion factor . Then there exists a constant such that for all -edges and all -tiles with , we have .
In addition, we will need the fact that a visual metric induces the standard topology on ([BM10, Proposition 8.9]) and the fact that the metric space is linearly locally connected ([BM10, Proposition 16.3]).
A Jordan curve is -invariant if . For each -invariant Jordan curve containing , the partition is a cellular Markov partition for (see [Li14, Definition 3.4]). M. Bonk and D. Meyer [BM10, Theorem 1.2] proved that there exists an -invariant Jordan curve containing for each sufficiently large depending on . We proved a slightly stronger version of this result in [Li13, Lemma 3.12] which we record in the following lemma.
Let be an expanding Thurston map, and be a Jordan curve with . Then there exists an integer such that for each there exists an -invariant Jordan curve isotopic to rel. such that no -tile in joins opposite sides of .
Definition 3.7 (Joining opposite sides).
Fix a Thurston map with and an -invariant Jordan curve containing . A set joins opposite sides of if meets two disjoint -edges when , or meets all three -edges when .
Note that for each expanding Thurston map [BM10, Corollary 6.4].
We proved in [Li13, Lemma 3.14] the following easy lemma.
Let be an expanding Thurston map. Then for each , the set is dense in , and
Expanding Thurston maps are Lipschitz with respect to a visual metric [Li14, Lemma 3.12].
Let be an expanding Thurston map, and be a visual metric on for . Then is Lipschitz with respect to .
Let be an expanding Thurston map, and be a Jordan curve that satisfies and for some . Let be a visual metric on for with expansion factor . Then there exists a constant , depending only on , , , and , with the following property:
If , , and , then
4. The Assumptions
We state below the hypothesis under which we will develop our theory in most parts of this paper. We will repeatedly refer to such assumptions in the later sections.
is an expanding Thurston map.
is a Jordan curve containing with the property that there exists such that and for each .
is a visual metric on for with expansion factor and a linear local connectivity constant .
is a real-valued Hölder continuous function with an exponent .
Observe that by Lemma 3.6, for each in (1), there exists at least one Jordan curve that satisfies (2). Since for a fixed , the number is uniquely determined by in (2), in the remaining part of the paper we will say that a quantity depends on even if it also depends on .
Recall that the expansion factor of a visual metric on for is uniquely determined by and . We will say that a quantity depends on and if it depends on .
Note that even though the value of is not uniquely determined by the metric , in the remainder of this paper, for each visual metric on for , we will fix a choice of linear local connectivity constant . We will say that a quantity depends on the visual metric without mentioning the dependence on , even though if we had not fixed a choice of , it would have depended on as well.
In the discussion below, depending on the conditions we will need, we will sometimes say “Let , , , , satisfy the Assumptions.”, and sometimes say “Let and satisfy the Assumptions.”, etc.
5. Asymptotic -Expansiveness
5.1. Basic concepts
Let be a compact metric space and a continuous map.
A cover of is a collection of subsets of with the property that , where is an index set. The cover is an open cover if is an open set for each . The cover is finite if the index set is a finite set.
A measurable partition of is a cover of consisting of countably many mutually disjoint Borel sets , , where is a countable index set.
Let and be two covers of , where and are the corresponding index sets. We say is a refinement of if for each , there exists such that . The common refinement of and defined as
is also a cover. Note that if and are both open covers (resp., measurable partitions), then is also an open cover (resp., a measurable partition). Define , and denote for ,
We adopt the following definition from [