On the pullback relation on curves induced by a Thurston map
OOn the pullback relation on curves induced by aThurston map
Kevin M. PilgrimIndiana University, Bloomington [email protected]
February 25, 2021
Abstract
Via taking connected components of preimages, a Thurston map f : ( S , P f ) → ( S , P f )induces a pullback relation on the set of isotopy classes of curves in the complement of itspostcritical set P f . We survey known results about the dynamics of this relation, and posesome questions. An orientation-preserving branched covering f : S → S of degree at least two is a Thurstonmap if its postcritical set P f = ∪ n> f n ( C f ) is finite, where C f is the finite set of branch(critical) points at which f fails to be locally injective.A fundamental theorem in complex dynamics–Thurston’s Characterization and RigidityTheorem [DH]–asserts that apart from a well-known and ubiquitous set of counterexamples,the dynamics of rational Thurston maps is determined, up to holomorphic conjugacy, by itsconjugacy-up-to-isotopy-relative-to- P f class.Suppose P ⊂ S is finite. The set of isotopy classes relative to P of Thurston maps f forwhich P f = P admits the structure of a countable semigroup under composition; we denotethis by BrMod( S , P ). Pre- and post-composition with homeomorphisms fixing P gives thissemigroup the additional structure of a biset over the mapping class group Mod( S , P ). Inthis way, BrMod( S , P ) may be fruitfully thought of as a generalization of the mapping classgroup. This perspective is useful in developing intuition for the range of potential behavior ofand structure theory for Thurston maps.The mapping class group of a surface acts naturally on the countably infinite set of isotopyclasses of curves on the surface. Even better, it acts on the associated curve complex; see[Har]. It is natural to try to do something similar for Thurston maps. Since the set P f containsthe branch values of f , the restriction f : S − f − ( P f ) → S − P f is a covering map. Itfollows that a component (cid:101) γ of the inverse image f − ( γ ) of a simple closed curve γ in S − P f is a simple closed curve in S − f − ( P f ). Since P f is forward-invariant, we have an inclusion S − f − ( P f ) (cid:44) → S − P f , so the curve (cid:101) γ is again a simple closed curve in S − P f . Abusingterminology, we’ll call (cid:101) γ a preimage of γ , or sometimes say γ lifts , or pulls back , to (cid:101) γ . Bylifting isotopies, we obtain a pullback relation f ← on the set of such simple closed curves C upto isotopy. The curve γ might have several preimages, so we obtain an induced relation insteadof a function. A preimage of an inessential curve is again inessential. Similarly, a preimage of a r X i v : . [ m a t h . D S ] F e b peripheral curve–one which is isotopic into any small neighborhood of a single point in P f –iseither again peripheral, or is inessential. We call inessential and peripheral curves trivial , andnote that the set of trivial curves is invariant under the pullback relation.When P f = 4, the pullback relation induces–almost–a function on the set of nontrivialcurves. On the one hand, distinct nontrivial curves in this case must intersect. On the otherhand, distinct components of f − ( γ ) are in general disjoint. It follows that there can be at mostone class of nontrivial preimage, and we almost get a function in this case. Why “almost”?Typical examples have the property that for some curve, each of its preimages are trivial. Sowhile the mapping class group acts naturally on e.g. the infinite diameter curve complex, itis less clear how to construct a nice complex related to curves on which a Thurston map actsvia pullback. This relative lack of preserved structure makes answering even basic questionschallenging.This survey presents some known results about the dynamical behavior of taking iteratedpreimages of curves under a given Thurston map. It assumes the basic vocabulary relatedto Thurston maps from [DH], and the reader may find [KPS] also useful for more detailedexplanations and references to some terminology encountered along the way.Here are some highlights, to convince you that this is interesting. When f ( z ) = z + c isthe so-called Douady Rabbit polynomial, where c is chosen so that (cid:61) ( c ) > f ( z ) = z + i , every curve pulls back eventually to a trivial curve. See[Pil3]. In these two cases, we see that there is a finite global attractor for the pullback relation.Among obstructed Thurston maps, though, it is easy to manufacture examples with wanderingcurves and infinitely many fixed curves. A basic conjecture is Conjecture 1.1 If f is rational and not a flexible Latt`es example, then the pullback relationon curves has a finite global attractor. Here, informally, is the source of the tension. Represent a curve class by the unique geodesic γ in the complement of P f equipped with its hyperbolic metric. Pulling back and lifting themetric to (cid:98) C − f − P f , the lifted curve (cid:101) γ may unwind and become up to deg( f ) times as long as γ . But when including the curve (cid:101) γ back into (cid:98) C − P f , the Schwarz lemma implies the length of (cid:101) γ shrinks. It is unclear which force–lengthening or shortening–dominates in the long run. Andsince there exist expanding non-rational examples with wandering curves, the exact mechanismthat would imply the conjecture remains mysterious. Acknowledgements
This work was partially supported by Simons Foundation collaboration grants 4429419 and615022.
