aa r X i v : . [ m a t h . D S ] F e b LATT `ES MAPS AND COMBINATORIAL EXPANSION
QIAN YIN
Abstract.
A Latt`es map f : b C → b C is a rational map that is ob-tained from a finite quotient of a conformal torus endomorphism.We characterize Latt`es maps by their combinatorial expansion be-havior. Contents
1. Background 12. Summary of Results 43. Expanding Thurston maps and Cell Decompositions 64. Latt`es and Latt`es-type Maps 125. Combinatorial Expansion Factor and D n Background A rational map f : b C → b C is a special type of analytic map on theRiemann sphere b C = C ∪ {∞} . It can be written as a quotient of tworelatively prime complex polynomials p ( z ) and q ( z ), with q ( z ) = 0, f ( z ) = p ( z ) q ( z ) = a z m + . . . + a m b z l + . . . + b l , (1)where a i , b j ∈ C for i = 0 , . . . , m and j = 0 , . . . , l . The fundamentalproblem in dynamics is to understand the behavior of the iterates of f , f n ( z ) := f ◦ f ◦ · · · ◦ f | {z } n factors ( z ) . The study of the dynamics of rational maps originated in 1917 byPierre Fatou and Gaston Julia, who developed the foundations of com-plex dynamics. In particular, they applied Montel’s theory of normalfamilies to develop the fundamental theory of iteration (see [F] and [J]).Their work was more or less forgotten for over half a century. Then
The author was partially supported by NSF grants DMS 0757732, DMS 0353549,DMS 0456940, DMS 0652915, DMS 1058772, and DMS 1058283.
Benoit Mandelbrot rekindled interest in the field in the 1970s by gen-erating beautiful and intriguing graphic images that naturally appearunder iteration of rational maps through his computer experiments(see [M1] and [M2]). In recent years, the study of dynamics of ratio-nal maps has attracted considerable interest, not only because complexdynamics itself is an intriguing and rich subject, but also because of itslinks to other branches of mathematics, such as quasi-conformal map-pings, Kleinian groups, potential theory and algebraic geometry. Forinstance, the study of the dynamical systems arising from polynomialsand those that arise from Kleinian groups that depend on holomorphicmotions are connected by the dictionary introduced by Sullivan (see[S2]), which led to his seminal work on the non-existence of wanderingdomains for rational maps.Given a rational map f : b C → b C , the degree deg( f ) of f is themaximal degree of the polynomials p ( z ) and q ( z ) as in equation (1).The degree of f can also be defined topologically as the cardinality ofthe preimage over a generic (non-critical) value.A rational map f with deg( f ) > We will assumethat the rational map f has deg( f ) > from now on. A point z ∈ b C is periodic if f n ( z ) = z for some n ≥
1. In this case, it is calledattracting if | ( f n ) ′ ( z ) | < | ( f n ) ′ ( z ) | = 1;repelling if | ( f n ) ′ ( z ) | > . For example, if we let f ( z ) = z , then z = 0 is an attracting periodicpoint of f , and f is contracting near 0; z = 1 is a repelling periodicpoint, and f is expanding near 1.The Julia set J ( f ) of f is the closure of the set of repelling periodicpoints. It is also the smallest closed set containing at least three pointswhich is completely invariant under f − . For the example f ( z ) = z ,the Julia set of f is the unit circle. The complement F ( f ) = b C \ J ( f )of the Julia set, called the Fatou set , is the largest open set such thatthe iterates of f restricted to it form a normal family. The Julia setand Fatou set are both invariant under f and f − .The postcritical set post( f ) of f is the forward orbits of the criticalpoints post( f ) = [ n ≥ { f n ( c ) : c ∈ crit( f ) } . The postcritical set plays a crucial role in understanding the expand-ing and contracting features of a rational map. If the postcritical setpost( f ) is finite, we say that the map f is postcritically finite . ATT`ES MAPS AND COMBINATORIAL EXPANSION 3
In 1918, Samuel Latt`es described a special class of rational mapswhich have a simultaneous linearization for all of their periodic points(see [L1]). This class of maps is named after Latt`es, even though similarexamples had been studied by Ernst Schr¨oder much earlier (see [S1]).A
Latt`es map f : b C → b C is a rational map that is obtained from a finitequotient of a conformal torus endomorphism, i.e., the map f satisfiesthe following commutative diagram:(2) T ¯ A −−−→ T Θ y y Θ b C f −−−→ b C where ¯ A is a map of a torus T that is a quotient of an affine map ofthe complex plane, and Θ is a finite-to-one holomorphic map. Latt`esmaps were the first examples of rational maps whose Julia set is thewhole sphere b C , and the postcritical set of a Latt`es map is finite. Moreimportantly, Latt`es maps play a central role as exceptional examples incomplex dynamics. We will discuss this further in the following section.Observing that much information about the dynamics of a rationalmap can be deduced from the postcritical set, Thurston introduced atopological analog of a postcritically finite rational map, now known asa Thurston map . A
Thurston map f : S → S is a branched coveringmap with finite postcritical set post( f ). Thurston characterized Latt`esmaps among Thurston maps up to Thurston equivalence (see Definition3.3) in terms of associated orbifolds and the derivatives of associatedtorus automorphisms (see Section 9 in [DH]).The notion of an expanding Thurston map was introduced in [BM] asa topological analog of a postcritically finite rational map whose Juliaset is the whole sphere b C . Roughly speaking, a Thurston map is called expanding if all the connected components of the preimage under f − n ofany open Jordan region disjoint from post( f ) become uniformly smallas n tends to infinity. We refer the reader to Definition 3.1 for a moreprecise statement. A related and more general notion of expandingThurston maps was introduced in [HP]. Latt`es maps are among thesimplest examples of expanding Thurston maps.Latt`es maps are distinguished among all rational maps in variousways. For instance, Latt`es maps are the only rational maps for whichthe measure of maximal entropy is absolutely continuous with respectto Lebesgue measure (see [Z]).Many different characterizations of Latt`es maps have been both givenand conjectured (e.g. [M3]). For example, a fundamental conjecturein complex dynamics states that the flexible Latt`es maps are the onlyrational maps that admit an “invariant line field” on their Julia set.The significance of this conjecture is demonstrated by a theorem of QIAN YIN
Ma˜n´e, Sad and Sullivan (see [MSS]). It states that if the fundamentalconjecture above is true, then hyperbolic maps are dense among ratio-nal maps. We refer the reader to [M4] for a nice exposition on Latt`esmaps. 2.
Summary of Results
Let f be an expanding Thurston map, and let C be a Jordan curvecontaining post( f ). The Jordan Curve Theorem implies that S \ C hasprecisely two connected components, whose closures we call 0 -tiles . Wecall the closure of each connected component of the preimage of S \ C under f n an n -tile . In Section 5 of [BM], it is proved that, for every n ≥
0, the collection of all n -tiles gives a cell decomposition of S . Thepoints in post( f ) divide C into several subarcs. Let D n = D n ( f, C )be the minimum number of n -tiles needed to join two of these subarcsthat are non-adjacent (see Definition 5.1 and (13)). For any Thurstonmap f without periodic critical points, there exists C > D n ≤ C (deg f ) n/ for all n > Theorem 2.1.
A map f : S → S is topologically conjugate to a Latt`esmap if and only if the following conditions hold: • f is an expanding Thurston map; • f has no periodic critical points; • there exists c > such that D n ≥ c (deg f ) n/ for all n > . There is an interpretation (see Theorem 5.3 in [Y]) of Theorem 2.1 inthe Sullivan dictionary corresponding to Hamenst¨adt’s entropy rigiditytheorem (see [H]).Let f be an expanding Thurston map. Even though D n = D n ( f, C )depends on the Jordan curve C , its growth rate is independent of C .Hence the limit(4) Λ ( f ) = lim n →∞ (cid:0) D n ( f, C ) (cid:1) /n exists and only depends on the map f itself (see [BM, Prop. 17.1]).We call this limit Λ ( f ) the combinatorial expansion factor of f . Thisquantity Λ ( f ) is invariant under topological conjugacy and is multi-plicative in the sense that Λ ( f ) n is the combinatorial expansion factorof f n . Inequality (3) implies thatΛ ( f ) ≤ (deg f ) / . ATT`ES MAPS AND COMBINATORIAL EXPANSION 5
The combinatorial expansion factor is closely related to the notion of visual metrics and their expansion factors . Every expanding Thurstonmap f : S → S induces a natural class of metrics on S , called visualmetrics (see Definition 3.10), and each visual metric d has an associated expansion factor Λ >
1. This visual metric is essentially characterizedby the geometric property that the diameter of an n -tile is about Λ − n ,and the distance between two disjoint n -tiles is at least about Λ − n .The supremum of the expansion factors of all visual metrics is equalto the combinatorial expansion factor Λ (see [BM, Theorem 1.5]). ForLatt`es maps, the supremum is obtained. In general, the supremum isnot obtained. For examples, the supremum is not obtained for Latt`es-type maps that are not Latt`es maps (see Section 4). We will show inProposition 6.13 that Theorem 2.1 remains true if we replace the thirdcondition by the requirement that there exists a visual metric on S with expansion factor Λ = (deg f ) / .Here we outline the sufficiency of the three conditions in Theorem2.1. These three conditions imply the existence of a visual metric d on S with expansion factor Λ = (deg f ) / (see Proposition 6.13). Thisis the most technical part of the paper. The way that we constructthe visual metric uses the idea that any quasi-visual metric can bemodified to be a visual metric. The existence of this visual metricimplies that ( S , d ) is Ahlfors 2-regular, which means any ball with ra-dius r has Hausdorff 2-measure roughly r (see Proposition 7.2). Usingthis 2-regularity together with the linear local connectivity condition of( S , d ), we obtain that ( S , d ) is quasisymmetrically equivalent to theRiemann sphere b C by [BK, Theorem 1.1] (see Proposition 7.2). Wededuce that f is topologically conjugate to a rational map from thequasisymmetrical equivalence of ( S , d ) to the Riemann sphere b C (seeProposition 7.4). Now we can focus on the rational maps with threeconditions satisfied in Theorem 2.1. In order to invoke the characteri-zation of Latt`es maps among rational maps by [M3], we need that theHausdorff measure with respect to the visual metric and with respectto the standard chordal metric d on b C are essentially the same. Thisfollows from a theorem by Juha Heinonen and Pekka Koskela (see The-orem 7.6), and it implies that the dimension of Lebesgue measure withrespect to the visual metric d is equal to 2 (see Theorem 7.8). Weconclude that the map f is topologically conjugate to a Latt`es map.We define Latt`es-type maps so as to include non-rational maps thatare quotients of affine maps and share many desired properties of Latt`esmaps. Comparing with diagram 2, we have a commutative diagram T ¯ A −−−→ T Θ y y Θ S f −−−→ S . QIAN YIN where Θ is essentially the same Θ as in diagram 2, and we require ¯ A tobe a quotient of an affine map on the real plane rather than the complexplane. A map f : S → S obtained by the above commutative diagramis called a Latt`es-type map (see Definition 4.2). If a Latt`es-type mapis rational, then the map is a Latt`es map.Latt`es-type maps are examples of expanding Thurston maps, andthey have the same orbifold structures as Latt`es maps (see Proposition5.9).
Proposition 2.2.
Let f be a Latt`es-type map with orbifold type (2 , , , .Let A be its corresponding linear map from R to R and let ℘ : R → S be the Weierstrass function with the lattice Z . We have k A − n k ∞ ≤ D n ( f, C ) ≤ k A − n k ∞ + 1 , where the Jordan curve C is the image of the boundary of the unit square [0 , × [0 , under ℘ . Here k B k ∞ denotes the operator norm of a linear map B on R withrespect to the ℓ ∞ -norm. As a corollary of this proposition and equation(4), we have the following result (see Corollary 5.10). Corollary 2.3.
Let f be a Latt`es-type map with orbifold type (2 , , , ,and let A be the corresponding linear map from R to R . Then the com-binatorial expansion factor Λ ( f ) equals the minimum absolute valueof the eigenvalues of A . Acknowledgements.
This paper is part of the author’s PhD thesisunder the supervision of Mario Bonk. The author would like to thankMario Bonk for introducing her to and teaching her about the subjectof Thurston maps and its related fields. The author is inspired by hisenthusiasm and mathematical wisdom, and is especially grateful forhis patience and encouragement. The author would like to thank Den-nis Sullivan for valuable conversations and sharing his mathematicalinsights. The author also would like to thank Kyle Kinneberg, AlanStapledon and Michael Zieve for useful comments and feedback.3.
Expanding Thurston maps and Cell Decompositions
In this section we review some definitions and facts on expandingThurston maps. We refer the reader to Section 3 in [BM] for moredetails. We write N for the set of positive integers, and N for the setof non-negative integers. We denote the identity map on S by id S .Let S be a topological 2-sphere with a fixed orientation. A contin-uous map f : S → S is called a branched covering map over S if f can be locally written as z z d ATT`ES MAPS AND COMBINATORIAL EXPANSION 7 under certain orientation-preserving coordinate changes of the domainand range. More precisely, we require that for any point p ∈ S , thereexists some integer d >
0, an open neighborhood U p ⊂ S of p , anopen neighborhood V q ⊆ S of q = f ( p ), and orientation-preservinghomeomorphism φ : U p → U ⊆ C and ψ : V p → V ⊆ C with φ ( p ) = 0 and ψ ( q ) = 0 such that( ψ ◦ f ◦ φ − )( z ) = z d for all z ∈ U . The positive integer d = deg f ( p ) is called the local degree of f at p and only depends on f and p . A point p ∈ S is called a critical point of f if deg f ( p ) ≥
2, and a point q is called critical value of f if there is a critical point in its preimage f − ( q ). If f is a branchedcovering map of S , f is open and surjective. There are only finitelymany critical points of f and f is finite-to-one due to the compactnessof S . Hence, f is a covering map away from critical values in the rangeand the preimages of critical values in the domain. The degree deg( f )of f is the cardinality of the preimage over a non-critical value. Inaddition, we have deg( f ) = X p ∈ f − ( q ) deg f ( p )for every q ∈ S . For n ∈ N , we denote the n -th iterate of f as f n = f ◦ f ◦ · · · ◦ f | {z } n factors . We also set f = id S . If f is a branched cover of S , so is f n , anddeg( f n ) = deg( f ) n . Let crit( f ) be the set of all the critical points of f . The set of postcriticalpoints of f is defined aspost( f ) = [ n ∈ N { f n ( c ) : c ∈ crit( f ) } . We call a map f postcritically-finite if the cardinality of post( f ) isfinite. Sincecrit( f n ) = crit( f ) ∪ f − (crit( f )) ∪ · · · ∪ f − ( n − (crit( f )) , one can verify that post( f ) = post( f n ) for any n ∈ N . So f ispostcritically-finite if and only if there is some n ∈ N for which f n is postcritically-finite.Let C ⊂ S be a Jordan curve containing post( f ). We fix a metric d on S that induces the standard metric topology on S . Denote by QIAN YIN mesh( f, n, C ) the supremum of the diameters of all connected compo-nents of the set f − n ( S \ C ). Definition 3.1.
A branched covering map f : S → S is called a Thurston map if deg( f ) ≥ f is postcritically-finite. A Thurstonmap f : S → S is called expanding if there exists a Jordan curve C ⊂ S with C ⊃ post( f ) and(5) lim n →∞ mesh( f, n, C ) = 0 . The relation (5) is a topological property, as it is independent ofthe choice of the metric, as long as the metric induces the standardtopology on S . Lemma 8.1 in [BM] shows that if the relation (5) issatisfied for one Jordan curve C containing post( f ), then it holds forevery such curve. One can essentially show that a Thurston map isexpanding if and only if all the connected components in the preimageunder f − n of any open Jordan region not containing post( f ) becomeuniformly small as n goes to infinity.The following theorem (Theorem 1.2 in [BM]) says that there existsan invariant Jordan curve for some iterate of f . Theorem 3.2. If f : S → S is an expanding Thurston map, then forsome n ∈ N there exists a Jordan curve C ⊂ S containing post( f ) such that C is invariant under f n , i.e., f n ( C ) ⊆ C . In the following, it is not assumed that the Jordan curve C is invariantunless stated otherwise.Recall that an isotopy H between two homeomorphisms is a homo-topy so that at each time t ∈ [0 , H t is a homeomorphism.An isotopy H relative to a set A is an isotopy satisfying H t ( a ) = H ( a ) = H ( a )for all a ∈ A and t ∈ [0 , Definition 3.3.
Consider two Thurston maps f : S → S and g : S → S , where S and S are 2-spheres. We call the maps f and g (Thurston)equivalent if there exist homeomorphisms h , h : S → S that areisotopic relative to post( f ) such that h ◦ f = g ◦ h . We call themaps f and g topologically conjugate if there exists a homeomorphism h : S → S such that h ◦ f = g ◦ h .For equivalent Thurston maps, we have the following commutativediagram S h −−−→ S f y y g S h −−−→ S . If f : S → S is an expanding Thurston map and g : S → S istopologically conjugate to f , then g is also expanding. If f and g are ATT`ES MAPS AND COMBINATORIAL EXPANSION 9 equivalent Thurston maps and one of them is expanding, then the otherone is not necessarily expanding as well. Thus, topological conjugacyis a much stronger condition than Thurston equivalence. The followingtheorem (see Theorem 9.2 in [BM]) shows that under the conditionthat both maps are expanding, these two relations are the same.
Theorem 3.4.
Let f : S → S and g : S → S be equivalent Thurstonmaps that are expanding. Then they are topologically conjugate. We now consider the cardinality of the postcritical set of f . In Re-mark 5.5 in [BM], it is proved that there are no Thurston maps with f ) ≤
1. Proposition 6.2 in [BM] shows that all Thurston mapswith f ) = 2 are Thurston equivalent to a power map on theRiemann sphere, z z k , for some k ∈ Z \ {− , , } . Corollary 6.3 in [BM] states that if f : S → S is an expanding Thurstonmap, then f ) ≥ f : S → S be a Thurston map, and let C ⊂ S be a Jordancurve containing post( f ). By the Sch¨onflies theorem, the set S \ C hastwo connected components, which are both homeomorphic to the openunit disk. Let T and T ′ denote the closures of these components. Theyare cells of dimension 2, which we call 0 -tiles . The postcritical pointsof f are called 0 -vertices of T and T ′ , which are cells of dimension 0.The closed arcs on C between vertices 0 -edges of T and T ′ , which arecells of dimension 1. These 0-vertices, 0-edges and 0-tiles form a celldecomposition of S , denoted by D = D ( f, C ). We call the elementsin D D = D ( f, C ) be the set of connected subsets c ⊂ S such that f ( c ) is a cell in D and f | c is a homeomorphism of c onto f ( c ).Call c a 1-tile if f ( c ) is a 0-tile, call c a 1-edge if f ( c ) is a 0-edge, and call c a 1-vertex if f ( c ) is a 1-vertex. Lemma 5.4 in [BM] states that D is acell decomposition of S . Continuing in this manner, let D n = D n ( f, C )be the set of all connected subsets of c ⊂ S such that f ( c ) is a cellin D n − and f | c is a homeomorphism of c onto f ( c ), and call theseconnected subsets n -tiles, n -edges and n -vertices correspondingly, for n ∈ N . By Lemma 5.4 in [BM], D n is a cell decomposition of S , foreach n ∈ N , and we call the elements in D n n -cells. The followinglemma lists some properties of these cell decompositions. For moredetails, we refer the reader to Proposition 6.1 in [BM]. Lemma 3.5.
Let k, n ∈ N , let f : S → S be a Thurston map, let C ⊂ S be a Jordan curve with C ⊃ post( f ) , and let m = f ) .Consider the associated cell decompositions of S described above. (1) If τ is any ( n + k ) -cell, then f k ( τ ) is an n -cell, and f k | τ is ahomeomorphism of τ onto f k ( τ ) . (2) Let σ be an n -cell. Then f − k ( σ ) is equal to the union of all ( n + k ) -cells τ with f k ( τ ) = σ . (3) The number of n -vertices is less than or equal to m deg( f ) n ,the number of n -edges is m deg( f ) n , and the number of n -tilesis f ) n . (4) The n -edges are precisely the closures of the connected compo-nents of f − n ( C ) \ f − n (post( f )) . The n -tiles are precisely theclosures of the connected components of S \ f − n ( C ) . (5) Every n -tile is an m -gon, i.e., the number of n -edges and n -vertices contained in its boundary is equal to m . Let σ be an n -cell. Let W n ( σ ) be the union of the interiors of all n -cells intersecting with σ , and call W n ( σ ) the n -flower of σ . In general, W n ( σ ) is not necessarily simply connected. The following lemma (fromLemma 7.2 in [BM]) says that if σ consists of a single n -vertex, then W n ( σ ) is simply connected. Lemma 3.6.
Let f : S → S be a Thurston map. Let C be a Jordancurve containing post( f ) and consider the corresponding cell decompo-sitions of S . If σ is an n -vertex, then W n ( σ ) is simply connected. Inaddition, the closure of W n ( σ ) is the union of all n -tiles containing thevertex σ . We obtain a sequence of cell decompositions of S from a Thurstonmap and a Jordan curve on S . In many instances it is desirable thatthe local degrees of the map f at all the vertices are bounded, and thiscan be obtained using the assumption that f has no periodic criticalpoints (see [BM, Lemma 17.1]). Lemma 3.7.
Let f : S → S be a branched covering map. Then f hasno periodic critical points if and only if there exists N ∈ N such that deg f n ( p ) ≤ N, for all p ∈ S and all n ∈ N . It can be shown using the lemma above as well as Proposition 12.5and 13.1 in [BM] that there exist expanding Thurston maps with pe-riodic critical points. However, from now on we will only considerThurston maps that do not have periodic critical points.
Definition 3.8.