Throughout, f denotes a Thurston map, P its postcritical set, and d its degree. To avoidrepeated mention of special cases, unless otherwise stated, f has hyperbolic orbifold and P ≥
4. We denote by • (cid:39) the equivalence relation of isotopy-through-Thurston-maps-with-fixed-postcritical-set P ; • C , the countably infinite set of istopy classes of unoriented, essential, simple, nonperipheralcurves in S − P (we will often call such elements simply “curves”, abusing terminology);on C we have the geometric intersection number ι ( α, β ) which counts the minimum numberof intersection points among representatives; o , the union of the P + 1 isotopy classes of unoriented, simple, closed, peripheral andinessential curves in S − P , i.e. the trivial ones; • C := C ∪ { o } ; • f ← , the pullback relation on C induced by γ (cid:55)→ δ ⊂ f − ( γ ), where [ γ ] ∈ C and δ is acomponent of f − ( γ ); • A and A , the set of curves contained in cycles of f ← in C and C , respectively; • W ⊂ C , the set of “wandering” curves γ , namely, those for which there is an infinitesequence γ n , n ≥
0, of distinct nontrivial curves satisfying γ n f ← γ n +1 , n ≥ • the relation f ← has a finite global attractor if W is empty and A is finite; • Teich( S , P ), the Teichm¨uller space of the sphere marked at the set P ; • σ f : Teich( S , P ) → Teich( S , P ), the holomorphic self-map obtained by pulling backcomplex structures; it is the lift to the universal cover of an algebraic correspondence onmoduli space X ◦ Y − , where Y is a finite cover and X is holomorphic. See Figure 1. Teich( S , P ) π (cid:15) (cid:15) σ f (cid:47) (cid:47) ω f (cid:37) (cid:37) Teich( S , P ) π (cid:15) (cid:15) W fY f (cid:121) (cid:121) X f (cid:37) (cid:37) M P M P Figure 1:
The fundamental diagram. f ← . A Thurston map f : ( S , P ) → ( S , P ) may factor as a composition of maps of pairs( S , P ) f → ( S , C ) f → ( S , P )where each f i is admissible in the sense that its set of branch values is contained in thedistinguished subset appearing in its codomain. This motivates studying properties of so-called admissible branched covers f : ( S , A ) → ( S , B ) where domain and codomain are nolonger identified; this perspective was introduced by S. Koch. Instead of a pullback self-relationon curves C , we have a pullback relation C B f ← C A from classes of curves in S − B to classesin S − A . Thinking non-dynamically first, we have the following known results about the pullback relation f ← . . When A = B , each nonempty fiber is dense in the Thurston boundary; in particular,each nontrivial fiber is infinite [KPS]. Here, by the fiber over β , we mean { α : α f ← β } .2. The relation f ← can be trivial in the sense that the only pairs are of the form γ f ← δ where δ is trivial. Equivalently, σ f is constant. See [KPS], correcting an argument appearingoriginally in [BEKP].3. The relation f ← satisfies a Lipschitz-type inequality related to intersection numbers: ι ( (cid:101) α, (cid:101) β ) ≤ d · ι ( α, β ) whenever α f ← (cid:101) α, β f ← (cid:101) β .The study of the interaction between intersection numbers and the geometry of σ f seemsto be just beginning. Implicit use of such interactions appears in the analysis of thefamily of so-called Nearly Euclidean Thurston (NET) maps by W. Floyd, W. Parry, andthis author; see [CFPP], [FPP]. Parry develops this intersection theory further in [Par].A careful study is also applied in [MBI] to give new methods for analyzing the effect ofcertain surgery operations on possible obstructions.4. Proper multicurves are in natural bijective correspondence with boundary strata in theaugmented Teichm¨uller space, which by a theorem of Masur is known to be the comple-tion of Teichm¨uller space in the Weil-Petersson metric [Mas]. A result of Selinger [Sel]shows that σ f : Teich( S , A ) → Teich( S , B ) extends to the this completion, sending thestratum corresponding to a multicurve Γ to the stratum corresponding to the multicurve f − (Γ). It follows that analytical tools for studying σ f can be used to study propertiesof the combinatorial relation f ← [Pil3], [KPS]. There is thus a rich interplay between theanalytic and algebro-geometric properties of the correspondence on moduli space, andthe combinatorial properties of the pullback relation; see recent work of R. Ramadas, forexample.5. Associated to a proper multicurve Γ is the free abelian group Tw(Γ) of products ofpowers of Dehn twists about the curves in Γ. The pullback function can be encodedusing the associated induced virtual endomorphism on the mapping class group φ f :Mod( S , P f ) (cid:57)(cid:57)(cid:75) Mod( S , P f ) induced by lifting. It follows that group-theoretic toolscan also be used to study properties of the pullback relation on curves; see [Pil3] and[KL]. Question 3.1
If the pullback relation f ← is not trivial, must it be surjective? It seems very likely that the answer is no, for the following reason. The Composition Trick,introduced in the next subsection, should allow one to build examples where the image of σ f has positive dimension and codimension, so that its image misses many strata. f ← There seem to be three or four mechanisms via which f ← can be trivial.1. Composition trick.