Let f : S → S be an expanding Thurston map, andlet C ⊂ S be a Jordan curve containing post( f ). Let x, y ∈ S . For x = y we define m f, C ( x, y ) := min { n ∈ N : there exist disjoint n -tiles X and Y for ( f, C ) with x ∈ X and y ∈ Y } . If x = y , we define m f, C ( x, x ) = ∞ .The minimum in the definition above is always obtained since thediameters of n -tiles go to 0 as n → ∞ . We usually drop one or both ATT`ES MAPS AND COMBINATORIAL EXPANSION 11 subscripts in m f, C ( x, y ) if f or C is clear from the context. If we definefor x, y ∈ S and x = y , m ′ f, C ( x, y ) = max { n ∈ N : there exist nondisjoint n -tiles X and Y for ( f, C ) with x ∈ X and y ∈ Y } , then m f, C and m ′ f, C are essentially the same up to a constant (seeLemma 8.6 (v) in [BM]). Note that our notation for m and m ′ isswitched from that used in [BM]. Lemma 3.9.
Let m f, C and m ′ f, C be defined as above. There exists aconstant k > , such that for any x, y ∈ S and x = y , m ′ f, C ( x, y ) − k ≤ m f, C ( x, y ) ≤ m ′ f, C ( x, y ) + 1 . Definition 3.10.
Let f : S → S be an expanding Thurston map andlet d be a metric on S . The metric d is called a visual metric for f ifthere exists a Jordan curve C ⊂ S containing post( f ), constants Λ > C ≥ C Λ − m f, C ( x,y ) ≤ d ( x, y ) ≤ C Λ − m f, C ( x,y ) for all x, y ∈ S . The number Λ is called the expansion factor of thevisual metric d .Proposition 8.9 in [BM] states that for any expanding Thurston map f : S → S , there exists a visual metric for f that induces the standardtopology on S . Lemma 8.10 in the same paper gives the followingcharacterization of visual metrics. Lemma 3.11.
Let f : S → S be an expanding Thurston map. Let C ⊂ S be a Jordan curve containing post( f ) , and d be a visual metricfor f with expansion factor Λ > . Then there exists a constant C > such that (1) d ( σ, τ ) ≥ (1 /C )Λ − n whenever σ and τ are disjoint n -cells, (2) (1 /C )Λ − n ≤ diam( τ ) ≤ C Λ − n for τ any n -edge or n -tile.Conversely, if d is a metric on S satisfying conditions (1) and (2) forsome constant C > , then d is a visual metric with expansion factor Λ > . Let (
X, d ) be a metric space. For α ≥ S ⊆ X , the α -dimensional Hausdorff measure H α ( S ) of S is defined as H α ( S ) := lim ǫ → H αǫ ( S ) , where H αǫ ( S ) = inf ( ∞ X i =1 diam( U i ) α : S ⊆ ∞ [ i =1 U i and diam( U i ) < ǫ ) where the infimum is taken over all countable covers { U i } of S . The Hausdorff dimension dim H ( X ) of a metric space X is the infimum of the set of α ∈ [0 , ∞ ) such that α -dimensional Hausdorff measure of X is zero: dim H ( X ) := inf { α ≥ H α ( X ) = 0 } . The dimension of a probability measure µ on X isdim µ := inf { dim H ( E ) : E ⊂ X is measurable and µ ( E ) = 1 } . The following theorem ([M3, Theorem 4]) gives a characterization ofLatt`es maps among among all expanding rational Thurston maps.
Theorem 3.12.
Let f : b C → b C be an expanding rational Thurstonmap. The map f is a Latt`es map if and only if there exists a visualmetric d on b C such that the dimension of the (normalized standard)Lebesgue measure with respect to the metric d is equal to . Latt`es and Latt`es-type Maps
In this section, we introduce Latt`es-type maps and establish some oftheir properties. We also briefly review the concept of the orbifold O f of a Thurston map f .Let L , L ′ ⊂ R be lattices. We will always assume that a latticehas rank 2. The quotients T = R / L and T ′ = R / L ′ are tori. Let A : R → R be an affine orientation-preserving map such that for anytwo points p, q ∈ R with p − q ∈ L , we have A ( p ) − A ( q ) ∈ L ′ . Thequotient of the map A , ¯ A : T → T ′ , is called an (orientation-preserving) torus homomorphism . If the map¯ A is also bijective, we call the map ¯ A : T → T ′ a torus isomorphism between T and T ′ . If L = L ′ , we call the induced map ¯ A : T → T a torus endomorphism . If in addition, the map ¯ A is a torus isomorphism,then we call ¯ A a torus automorphism of T . If L = Z , then an affinemap A that induces a torus endomorphism has the form(6) A (cid:18) xy (cid:19) = L (cid:18) xy (cid:19) + (cid:18) x y (cid:19) for (cid:18) xy (cid:19) ∈ R , where L is a 2 × x , y ∈ Z . In this case, the map A is a torus automorphism if andonly if L ∈ SL(2 , Z ).The matrix L is uniquely determined by ¯ A . Indeed, if affine maps A and A ′ induce the same torus endomorphism, then A and A ′ differby a translation by λ according to equation (6), where λ ∈ L . So wecan uniquely define the determinant, trace and eigenvalues of a torusendomorphism ¯ A and the affine map A to be the determinant, traceand eigenvalues of the matrix L as in equation (6). Denotedet ¯ A = det A = det L, tr( ¯ A ) = tr( A ) = tr( L ) . ATT`ES MAPS AND COMBINATORIAL EXPANSION 13
Definition 4.1.
We call Θ :
T → S a branched cover induced by arigid action of a group G on T if every element of g ∈ G acts as a torusautomorphism and for any t, t ′ ∈ T , we have Θ( t ) = Θ( t ′ ) if and onlyif there exists g ∈ G such that t = g ( t ′ ) . An equivalent formulation is that Θ induces a canonical homeomor-phism from the quotient space T /G onto S . Definition 4.2.
Let
L ⊂ R be a lattice. Let ¯ A be a torus endomor-phism of T = R / L whose eigenvalues have absolute values greaterthan 1. Let Θ : T → S be a branched covering map induced by a rigidaction of a finite cyclic group on T . A map f : S → S is called a Latt`es-type map (with respect to a lattice L ) if there exists ¯ A as abovesuch that the semi-conjugacy relation f ◦ Θ = Θ ◦ ¯ A is satisfied, i.e.,the following diagram commutes: T ¯ A −−−→ T Θ y y Θ S f −−−→ S .. In addition, if a Latt`es-type map f is rational, then the map f is calleda Latt`es map .We remark that this definition of Latt`es maps is equivalent to thedefinition of Latt`es maps in [M4].
Example 4.3.
Let A : C → C be the C -linear map defined by z z ,and let ℘ : C → b C be the Weierstrass elliptic function with respect tothe lattice 2 Z . Let ¯ A : T → T = C / Z be induced by A , and letΘ : T → b C be induced by ℘ . Then the map f satisfying the followingdiagram is well-defined and is a Latt`es-type map: T ¯ A −−−→ T Θ y y Θ b C f −−−→ b C .. In fact, the map f is a Latt`es map (see Example 7.9). We can thinkof the map f as follows (see the picture below): observe that the unitsquare [0 , in C can be conformally mapped to the upper half planein b C ; we glue two unit squares [0 , together along their boundaries, and get a pillow-like space which is homeomorphic to b C ; we colorone of the squares black and the other white; we divide each of thesquares into 4 smaller squares of half the side length, and color themwith black and white in checkerboard fashion; we map one of the smallblack pillows to the bigger black pillow by Euclidean similarity, and extend the map to the whole pillow-like space by reflection. We referthe reader to Section 1.2 in [BM] for further discussion of this example. > f Example 4.4.
Let A : R → R be the R -linear map defined by A (cid:18) xy (cid:19) = (cid:18) (cid:19) (cid:18) xy (cid:19) for (cid:18) xy (cid:19) ∈ R , and let ℘ be the Weierstrass elliptic function with respect to the lattice2 Z . Let ¯ A : T → T = R / Z be induced by A , and Θ : T → S be induced by ℘ . Then the map g satisfying the following diagram iswell-defined and is a Latt`es-type map: T ¯ A −−−→ T Θ y y Θ S g −−−→ S .. In fact, the map g is not topologically conjugate to a Latt`es map (seeExample 7.9). See the picture below:We refer the reader to Example 12.13 in [BM] for further discussionof this example. ATT`ES MAPS AND COMBINATORIAL EXPANSION 15
Lemma 4.5.
A Latt`es-type map f over any lattice L is a Latt`es-typemap over the integer lattice Z .Proof. For any Latt`es-type map f over a lattice L , let T = R / L .There exist a torus endomorphism¯ A : T → T and a branched covering map Θ :
T → S induced by a rigid action ofa fixed cyclic group on T , such that f ◦ Θ = Θ ◦ ¯ A . Let T = R / Z .Since L is a lattice with rank 2, there is an orientation-preserving iso-morphism L : Z → L . This isomorphism L can be extended to an R -linear map of R , still denoted by L , which induces a torus isomor-phism ¯ L : T → T . Define a map¯ A : T → T by ¯ A = ¯ L − ◦ ¯ A ◦ ¯ L , and a branched covering map Θ : T → S byΘ = Θ ◦ ¯ L . Then ¯ A is a torus endomorphism and the branchedcovering map Θ is induced by a rigid action of a finite cyclic group on T . In addition, f ◦ Θ = f ◦ Θ ◦ ¯ L = Θ ◦ ¯ A ◦ ¯ L = (Θ ◦ ¯ L ) ◦ ( ¯ L − ◦ ¯ A ◦ ¯ L ) = Θ ◦ ¯ A (see the diagram below). It follows that the map f is a Latt`es-typemap over the lattice Z . T L w w ♦♦♦♦♦♦♦ ¯ A / / Θ (cid:4) (cid:4) ✡✡✡✡✡✡✡✡✡✡✡✡✡ T (cid:4) (cid:4) ✡✡✡✡✡✡✡✡✡✡✡✡✡ ¯ L w w ♦♦♦♦♦♦♦ T ¯ A / / Θ (cid:15) (cid:15) T Θ (cid:15) (cid:15) S f / / S (cid:3) Remark.
Notice that the proof of this lemma works for any rank-2lattice besides the integer lattice Z . Hence, we can choose the latticefor our convenience. Lemma 4.6.
If a branched covering map
Θ :
T → S is induced bya rigid action of a finite cyclic group G on T , then G acts on T byrotation around a fixed point with order of G either , , or . Here by G acting on T by rotation around a fixed point, we meanthat if we identify the fixed point with the origin in R , and T with thefundamental domain in R , then G acts as a rotation on the Euclideanspace R . Proof.
By Lemma 4.5, we may assume that T = R / Z . Let g be agenerator of G with order n . The order n is greater than 1 since T /G = S . The element g is an automorphism of the torus T ; so g is induced byan affine map A g on R of the form, A g (cid:18) xy (cid:19) = L g (cid:18) xy (cid:19) + (cid:18) x g y g (cid:19) for (cid:18) xy (cid:19) ∈ R , where L g ∈ SL(2 , Z ) and x g , y g ∈ R . Since L g ∈ SL(2 , Z ) and L ng = I , where I ∈ SL(2 , Z ) is the identity element, the matrix L g is conjugateto a rotation of R , (cid:18) cos πn sin πn − sin πn cos πn (cid:19) . In addition, since the trace of L g is an integer, we must have2 cos 2 πn ∈ Z . Hence, the order n of the group G can only be 2 , , A ng = id R ,( L ng + L n − g + . . . + L g + I ) (cid:18) x g y g (cid:19) = (cid:18) ab (cid:19) , (7)where a, b ∈ Z . Since L g is conjugate to a non-trivial rotation, ( L g − I )is invertible. Multiplying equation (7) by ( L g − I ), we have( L g − I )( L ng + L n − g + . . . + L g + I ) (cid:18) x g y g (cid:19) = ( L g − I ) (cid:18) ab (cid:19) , so ( L g − I ) (cid:18) x g y g (cid:19) = ( L n +1 g − I ) (cid:18) x g y g (cid:19) = ( L g − I ) (cid:18) ab (cid:19) . Hence, we have (cid:18) x g y g (cid:19) = (cid:18) ab (cid:19) ∈ Z , and there exists a fixed point on T = R / Z under g . Therefore, thegroup G acts on T by rotation around a fixed point with order of G either 2 , , (cid:3) Lemma 4.7.
Every Latt`es-type map f is a Thurston map.Proof. By Lemma 4.5, we know that f is a Latt`es-type map over thelattice Z . Let T = R / Z . There exists a torus endomorphism¯ A : T → T and a branched covering map Θ :
T → S induced by a rigid action ofa finite cyclic group G on T , such that f ◦ Θ = Θ ◦ ¯ A . Let A : R → R be an affine map inducing ¯ A . ATT`ES MAPS AND COMBINATORIAL EXPANSION 17
The map Θ ◦ ¯ A is a branched covering map since locally ¯ A is ahomeomorphism and Θ is a branched covering map. Since f ◦ Θ = Θ ◦ ¯ A and ¯ A has local degree 1 on every point, we havedeg f (Θ( z )) deg Θ ( z ) = deg Θ ( ¯ A ( z )) , and f can be locally written as w w d , where w = Θ( z ) and d = deg f ( w ) = deg Θ ( ¯ A ( z )) / deg Θ ( z ) . Hence, a Latt`es-type map f is a branched covering map.Let V f and V Θ be the sets of critical values of f and Θ, respectively.We claim that post( f ) = V Θ and these sets have finite cardinality. Theclaim is proved similarly to Lemma 3.4 in [M4]. We give the detailsof the argument for the convenience of the reader. Since ¯ A is a localhomeomorphism and ¯ A and Θ are both surjective, a point p ∈ S is acritical value of Θ if and only if either p is a critical value of f , or p hasa preimage in f − ( p ) that is a critical value of Θ. So V Θ = V f ∪ f ( V Θ ).Hence f ( V f ) ⊆ f ( V Θ ) ⊆ V Θ and inductively, we have post( f ) ⊆ V Θ .The set V Θ is finite due to the compactness of S , and hence the set ofcritical points of Θ is also finite.In order to show that V Θ is a subset of post( f ), we argue by con-tradiction. Then there exists a critical point t ∈ T of Θ such thatΘ( t ) post( f ). There exists t = t in the preimage of t under ¯ A ,and there exists t = t , t in the preimage of t under ¯ A . Continuing inthis manner, we get a sequence { t i } and the cardinality of { t i } is notfinite. For all i ≥
1, we have(8) deg f (Θ( t i )) deg Θ ( t i ) = deg Θ ( ¯ A ( t i )) = deg Θ ( t i − )On the other hand, since Θ( t ) post( f ), every element of f − i (Θ( t )) isa non-critical point for f and has degree 1. In particular, deg f (Θ( t i )) =1 for all i ≥
0, and equation (8) implies thatdeg Θ ( t i ) = deg Θ ( t i − ) = . . . = deg Θ ( t ) > . Hence, each t i is a critical point of Θ. This is a contradiction to thefiniteness of the critical set of Θ.We claim that deg( f ) = det( A ). Since f ◦ Θ = Θ ◦ ¯ A and deg(Θ) < ∞ ,we have deg( f ) = deg( ¯ A ) . The map ¯ A carries a small region of area ǫ to a region of area det( ¯ A ) ǫ ,so deg( ¯ A ) = det( ¯ A ) . The claim follows. Since deg( f ) = det( A ) >
1, the Latt`es map f is aThurston map. (cid:3) For a, b ∈ N ∪ {∞} , we use the convention that ∞ is a multipleof any positive integer or itself. If a is a multiple of b , we write b | a .We also use the notation gcd { a, b } as the greatest common divisor for a, b ∈ N ∪ {∞} (defined in the obvious way). Recall that in a set X with partial order ≤ , an element x ∈ X is called a minimal element iffor all y ∈ X we have that y ≤ x implies that y = x ; an element x ∈ X is called the minimum if for all y ∈ X we have that x ≤ y . It is easyto see that if the minimum exists, then it is unique. Lemma 4.8.
For any Thurston map f , there exists a function ν f thatis the minimum among functions ν : S → N ∪ {∞} such that (9) ν ( p ) deg f ( p ) (cid:12)(cid:12) ν ( f ( p )) for all p ∈ S .Proof. We have a natural partial order for functions satisfying (9). If ν and ν are such functions, then we set ν ≤ ν iff ν ( p ) | ν ( p )for all p ∈ S . In order to show the existence of such a minimal functionsatisfying (9), we set ν ( p ) = 1 if p is not a postcritical point of f . Weonly need to assign a value to the finitely many postcritical points of f . If we let ν ( p ) = ∞ when p ∈ post( f ), this shows the existence ofa such function ν . The existence of a minimal function follows fromassigning values over a fixed finite set.To show uniqueness of a minimal function, suppose that ν and ν are both minimal functions satisfying condition (9). Let ν ( p ) := gcd { ν ( p ) , ν ( p ) } . We claim that ν satisfies condition (9). Indeed,gcd { ν ( f ( p )) , ν ( f ( p )) } = ν ( f ( p ))is a multiple ofgcd { ν ( p ) deg f ( p ) , ν ( p ) deg f ( p ) } = gcd { ν ( p ) , ν ( p ) } deg f ( p )= ν ( p ) deg f ( p ) . Hence, we have ν ≤ ν , ν . Since ν and ν are both minimal functions,we conclude that ν = ν = ν . We claim that this unique minimal function ν f is the minimum withrespect to the order ≤ . Indeed, let ν be a function satisfying (9). Thenwe have that ν = gcd( ν f , ν ) ≤ ν f also satisfies (9). Hence, ν = ν f by the minimality of ν f . Therefore, ν f = gcd( ν f , ν ) ≤ ν . (cid:3) Thurston associated an orbifold O f = ( S , ν f ) to a Thurston map f through the smallest ν f function in Lemma 4.8 (see [DH]). Moreprecisely, for each p ∈ S with ν f ( p ) = 1, the point p is a conepoint with cone angle π/v f ( p ). For post( f ) = { p , . . . , p m } , use ATT`ES MAPS AND COMBINATORIAL EXPANSION 19 ( ν ( p ) , . . . , ν ( p m )) to denote the type of O f . We will not elaborateon the geometric significance of the orbifold here, but instead refer thereader to Chapter 13 in [T]. Definition 4.9.
For any Thurston map f and the smallest function ν f : S → N ∪ {∞} associated to f satisfying condition (9), let χ ( O f ) = 2 − X p ∈ post( f ) (cid:18) − ν f ( p ) (cid:19) . • If χ ( O f ) = 0, we say that the orbifold O f is parabolic ; • If χ ( O f ) <
0, we say that the orbifold O f is hyperbolic .We call χ ( O f ) the Euler characteristic of the orbifold O f associated to f . Remark.
By Proposition 9.1 (i) in [DH], χ ( O f ) ≤ Lemma 4.10.
For a Latt`es-type map f , the orbifold O f is parabolic.In particular, the number of cone points must be either three or four.Hence, the cardinality of the postcritical set of f is either three or four.Proof. There exist a torus endomorphism ¯ A : T → T and a branchedcovering map Θ :
T → S induced by a group action on T as a rotationaround some base point in T , such that f ◦ Θ = Θ ◦ ¯ A . For any points t ∈ ¯ A − ( t ), t i ∈ T , and p ∈ f − ( p ), p i ∈ S such that Θ( t i ) = p i , i = 0 ,
1, we have thatdeg Θ ( t ) = deg f ( p ) deg Θ ( t ) . Define ν (Θ( t )) = deg Θ ( t ), and ν ( p ) = 1 if p post( f ). Since Θ isinduced by a group action, different preimages of Θ( t ) under Θ all havethe same degree. Explicitly, for any t, t ′ ∈ T such that Θ( t ) = Θ( t ′ ),there is a torus automorphism g such that g ( t ) = t ′ and Θ( g ( x )) = Θ( x )for all x ∈ T , sodeg Θ ( t ) = deg Θ ( g ( t )) deg g ( t ) = deg Θ ( t ′ ) . In the proof of Lemma 4.7, we showed that post( f ) is equal to the setof critical values of Θ, so ν is well-defined on S . In addition, ν ( p ) = ν ( f ( p )) = deg Θ ( t ) = deg f ( p ) deg Θ ( t ) = deg f ( p ) ν ( p ) . So ν is a function satisfying condition (9).We claim that ν is the smallest function satisfying condition (9).Indeed, suppose that ν ′ satisfying condition (9) is smaller than ν . If p post( f ), then ν ′ ( p ) = ν ′ ( f ( p )) = deg f ( p ) = ν ( f ( p )) = ν ( p ) . If p ∈ post( f ), then there exists n > p ∈ f − n ( p ), such that p post( f ). By induction on n , we get that ν ′ ( p ) = ν ( p ) and hence ν ′ ( p ) = ν ′ ( f ( p )) = deg f ( p ) ν ′ ( p ) = deg f ( p ) ν ( p ) = ν ( f ( p )) = ν ( p ) . Thus, ν ′ = ν and our claim is proved.By the proof of Proposition 9.1 (i) in [DH], ν ( f ( p )) = deg f ( p ) ν ( p )implies that f is a covering map of orbifolds f : O f → O f , and againby Proposition 9.1 (ii), χ ( O f ) = 0 and O f is parabolic. All the para-bolic orbifolds are classified in Section 9 in [DH], and they have type(2 , , , , (3 , , , (2 , ,
6) and (2 , , (cid:3) Proposition 4.11.
Every Latt`es-type map f is an expanding Thurstonmap.Proof. By Lemma 4.7, we know that f is a Thurston map. Given aLatt´es-type map f , there exists a torus endomorphism¯ A : T → T and a branched covering map Θ :
T → S induced by a rigid action ofa finite cyclic group G on T , such that f ◦ Θ = Θ ◦ ¯ A .Let C ⊂ S be a Jordan curve containing post( f ). The torus T carries a flat metric induced by the Euclidean metric R , and the mapΘ induces a flat orbifold metric on T /G ∼ = S . Observe that the interior T of a 0-tile on S under the cell decomposition of ( f, C ) does notintersect with post( f ) = V Θ , so Θ restricted to one of the connectedcomponents T ′ of Θ − ( T ) is a homeomorphism. In addition, since S is obtained by a finite quotient of T by G ,diam( T ′ ) ≤ diam( T ) ≤ | G | diam( T ′ ) . Each connected component of ¯ A − ( T ′ ) has diameter λ diam( T ′ ), where λ = | λ | − < A .Hence, each connected component of the preimage of T under f n hasdiameter bounded by 2 | G | λ n diam( T ), where λ <
1. Therefore, wehave mesh( f, n, C ) ≤ | G | λ n mesh( f, , C ) → n → ∞ , and the map f is expanding. (cid:3) Combinatorial Expansion Factor and D n In this section, we first review the definitions and some properties ofthe quantity D n and the related combinatorial expansion factor of anexpanding Thurston map. Then we prove a relation between D n andthe operator norm of the associated torus map for Latt`es-type maps,which gives the necessity of the third condition in Theorem 8.1.Let f : S → S be an expanding Thurston map and let C be aJordan curve containing post( f ). First, we review some definitionsand propositions from [BM]. ATT`ES MAPS AND COMBINATORIAL EXPANSION 21
Definition 5.1.