The map f : ( S , A ) → ( S , B ) may factor through ( S , C ) with C = 3 (C. McMullen, [BEKP]). Such maps have the property that σ f and f ← are trivial.2. NET maps.
A. Saenz [Mal] found an example of a Thurston map f for which σ f isconstant but for which f does not decompose as in the Composition Trick. Here is hisexample, from a different point of view.Let E be an elliptic curve over C . Regarded as an abelian group, there are 8 distinctpoints of order 3; under the involution z (cid:55)→ − z these 8 points descend to a set A of 4points on P = E/ ± E varies. Now ake E to be the square torus and let f : P → P be the degree 9 flexible Latt`es mapinduced by the tripling map on E , and let B = f ( A ). As E varies, the conformal shapeof B varies, but that of A does not. Thus σ f is constant and so f ← is trivial. One cansee this triviality directly by observing that the action of PSL ( Z ) is transitive on curves(since it acts transitively on extended rationals regarded as slopes), that A is invariantunder this action (since points of order 3 are invariant under the induced group-theoreticautomorphisms), and that the horizontal curve has all preimages inessential or peripheral(as drawing a single easy picture shows).3. Sporadic examples.
Let f be the unique (up to pre- and post-composition by indepen-dent automorphisms) degree four rational map with three double critical points ( c , c , c )mapping to necessarily distinct critical values ( v , v , v ). Choose w a point distinct fromthe v i ’s, let B = { v , v , v , w } and A = R − ( w ) = { z , z , z , z } . Then the j -invariant(obtained from the cross-ratio by applying a certain degree six rational function) of the z i ’s is constant in w , whence σ f is constant and so f ← is trivial. To see this, note that as w approaches some v i , three of the z i ’s converge to c i , and the remaining one convergeto some other point, call it c (cid:48) i . Normalizing so c i = 0 and c (cid:48) i = ∞ and scaling via mul-tiplication with a nonzero complex constant shows that the conformal shape of the fiber R − ( w ) converges to that of the cube roots of unity together with the point at infinity.Thus the j -invariant of R − ( w ) is a bounded holomorphic function, hence constant.4. Combinations of the above.Question 3.2
Do there exist examples f with σ f constant and deg( f ) prime? Cui G. has thought about the general case, see [Cui]. It is natural to look for the simplestsuch examples. By cutting along a maximal multicurve, one may restrict to the case A = B = 4; let’s call these “minimal”. It is natural to look for examples which do not factor asin the Composition Trick; let’s call these “primitive”. Question 3.3
What are the minimal primitive branched covers f : ( S , A ) → ( S , B ) forwhich f ← is trivial? f ← Though the set of curves C is complicated, it is conveniently described by a variety of coordinatesystems. For example, Dehn-Thurston coordinates record intersection numbers of curves withedges in a fixed triangulation with vertex set P [FLP]. Train tracks and measured foliationsgive other methods. But expressing the pullback relation in these coordinates can be verycomplicated. The effect on Dehn-Thurston coordinates of pulling back from S − P f to S − f − ( P f ) is indeed easy to compute, since one can just lift intersection numbers. But theexpression for the induced “erasing map” is hard to write down in closed form and leads tocontinued-fraction-like cases.When P f = 4, though, the set of curves C can be encoded by “slopes” in the extendedrationals Q ∪ { / } , the pullback relation f ← is a almost a function, and things are a bit easierbut are still quite complicated.A Thurston map is an NET map if P f = 4 and each critical point has local degree two;see [CFPP]. If f is an NET map, there is an algorithm that computes the image of a slopeunder pullback. This can be done easily by hand, and has been implemented. NET maps canbe easily encoded by combinatorial input. W. Parry has written a computer program that im-plements this algorithm. The website http://intranet.math.vt.edu/netmaps/ , maintained y W. Floyd, contains a database of tens of thousands of examples. For NET maps, it ap-pears that this ability to calculate the pullback map on curves (and related invariants, such asthe degree by which preimages map, and how many preimages there are) leads to an effectivealgorithm for determining whether a given example is, or is not, equivalent to a rational map.When P f = 4 and f is the subdivision map of a subdivision rule of the square pillowcase(like in Figure ?? ), W. Parry has written a program for computing the image of a slope underpullback (personal communication). Question 3.4