A set K ⊆ S joins opposite sides of C if f ) ≥ K meets two disjoint 0-edges, or if f ) = 3 and K meets allthree 0-edges.Let D n = D n ( f, C ) be the minimum number of n -tiles needed to joinopposite sides of a Jordan curve C . More precisely, D n := min { N ∈ N : there exist n -tiles X , ..., X N such that(10) N [ j =1 X j is connected and joins opposite sides of C} . We will often abuse notation and write D n rather than D n ( f, C ). Example 5.2.
Recall the Latt`es-type map f in Example 4.3 which isinduced by the map z z . The postcritical set post( f ) consists ofthe four common corner points of the two big squares. If we let C bethe common boundary of the two big squares, then C contains post( f )and D n ( f, C ) = 2 n for all n ≥
0. Similarly, consider the Latt`es-type map g in Example4.4. If we let C ′ be the boundary of the common big squares, then C ′ contains post( g ) which consists of the four corner points, and D n ( g, C ′ ) = 2 n for all n ≥ Lemma 5.3.
Let n ∈ N , and let K ⊂ S be a connected set. If thereexist two disjoint n -cells σ and τ with K ∩ σ = ∅ and K ∩ τ = ∅ , then f n ( K ) joins opposite sides of C . Lemma 7.10 in [BM] states that:
Lemma 5.4.
For n, k ∈ N , every set of ( n + k ) -tiles whose union isconnected and meets two disjoint n -cells contains at least D k elements. Proposition 17.1 in [BM] says that:
Proposition 5.5.
For an expanding Thurston map f : S → S , and aJordan curve C containing post( f ) , the limit Λ ( f ) := lim n →∞ D n ( f, C ) /n exists and is independent of C . We call Λ ( f ) the combinatorial expansion factor of f .Proposition 17.2 in [BM] states that: Proposition 5.6. If f : S → S and g : S → S are expandingThurston maps that are topologically conjugate, then Λ ( f ) = Λ ( g ) . Let f be an expanding Thurston map. For any two Jordan curves C and C ′ with post( f ) ⊂ C , C ′ , inequality (17.1) in [BM] states that thereexists a constant c > n > c D n ( f, C ) ≤ D n ( f, C ′ ) ≤ cD n ( f, C ) . We obtain the following lemma:
Lemma 5.7.
With the notation above, there exists a constant c > such that D n ( f, C ) ≥ c (deg f ) n/ if and only if there exists a constant c ′ > such that D n ( f, C ′ ) ≥ c ′ (deg f ) n/ for all n > . So we may say that D n ≥ c (deg f ) n/ for some c > Lemma 5.8.
Let f and g be two expanding Thurston maps that aretopologically conjugate via a homeomorphism h . Let C be a Jordancurve on S containing post( f ) , and let C ′ be the image of C under h .Then D n ( f, C ) = D n ( g, C ′ ) for all n ≥ . This lemma follows directly from the definitions of D n and topolog-ical conjugacy.Recall that the maximum norm (or l ∞ norm ) of a vector v = ( x , . . . , x n ) ∈ R n is k v k ∞ = max {| x | , . . . , | x n |} . Let A : R n → R n be an R -linear map. Then the ( l ∞ -)operator norm is k A k ∞ := max {k Av k ∞ : v ∈ R n , k v k ∞ = 1 } . Let f be a Latt`es-type map over a lattice L with orbifold type(2 , , , A : T → T and a branchedcovering map Θ :
T → S induced by a group action on T such that f ◦ Θ = Θ ◦ ¯ A , where T = R / L . We use A to denote an affine maplifted to the covering of T with L as the corresponding linear map. Bythe remark after Lemma 4.5, we may assume L = 2 Z . For a Latt`es-type map with orbifold type (2 , , , ◦ p : R → S ,where p : R → R / (2 Z ) is the quotient map, with the Weierstrassfunction ℘ : R → S with the lattice 2 Z . See the diagram below. R ℘ (cid:15) (cid:15) A / / R ℘ (cid:15) (cid:15) S f / / S . ATT`ES MAPS AND COMBINATORIAL EXPANSION 23
Let the Jordan curve C on S be the image of the boundary of the unitsquare [0 , × [0 ,
1] under ℘ . Proposition 5.9.
Let f be a Latt`es-type map with orbifold type (2 , , , .Let A be its affine map from R to R with L as the corresponding lin-ear map and ℘ : R → S be the Weierstrass function with the lattice Z (as in the remark above). We have k L − n k ∞ ≤ D n ( f, C ) ≤ k L − n k ∞ + 1 , where the Jordan curve C is the image of the boundary of the unit square [0 , × [0 , under ℘ .Proof. The idea of the proof is to lift everything to R . Since the unitsquare is homeomorphic to a 0-tile, D n ( f, C ) is the same as the numberof pre-images of the unit squares under A n needed to join the oppositesides of the unit square. We present the details below.Notice that the pre-image of C under ℘ is the whole grid of Z (i.e.,the union of all the horizontal and vertical lines containing an integer-valued point), and C contains all the post-critical points of f . Therestriction of ℘ to the interior of the rectangle R := [0 , × [0 ,
1] isa homeomorphism onto its image, which is the union of the interiorsof the 0-tiles of S and one edge of a 0-tile. Notice that the sameholds for any rectangle obtained from two adjacent unit squares. Thepre-images of unit squares under A n are parallelograms, which we call n -parallelograms .The n -tiles of ( f, C ) (i.e., the pre-images of 0-tiles under f n ) are theimages of n -parallelograms under ℘ . Let D v be the minimum numberof n -parallelograms connecting the line { } × ( −∞ , + ∞ ) and { } × ( −∞ , + ∞ ), and let D h be the minimum number of n -parallelogramsconnecting the lines ( −∞ , + ∞ ) × { } and ( −∞ , + ∞ ) × { } . We define D ′ n := min { D v , D h } . We claim that D n = D ′ n . Let T , T , . . . , T D n be a sequence of n -tiles with the minimum number of n -tiles joining opposite sides of a0-tile. Without loss of generality, we may assume that this 0-tile isthe image of [0 , × [0 ,
1] under ℘ , and the opposite sides of the 0-tile are the images of the sides [0 , × { } and [0 , × { } . Let T ′ be the connected component of ℘ − ( T ) intersecting with [0 , × [0 , n -parallelogram. Let T ′ be the component of ℘ − ( T )intersecting with T ′ , which is also an n -parallelogram. Let T ′ be thecomponent of ℘ − ( T ) intersecting with T ′ , and so on. We obtain asequence of n -parallelograms T ′ , . . . , T ′ D n connecting ( −∞ , + ∞ ) × { } and ( −∞ , + ∞ ) ×{ } , and hence D ′ n ≤ D n . On the other hand, supposethat a sequence of n -parallelograms P , . . . , P m connects ( −∞ , + ∞ ) ×{ } and ( −∞ , + ∞ ) × { } , or connects { } × ( −∞ , + ∞ ) and { } × ( −∞ , + ∞ ). Then the sequence ℘ ( P ) , . . . , ℘ ( P m ) of n -tiles connects apair of opposite sides of a 0-tile. We conclude that D ′ n = D n as desired.Since A and L differ be a translation of an element in 2 Z , every n -parallelogram with respect to A is an n -parallelogram with respectto L , and vice versa. So we may assume that A = L . Without loss ofgenerality, we may assume that D h ≤ D v , so that D ′ n = D h . Observethat we need at least m parallelograms to connect a pair of oppositesides of an ( m × m )-grid of parallelograms. Notice that L − n ([ − m, m ] × [ − m, m ]) ∩ ( −∞ , + ∞ ) × { } 6 = ∅ (11)if and only if there exist m n -parallelograms connecting ( −∞ , + ∞ ) ×{ } and ( −∞ , + ∞ ) × { } . Hence, D ′ n is equal to the smallest positiveinteger m such that y = max { y ( ± m, ± m ) } is greater than 1, where (cid:18) x ( ± m, ± m ) y ( ± m, ± m ) (cid:19) = L − n (cid:18) ± m ± m (cid:19) , and ( ± m, ± m ) varies over ( m, m ) , ( m, − m ) , ( − m, m ) and ( − m, − m ).Since the image of { v : k v k ∞ = 1 } = ∂ [ − , under L − n is the bound-ary of a parallelogram, k L − n k ∞ = max (cid:13)(cid:13)(cid:13)(cid:13) L − n (cid:18) ± ± (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ , so D ′ n k L − n k ∞ = max (cid:13)(cid:13)(cid:13)(cid:13) L − n (cid:18) ± D ′ n ± D ′ n (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ = y ≥ . Hence, 1 k L − n k ∞ ≤ D ′ n ≤ k L − n k ∞ + 1 . Since D n = D ′ n , the proof is complete. (cid:3) Corollary 5.10.
Let f be a Latt`es-type map with orbifold type (2 , , , ,and let A be its affine map from R to R . Then the combinatorial ex-pansion factor Λ ( f ) equals the minimum absolute value of the eigen-values of A .Proof. Let L be the linear map of A . By the previous proposition,1 k L − n k ∞ ≤ D n ≤ k L − n k ∞ + 1 . Taking n -th roots gives (cid:18) k L − n k ∞ (cid:19) /n ≤ D n /n ≤ (cid:18) k L − n k ∞ + 1 (cid:19) /n , so by Gelfand’s formula (see Theorem 13 in [L2, Chapter 8]),lim n →∞ D n /n = lim n →∞ k L − n k /n ∞ = 1 ρ ( L − ) , ATT`ES MAPS AND COMBINATORIAL EXPANSION 25 where ρ ( L − ) is the spectral radius of A − . On the other hand, thespectral radius of L − is the maximal absolute value of the eigenvaluesof L − , which is equal to 1 / | λ | , where | λ | is the minimum absolutevalue of the eigenvalues of A (and L ). We conclude thatΛ ( f ) = lim n →∞ D /nn = | λ | . (cid:3) Proposition 5.11.
Let f be a Latt`es map over L and let C ⊂ S be aJordan curve containing all postcritical points of f . Then there existsa constant c > such that D n ( f, C ) ≥ c (deg f ) n/ for all n > .Proof. Let f be a Latt`es map induced by the linear map L : C → C defined by z λz . It follows from the claim at the end of the proof ofLemma 4.7 that(12) deg f = | λ | . First, assume that f is a Latt`es map with orbifold type (2 , , , D n ≥ k L − n k ∞ = 1 | λ − n | = | λ | n = (deg f ) n/ . For Latt`es maps with f ) = 3, there exists a Jordan curve C containing the postcritical set that lifts to a tiling of the plane byEuclidean triangles. We refer the reader to Page 13 in [M4] for moredetails. We will call the triangles in the tiling above unit triangles .The idea of the proof is similar to the proof of Proposition 5.9. Thatis, we will attempt to lift everything to C . Since a unit triangle isholomorphic to a 0-tile, D n ( f, C ) is the same as the number of pre-images of the unit triangles under L n needed to connect all edges ofthe unit triangle. We present the details below.The pre-images of the unit triangles under L are triangles similar tothe unit triangle by a ratio of | λ | n , which we call n -triangles . Sincethe image of a 1-triangle under Θ ◦ L = f ◦ Θ is a 0-tile (see thecommutative diagram below), and since the only connected set thatmaps onto a 0-tile under f is a 1-tile, we conclude that the image of a1-triangle under Θ is a 1-tile. C Θ (cid:15) (cid:15) L / / C Θ (cid:15) (cid:15) ˆ C f / / ˆ C . Similarly, the image of an n -triangle under Θ is an n -tile since the onlyconnected set that maps onto a 0-tile under f n is a n -tile.Let D ′ n be the minimum number of n -triangles needed to connectall three edges of a 0-triangle △ . Assume that D n n -tiles connect the three edges of the 0-tile T = Θ( △ ). We claim that D n = D ′ n . Let T , T , . . . , T D n be a sequence of n -tiles connecting the three edges ofthe 0-tile T . Let T ′ be the connected component of Θ − ( T ) intersectingwith △ , which is an n -triangle. Let T ′ be the connected componentof Θ − ( T ) intersecting with T ′ , which is also an n -triangle. Let T ′ be the connected component of Θ − ( T ) intersecting with T ′ , and soon. We obtain a sequence of n -triangles T ′ , . . . , T ′ D n connecting thethree edges of △ , and hence D ′ n ≤ D n . On the other hand, supposethat a sequence of n -triangles T , . . . , T D ′ n connects the three edges of △ . Then the sequence Θ( T ) , . . . , Θ( T D ′ n ) of n -tiles connects the threeedges of the 0-tile T . We conclude that D ′ n = D n as desired.Let R denote the length of the longest edge of the unit triangle.Then the longest edge of an n -triangle is R | λ | − n . Let S be the unionof all the edges of the n -triangles T , . . . T D n . Then S is a union ofline segments and hence we may consider its length. On the one hand,the length of S is bounded above by 3 D n R | λ | − n , since R | λ | − n is thelongest edge of an n -triangle. On the other hand, the length of S isbounded below by the length r of the shortest connected union of linesegments connecting the three edges of a unit triangle. Explicitly, r is the minimum length of the line joining the longest edge of a unittriangle to the vertex containing the other two edges. Using (12), weconclude that D n ≥ r R | λ | − n = r R | λ | n = r R (deg f ) n/ . (cid:3) Existence of the Visual Metric
In this section, we prove that there exists a visual metric on S with ex-pansion factor equal to deg( f ) / under the three conditions in Theorem8.1. This will imply that the expanding Thurston map f is topologi-cally conjugate to a Latt`es map.We refer to the following assumptions as ( ∗ ):( ∗ ) The map f : S → S is an expanding Thurston map with noperiodic critical points, and C is a Jordan curve in S thatis invariant under f and satisfies post( f ) ⊂ C .Notice that the cell decompositions D n ( f, C ) of S induced by a Jor-dan curve as in ( ∗ ) are compatible with one another in the sense that D n +1 ( f, C ) is a subdivision of D n ( f, C ).Let λ := (deg f ) / . We refer to the following assumptions as ( ∗∗ ):( ∗∗ ) The map f : S → S is an expanding Thurston map with noperiodic critical points, and there exists a constant c > D n = D n ( f, C ) ≥ cλ n for all n >
0, where C is a Jordancurve in S that is invariant under f and satisfies post( f ) ⊂ C . ATT`ES MAPS AND COMBINATORIAL EXPANSION 27
First, let us review some definitions (see the proof of Theorem 17.3in [BM] for more details). Let f be an expanding Thurston map. BySection 3, we have a sequence of cell decompositions of the underlyingspace S by tiles. We define a tile chain P to be a finite sequence oftiles X , . . . , X N such that X j ∩ X j +1 = ∅ for j = 1 , . . . , N −
1. We also write P = X X . . . X N , and we use | P | to denote the underlying set S Ni =1 X i with the subspacetopology. In addition, if X n intersects with X , then we call the tilechain P a tile loop . For A, B ⊆ S , we say that the tile chain P joins the sets A and B if A ∩ X = ∅ and B ∩ X N = ∅ . We say that the tile chain P joins the points x and y if P joins { x } and { y } . A subchain of P = X X . . . X N is a tile chain of the form X j . . . X j s , where 1 ≤ j < . . . < j s ≤ N. We call a tile chain P = X X . . . X N simple if there is no subchainof P that joins X and X N . We call a tile chain P = X X . . . X N an n -tile chain if all the tiles X i are n -tiles 1 ≤ i ≤ N . An n -tile chain P = X X . . . X N is called an e -chain if there exists an n -edge e i with e i ⊆ X i ∩ X i +1 for i = 1 , . . . , N . The e -chain joins the tiles X and Y if X = X and X N = Y . A set M of n -tiles is e -connected if every twotiles in M can be joined by an e -chain consisting of n -tiles containedin M .The following lemma is from [BM, Lemma 14.4]. Lemma 6.1.
Let γ ⊂ S be a path in S defined on a closed interval J ⊂ R and M = M ( γ ) be the set of tiles having nonempty intersectionwith γ . Then M is e-connected. If P = X . . . X N is a tile chain, then we define the length of the tilechain to be the number of tiles in P :length( P ) = N. For n ≥
1, we define a function d n : S × S → R (13)as follows: for any x, y ∈ S , if x = y , then d n ( x, y ) = 0; otherwise, d n ( x, y ) = min { length( P ) } λ − n , where the minimum is taken over all n -tile chains P joining x and y .It is clear that d n is a metric on S .In the following, we will show that for any x, y ∈ S with x = y , theratio d n ( x, y ) /λ − m ( x,y )0 has a uniform upper and lower bound for all n > m ( x, y ), where m ( x, y ) = m f, C ( x, y ) is defined in Definition 3.8 (see Lemma 6.11 andLemma 6.12). Then we will define a distance function d = lim sup n →∞ d n , and we will see that this metric d is a visual metric on S with expansionfactor Λ (see Proposition 6.13). Definition 6.2.
For n ≥
3, we call a topological space X an n -gon if X is homeomorphic to the closed unit disk D ⊂ R with n pointsmarked on the boundary of X . Since the boundary of an n -gon ishomeomorphic to S , there is a natural cyclic order for the n markedpoints on the boundary. We call these n -points vertices of the n -gonand the parts of the boundary of X joining two consecutive vertices inthe cyclic order the edges of the n -gon.Now let us review some basic definitions from graph theory (we referthe reader to [D] for more details). A graph G is a pair ( V, E ) ofsets such that the edge set E = E ( G ) is a symmetric subset of theCartesian product V × V of the vertex set V = V ( G ). We call a graph G ′ = ( E ′ , V ′ ) a subgraph of G = ( V, E ) if E ′ ⊆ E and V ′ ⊆ V, written as G ′ ⊆ G . A (simple) path in a graph G = ( V, E ) is a non-empty subgraph P = ( V ′ , E ′ ) of the form V ′ = V ′ ( P ) = { x , x , . . . , x k } and E ′ = E ′ ( P ) = { x x , x x , . . . , x k − x k } , where the x i ∈ V ′ are all distinct, and uv denotes the edge with endpoints u, v ∈ V . We also write a path as P = x x . . . x k and call P a path from x to x k . Given sets A, B of vertices in G , wecall P = x x . . . x k an A - B path if x ∈ A and x k ∈ B . Given a graph G = ( V, E ), if
A, B, X ⊆ V are such that X is disjoint from A and B , and every A - B path in G contains a vertex from X , we say that X separates the sets A and B in G . We call X a separating set for A and B in the graph G . We will use the following theorem (see Theorem3.3.1 in [D]). In general, given a topological space T , let A, B ⊂ T suchthat A ∩ B = ∅ . We say that a set U ⊂ T separates A and B if for any path γ ⊆ T joining A and B , γ ∩ U = ∅ . ATT`ES MAPS AND COMBINATORIAL EXPANSION 29
We call U a separating set of A and B in T . For x, y ∈ T , we call U a separating set of x and y in T if U separates { x } and { y } in T . Theorem 6.3 (Menger’s theorem) . Let G = ( V, E ) be a finite graphand A, B ⊆ V . Then the minimal cardinality of a set separating A and B in G is equal to the maximal number of pairwise disjoint A - B pathsin G . Let X be a set of m -tiles, and denote the union by | X | . All thevertices and edges of m -tiles in X give a cell decomposition of | X | . Wedefine a graph G ( X ) with vertex set being the set of all m -tiles in X ,and with an edge between two vertices if and only if the corresponding m -tiles share a common edge. We call G ( X ) the dual graph associatedwith X . Note that a subset of X is e -connected if and only if thecorresponding vertex set in G ( X ) is path connected.Given an l -vertex v , recall that W l ( v ) is the union of the interior of n -cells intersecting with v , so W l ( v ) is connected. For n > l , let D n ( v ) bethe set of n -cells in W l ( v ). This gives us a cell decomposition of W l ( v ).Let G n ( v ) be the dual graph associated with the cell decomposition D n ( v ). Lemma 6.4.
Assume that ( ∗ ) holds and n > l ≥ . Then with thenotation above, the graph G n ( v ) is path connected.Proof. By Lemma 3.6, the l -flower W l ( v ) is simply connected. Thereexists a path γ ⊂ W l ( v ) containing all n -vertices in W l ( v ). Let V = V ( γ ) be the set of tiles having nonempty intersection with γ . Then V is the vertex set of the graph G n ( v ). Lemma 6.1 states that V is e -connected, which implies that G n ( v ) is path connected. (cid:3) Note that we do no need f to be expanding in Lemma 6.4. Lemma 6.5.
Assume that ( ∗ ) holds and n > . Let X be a set of n -tiles, and A, B, S ⊂ X . If S separates A and B in the graph G ( X ) ,then | S | separates | A | and | B | in | X | .Proof. Assume that the set | S | does not separate | A | and | B | in | X | .Then there exists a path γ ⊂ | X | joining | A | and | B | such that γ ∩ | S | = ∅ . By Lemma 6.1, the set M ( γ ) of n -tiles intersecting with γ is e -connected.In addition, we have that M ( γ ) ∩ S = ∅ since γ ∩ | S | = ∅ . This means that there exists an e -path in M ( γ ) ⊂ X \ S joining A and B . This is a contradiction of the definition of theseparating set S . Hence, the set | S | separates | A | and | B | in | X | . (cid:3) Continuing with the notation of Lemma 6.5 above, if in addition, S is a minimal separating set of A and B in the graph G ( X ), we would like to show that | S | is a connected set. This will follow from Lemma6.8. In order to prove it, we need some preparations.The following theorem is from [N, Page 110]. Theorem 6.6 (Janiszewski) . Let A and B be closed subset of S suchthat A ∩ B is connected. If neither A nor B separates two points x and y in S , then A ∪ B does not separate x and y either. As a corollary of Janiszewski Theorem, we have the following.
Corollary 6.7.
Let U be a closed subset of S with finitely many con-nected components. For two path-connected regions X, Y ⊂ S whichare disjoint from U , if the set U separates X and Y , then one of theconnected components of U separates X and Y .Proof. Fix x ∈ X and y ∈ Y . By induction on the number of con-nected components of U and by Janiszewski’s Theorem, there exists aconnected component U ′ of U that separates x and y . Consider a path γ connecting points x ′ ∈ X and y ′ ∈ Y . Let α ⊂ X be a path from x to x ′ , and let β ⊂ Y be a path from y ′ to y . Then the path αγβ joining x and y intersects U ′ . Hence, the path γ intersects U ′ , and U ′ separates x ′ and y ′ . We conclude that U ′ separates X and Y . (cid:3) Lemma 6.8.
Let W be a simply connected region in S . Let U bea closed subset of the closure W of W in S with finitely many con-nected components. For two path-connected regions X, Y ⊂ W whichare disjoint from U , if U separates X and Y in W , then there exists aconnected component of U separating X and Y .Proof. Without loss of generality, we may assume that U = ∪ Ii =1 U i where U i is a connected components of U , and U i ∩ U j = ∅ if i = j . Let ∂W be the boundary of W in S . For any path γ ⊆ S from X and Y ,if γ ⊂ W , then γ ∩ U = ∅ , and if γ W , then γ ∩ ∂W = ∅ . So the set U ∪ ∂W separates X and Y in S . Case 1:
None of the U i intersect with ∂W . Since ∂W does notseparate x and y in S , Corollary 6.7 implies that one of the U i separates X and Y in S . Case 2:
All of the U i intersect with ∂W . Let U ′ i = U i ∪ ∂W for 1 ≤ i ≤ I Notice that U ′ i ∩ U ′ j = ∂W is connected for any i = j . Weclaim that one of the U ′ i separates X and Y in S . If none of U ′ i separates X and Y in S , then by Janiszewski’s Theorem, theset ∪ Ii =1 U ′ i = U ∪ ∂W ATT`ES MAPS AND COMBINATORIAL EXPANSION 31 does not separate X and Y , which is a contradiction. Withoutloss of generality, assume that U ′ separates X and Y in S .Then U separates X and Y in W . Case 3:
Only some of the U i intersect with ∂W . Without loss ofgenerality, assume that U i ∩ ∂W = ∅ for 1 ≤ i ≤ J < I, and U i ∩ ∂W = ∅ for J < i ≤ I, Let U ′ = ∪ Ii = J +1 U i ∪ ∂W . By Corollary 6.7, either one of the U i for 1 ≤ i ≤ J or U ′ separates X and Y in S . If one of the U i for 1 ≤ i ≤ J separates X and Y , we are done. If U ′ separates X and Y , then it is Case 2. This implies that one of the U i for i ∈ I separates X and Y in W .Hence, one of the connected components of U separates X and Y in W . (cid:3) Proposition 6.9.
Let f be a Thurston map without periodic criticalpoints and let C ⊂ S be a Jordan curve that is invariant under f andcontains post( f ) . Then there exists a constant C > such that D n = D n ( f, C ) ≤ C deg( f ) n/ for all n ≥ .Proof. First, assume that m = f ) ≥
4. Let e , . . . , e m be the0-edges in cyclic order. Fixing a 0-tile, let X be the union of all n -tilesin this 0-tile, and let G ( X ) be the dual graph of X . Let A be the setof all n -tiles in X intersecting with e , and B be the set of all n -tilesin X intersecting with e .Let S be a minimal separating set between A and B in G ( X ). ByLemma 6.5, the set | S | separates | A | and | B | in | X | . Consider the sub-space topology on | A | and | B | . Since e ⊂ int( | A | ) and e ⊂ int( | B | ), e and e are both connected and disjoint from | S | . So by Lemma 6.8, oneof the connected components of | S | separates e and e . Since S is aminimal separating set of A and B , the separating set | S | is connected.Notice that there are two connected components in Q = ∂ | X | \ (cid:0) int( e ) ∪ int( e ) (cid:1) , which gives us two disjoint paths from e to e . Since | S | intersects withboth components of Q , the set | S | joins at least two disjoint 0-edges.Hence, there exist at least D n n -tiles in S .By Menger’s theorem, there are at least D n many disjoint A - B paths.Let N n be the minimum number of tiles in an A - B path, and since an A - B path is an n -tile chain joining opposite sides of the Jordan curve C , we have D n ≤ N n . We get D n ≤ D n N n ≤ f )) n , so D n ≤ C deg( f ) n/ for C = √ f ) = 3, we can cut along any two edges of the 3-gons,and we unfold it to get a 4-gon. Let X be the union of all n -tiles inthis 4-gon, and pick two non-adjacent edges in this 4-gon and call them e and e . Now we can apply the same argument as in the case when f ) = 4 above. (cid:3) Given an h -tile X h , and an ( h − X h − , if X h ⊂ X h − , thenwe call X h − the parent of X h . Lemma 6.10.
Assume that ( ∗∗ ) holds and n > h > . Let X h , Y h be h -tiles and let X h − , Y h − be their parents respectively. Assume X h − ∩ Y h − = ∅ . Then there exists an n -tile chain with at most c ′ λ n − h tiles joining X h and Y h , where c ′ > only depends on f .Proof. The lemma is trivial if X h ∩ Y h = ∅ . Now assume that X h and Y h are disjoint. Let v be an ( h − X h − ∩ Y h − , and let G n ( v )and G h ( v ) be the dual graphs associated to the cell decompositions of W h − ( v ) consisting of n -tiles and h -tiles respectively (see the paragraphbefore Lemma 6.4 for the meaning of the notation). By Lemma 6.4,the graphs G n ( v ) and G h ( v ) are both path-connected.Let A be the set of all n -tiles in X h , and B be the set of all the n -tilesin Y h . Let S be a minimal separating set between A and B . Consider A, B, S as vertices in G n ( v ). By Lemma 6.5, the set | S | separates X h and Y h in W h − ( v ). Since int( X h ) and int( Y h ) are both connectedregions and disjoint from | S | , by Lemma 6.8, one of the connectedcomponent of | S | separates int( X h ) and int( Y h ). Since S is a minimalseparating set, the separating set | S | is connected. X |S|X h e e e X X e =e l Y h =X X h- Y h- =X l =X ATT`ES MAPS AND COMBINATORIAL EXPANSION 33
Since G h ( v ) is path-connected, there is an e -chain P = X X X . . . X l of h -tiles in W h − ( v ) with X = X h and X l = Y h . After possibly re-placing P with a shorter e-chain, we may assume that X i = X j if i = j .Pick an h -edge in X i − ∩ X i and call it e i , for i = 1 , , . . . , l . Noticethat there are two connected components in Q i = ∂X i \ (cid:0) int( e i ) ∪ int( e i +1 ) (cid:1) , for i = 1 , . . . , l −
1. The union Q = ∪ l − i =1 Q i has two connected compo-nents, which gives us two disjoint paths from X h to Y h (See the figureabove). Since | S | intersects with both components of Q , the set | S | joins at least two disjoint h -edges. Hence, there exist at least D n − h n -tiles in S .Let N n be the minimal number of n -tiles in an A - B path. ByMenger’s Theorem, there are at least D n − h non-disjoint A - B pathsin G n ( v ). Thus N n D n − h ≤ K (deg f ) ( n − h ) , where K > f .Hence N n ≤ K (deg f ) ( n − h ) D n − h ≤ KC λ n − h = c ′ λ n − h , where c ′ only depends on f . (cid:3) Recall the function d n as defined in 13. Lemma 6.11.
Assume that ( ∗∗ ) holds and assume that λ > . Thereexists a constant C > depending only on f , such that for any x, y ∈ S with x = y and for any n > m ( x, y ) , d n ( x, y ) ≤ Cλ − m ( x,y )0 . Proof.
For simplicity of notation, let m = m ( x, y ). With the notationof Lemma 6.10, let A ( k ) = 2 k +1 λ + c ′ λ k (1 + 2 /λ + · · · + (2 /λ ) k − ) , for all non-negative integers k , and let A ( k ) = 2 for all negative integers k . Observe that A ( k ) = 2 A ( k − c ′ λ k for k >
0, and A ( k ) ≥ A ( k − k . We will first show that there exists an n -tile chain joining x and y of length at most A ( n − m ).By the definition of m , m > m − X m − and Y m − containing x and y respectively. If n < m , thenthere exists an n -tile chain of length A ( n − m ) = 2 joining x and y .If n = m , then since an ( n − λ n -tiles, theunion of all n -tiles in X m − and Y m − forms an n -tile chain joining x and y of length A (0) = 2 λ .Hence we may assume that n > m . We will argue by inductionon n − m . Fix m -tiles X m ⊆ X m − and Y m ⊆ Y m − containing x and y respectively. By Lemma 6.10, there exists an n -tile chain P of length at most c ′ λ n − m joining X m and Y m . Let x ′ = x and y ′ = y bepoints in the first and last n -tiles of the chain P respectively. Thenany m -tile containing x ′ also contains the first n -tile in P , and hencehas non-empty intersection with X m . Therefore, m ( x, x ′ ) > m and byinduction there exists an n -tile chain P x joining x and x ′ of length atmost A ( n − m ( x, x ′ )) ≤ A ( n − m − n -tilechain P y joining y and y ′ of length at most A ( n − m − P x , P y and P is an n -tile chain joining x and y oflength at most 2 A ( n − m −
1) + c ′ λ n − m = A ( n − m ) . Finally, for k >
0, since λ >
2, we have A ( k ) < k +1 λ + c ′ λ k / (1 − /λ ) < λ k [4 + c ′ λ / ( λ − . Hence the result follows by setting C = 4 + c ′ λ / ( λ − (cid:3) Lemma 6.12.
Assume that ( ∗∗ ) holds. For any x, y ∈ S with x = y ,and for any n > m ( x, y ) , we have d n ( x, y ) ≥ cλ − m ( x,y )0 , where c > is the same constant as in ( ∗∗ ) .Proof. Let m = m ( x, y ), and let X m and Y m be disjoint m -tiles con-taining x and y respectively. The length of any n -tile chain joining X m and Y m is at least D n − m . Hence, we have that d n ( x, y ) ≥ D n − m λ − n ≥ cλ n − m λ − n = cλ − m = cλ − m ( x,y )0 . (cid:3) Proposition 6.13.
Let f : S → S be an expanding Thurston mapwith no periodic critical points. Assume there exists c > such that D n = D n ( f, C ) ≥ c (deg f ) n/ for all n > , where C is a Jordancurve containing post ( f ) . Then there exists a visual metric with Λ =(deg f ) / as the expansion factor. See Definition 3.10, for the definition of a visual metric.
Proof.
By Theorem 3.2, for some n >
0, there exists a Jordan curve C containing post( f ) that is invariant under f n . Proposition 8.8 (v) in[BM] states that a metric is a visual metric for f n if and only if it is avisual metric for f . Hence, we may assume that there exists a Jordancurve C that is invariant under f . Since we can pass to an iterate of f ,we may assume that λ = (deg f ) / > . Let d = lim sup n →∞ d n , where d n is defined in equation (13). We will show that d is a visualmetric on S with expansion factor Λ . ATT`ES MAPS AND COMBINATORIAL EXPANSION 35
Fix x, y ∈ S such that x = y . By Lemma 6.11, d ( x, y ) = lim sup n →∞ d n ( x, y ) ≤ Cλ − m ( x,y )0 , where C > f . By Lemma 6.12, d ( x, y ) = lim sup n →∞ d n ( x, y ) ≥ λ − m ( x,y )0 . In addition, the function d is a metric since d n is metric on S for all n >
0. Therefore, the function d is a visual metric on S with expansionfactor Λ = (deg f ) / . (cid:3) The Sufficiency of the Conditions
In this section, we show that under the conditions in Theorem 8.1, theexpanding Thurston map f is topologically conjugate to a Latt`es map.For the next definition, we use the notion of continuum , which is acompact connected set consisting of more than one point. Definition 7.1.
A metric space (
X, d ) is called linearly locally con-nected (denoted
LLC ) if there exists some λ > (LLC1): If B ( a, r ) is a ball in X and x, y ∈ B ( a, r ) and x = y ,then there exists a continuum E ⊆ B ( a, λr ) containing x and y ; (LLC2): If B ( a, r ) is a ball in X and x, y ∈ X \ B ( a, r ) and x = y ,then there exists a continuum E ⊆ X \ B ( a, r/λ ) containing x and y .A metric space X is called Ahlfors Q -regular for Q >
0, if there existsa Borel measure µ and a constant C ≥ x ∈ X and0 < r ≤ diam( X ), 1 C r Q ≤ µ ( B ( x, r )) ≤ Cr Q , Two metric space (
X, d X ) and ( Y, d Y ) are quasisymmetriclly equivalent if there are homeomorphisms f : X → Y and η : [0 , ∞ ) → [0 , ∞ ) suchthat for all x, y, z ∈ X with x = z , we have d Y ( f ( x ) , f ( y )) d Y ( f ( x ) , f ( z )) ≤ η (cid:18) d X ( x, y ) d X ( x, z ) (cid:19) . We have a natural metric on b C = C ∪ {∞} by stereographic projec-tion, called the chordal metric , defined by δ ( z, w ) = 2 | z − w | p | z | p | w | ,δ ( z, ∞ ) = δ ( ∞ , z ) = 2 p | z | and δ ( ∞ , ∞ ) = 0for z, w ∈ C . Proposition 7.2.
If we let d be a visual metric that we get under theassumption of Proposition 6.13, then ( S , d ) is Ahlfors -regular andquasisymmetrically equivalent to the Riemann sphere b C .Proof. Proposition 19.10 in [BM] states that for an expanding Thurstonmap f : S → S without periodic critical points, if d is a visual metricwith expansion factor Λ, then ( S , d ) is Ahlfors Q -regular with Q = log(deg( f ))log Λ . Since our Λ = deg( f ) / , the metric space ( S , d ) is Ahlfors 2-regular.Proposition 16.3 (iii) in [BM] states that S , with a visual metric d for f ,is linearly locally connected. Now our proposition follows immediatelyfrom Theorem 1.1 in [BK], which states that for a metric space X homeomorphic to S , if X is linearly locally connected and Ahlfors2-regular, then X is quasisymmetrically equivalent to the Riemannsphere b C . (cid:3) Theorem 1.7 in [BM] states that:
Theorem 7.3 (Bonk-Meyer 2010) . For an expanding Thurston mapwith visual metric d , ( S , d ) is quasisymmetrically equivalent to the Rie-mann sphere b C if and only if f is topologically conjugate to a rationalmap. By this theorem and Proposition 7.2, there exists a rational map R : S → S and a homeomorphism φ such that φ ◦ f = R ◦ φ . See thediagram below: S φ −−−→ b C y f y R S φ −−−→ b C Since φ is a homeomorphism, deg( R ) = deg( f ). The Jordan curve C ′ = φ ( C ) contains all the post-critical points of R , where C ⊂ S isa Jordan curve containing post( f ). Also, X is an n -tile in the celldecomposition induced by ( f, C ) if and only if φ ( X ) is an n -tile inthe cell decomposition induced by ( R, C ′ ). So the minimal numbersof n -tiles needed to join opposite sides of the Jordan curves C and C ′ are the same, i.e., D n ( R, C ′ ) = D n ( f, C ). By Proposition 6.13 andProposition 7.2, there exists a visual metric d R on b C with expansionfactor Λ = (deg( R )) / and ( b C , d R ) is Ahlfors 2-regular. In addition,with the metric d R ( φ ( x ) , φ ( y )) = d ( x, y ) , ATT`ES MAPS AND COMBINATORIAL EXPANSION 37 φ is an isometry since m f, C ( x, y ) = m R, C ′ ( φ ( x ) , φ ( y )). Hence, we havethe following: Proposition 7.4.
Let f : S → S be an expanding Thurston mapwith no periodic critical points. If there exists c > such that D n ≥ c (deg f ) n/ for all n > , then f is topologically conjugate to a rationalmap R : b C → b C . In addition, there is a visual metric d on b C for R withexpansion factor Λ = (deg( R )) / such that ( b C , d ) is Ahlfors 2-regular. Corollary 18.4 in [BM] says that:
Lemma 7.5. If d is a visual metric for an expanding rational Thurstonmap R , then the identity map id : ( S , d ) → ( S , δ ) is a quasisymmetry,where δ is the chordal metric. To state our next lemma, let us recall some definitions on metricspaces. We refer the reader to [HK] for more details. Given a real-valued function u on a metric space X , a Borel function ρ : X → [0 , ∞ ]is said to be an upper gradient of u if | u ( x ) − u ( y ) | ≤ Z γ ρ ds for each rectifiable curve γ joining x and y in X . If u is a smoothfunction on R n , then its gradient |∇ u | is an upper gradient. We saythat a metric space X equipped with a (Borel) measure µ admits a(1 , p ) -Poincar´e inequality for p ≥
1, if there are constants 0 < λ ≤ C ≥ B in X , for all bounded continuousfunctions u on B , and for all upper gradients ρ of u on B , we have that1 µ ( λB ) Z λB | u − u λB | dµ ≤ C (diam B ) (cid:18) µ ( B ) Z B ρ p dµ (cid:19) /p , where λB is a scaling of the ball B by λ and u λB = 1 µ ( λB ) Z λB u dµ. Corollary 7.13 in [HK] states that:
Theorem 7.6 (Heinonen-Koskela 1998) . Let X and Y be two locallycompact Q -regular spaces, where X satisfies a (1 , p ) -Poincar´e inequalityfor p < Q . If g is a quasisymmetric map from X to Y , then g andits inverse are absolutely continuous with respect to the Hausdorff Q -measure (of each individual space). To formulate the next theorem, we call a metric space X linearlylocally contractible if there is a C ≥ x ∈ X and R < C − diam( X ), the ball B ( x, R ) can be contracted to a point in B ( x, CR ). Theorem 6.11 in [HK] states that: Theorem 7.7 (Heinonen-Koskela 1998) . Let X be a connected and n -regular metric space that is also an orientable n -manifold, with n ≥ .If X is linearly locally contractible, then X admits a (1 , p ) -Poincar´einequality for all p ≥ . By the previous theorem, the Riemann sphere with the chordal met-ric satisfies a (1 , p )-Poincar´e inequality.
Theorem 7.8.
Let f : S → S be an expanding Thurston map with noperiodic critical points. If there exists c > such that D n ≥ c (deg f ) n/ for all n > , then f is topologically conjugate to a Latt`es map.Proof. By Proposition 7.4, there exists a rational function R conjugateto f , and R has a visual metric d with expansion factor Λ = deg( f ) / such that ( S , d ) is Ahlfors 2-regular. Applying Lemma 7.5 to the ra-tional map R , the identity map id : ( b C , d ) → ( b C , δ ) is a quasisymmetry,where δ is the chordal metric.The standard Riemann sphere with chordal metric ( b C , δ ) satisfies a(1 , p = 1 and Q = 2 by Theorem 7.7. ByLemma 7.6, the (normalized) Hausdorff measure H d of the metric d andthe (normalized) Hausdorff measure H δ of the metric δ are mutuallyabsolutely continuous with each other. This implies that a set E ⊂ S has full measure under H δ if and only if E has full measure under H d .The dimension of the Lebesgue measure ∆ (i.e., the normalizedspherical measure of b C ) with respect to the metric d isdim(∆ , d ) = inf { dim H d ( E ) : ∆( E ) = 1 } = inf { dim H d ( E ) : H δ ( E ) = 1 } = inf { dim H d ( E ) : H d ( E ) = 1 } = 2 . Theorem 3.12 says that the dimension dim(∆ , d ) of the Lebesgue mea-sure ∆ with respect to the metric d is equal to 2 if and only if R is aLatt`es map. Hence, R is a Latt`es map and f is topologically conjugateto a Latt`es map. (cid:3) Example 7.9.
Recall the Latt`es-type maps f in Example 4.3 and g inExample 4.4. Let Jordan curves C and C ′ be the same as described inExample 5.2. Then D n ( f, C ) = 2 n = deg( f ) n/ and D n ( g, C ′ ) = 2 n < n/ = deg( g ) n/ for all n >
0. By Theorem 7.8, the map f is topologically conjugate toa Latt`es map while g is not.In the proof of Theorem 7.8, we also proved that Corollary 7.10.
Let f : S → S be an expanding Thurston map withno periodic critical points. If there exists a visual metric on f with ATT`ES MAPS AND COMBINATORIAL EXPANSION 39 expansion factor
Λ = deg( f ) / , then f is topologically conjugate to aLatt`es map. Conclusion
We get the following topological characterization of Latt`es maps:
Theorem 8.1.
A map f : S → S is topologically conjugate to a Latt`esmap if and only if the following conditions hold: • f is an expanding Thurston map; • f has no periodic critical points; • there exists c > , such that D n ≥ c (deg f ) n/ for all n > .Proof. Since all three conditions are preserved under topological con-jugacy, we only need to check them for Latt`es maps. If f is a Latt`esmap, then f is an expanding Thurston map without periodic criticalpoints. In addition, by Proposition 5.11, there exists c >
0, such that D n ≥ c (deg f ) n/ for all n > (cid:3) Corollary 8.2.
A map f : S → S is topologically conjugate to aLatt`es map if and only if the followings conditions hold: • f is an expanding Thurston map; • f has no periodic critical points; • there exists a visual metric on S with respect to f with expan-sion factor Λ = deg( f ) / .Proof. The sufficiency of these conditions follows directly from Corol-lary 7.10.If f is topologically conjugate to a Latt`es map, then f satisfies thethree conditions in Theorem 8.1. By Proposition 6.13, there exists avisual metric on S with expansion factor Λ = deg( f ) / . (cid:3) ATT`ES MAPS AND COMBINATORIAL EXPANSION 41
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