Are there any interesting settings in which one can effectively compute f ← when P f ≥ ? The example studied by Lodge [Lod] is, nondynamically speaking, the unique generic cubic, inthe following sense. We have f : ( S , A ) → ( S , B ) where A is the set of four simple criticalpoints and B = f ( A ). Each nontrivial curve has exactly one nontrivial curve in its preimage. Question 3.5
Suppose f : ( S , A ) → ( S , B ) has the property that A consists of d − simplecritical points, B = f ( A ) , and f | B is injective, so that B consists of the d − critical values.Does each nontrivial curve in S − B have a nontrivial preimage? Up to pre- and post-composition with homeomorphisms there is a unique such map [BE].If each γ has a nontrivial preimage, so does h ( γ ), where h : ( S , P f ) → ( S , P f ) is a homeo-morphism that lifts under f . Since the set of such h is a finite-index subgroup of Mod( S , P f ),checking this is a finite computation.In the case of four postcritical points, we have the following result from [FPP]. Theorem 3.6 If P = 4 , the pullback function on curves is either surjective, or trivial. The key insight: when P = 4, looking at the correspondence on moduli space, the space W f a Riemann surface whose set of ends consists of finitely many cusps, the map X : W f → M P is holomorphic, and M P is the triply-punctured sphere; thus if X is nonconstant, then eachcusp of M P is the image of cusp of W f . We first discuss in general some examples and known results about the possible dynamicalbehavior of f ← .1. Example:
Every curve iterates to the trivial curve.
This happens for z + i . Here is oneway to see this. Examining the possibilities for how the bounded region enclosed by acurve meets the finite postcritical set { i, i − , − i } , one sees that a curve must eventuallybecome trivial unless it surrounds both − i and i −
1. For this type of curve α , there isat most one nontrivial curve β with α f ← β and β a curve of the same type. Moreover,deg( α f ← β ) = 1. Equipping the complement of the postcritical set with the hyperbolicmetric, the Schwarz Lemma shows that the length of a geodesic representative of β isstrictly shorter than that of α . Iterating this process, it follows that such a curve cannotbe periodic under f ← : points in its orbit cannot get too complicated, since otherwise theywould have to get long, so they must eventually become a different type of curve and thusbecome trivial upon further iteration. he “airplane” quadratic polynomial f ( z ) = z + c , with the origin periodic of period 3and Im( c ) = 0, is another such example [KL].2. Question 4.1
Does there exist an example of a Thurston map f for which the pullbackrelation induced by f is nontrivial but that induced by some iterate f n is trivial? Theorem. If f is rational and non-Latt`es, then there are only finitely many fixed propermulticurves for which f − Γ = Γ; see [Pil3, Thm. 1.5]. The proof uses the decompositiontheory.4.
Conjecture 1.1: If f is rational and not a flexible Latt`es example then the pullbackrelation on curves has a finite global attractor. There is partial progress on this conjecture for special families of Thurston maps.(a) Kelsey and Lodge [KL] verify this for all quadratic non-Latt`es maps with four post-critical points.(b) Hlushchanka [Hlu] verifies this for critically fixed rational maps. As he shows, eachsuch map is obtained by blowing up, in the sense of [PT], edges of a planar multigraph G . If f is a critically fixed rational map, α an edge in the planar connected multigraph G that describes f via the blowing up construction [CGN + ], γ f ← (cid:101) γ , and ι ( γ, α ) >
0, then it is easy to see that unless γ is homotopic to the boundary of a regularneighborhood of an edge-path homeomorphic to an embedded arc in G and ι ( γ, α ) =1, we must have ι ( (cid:101) γ, α ) < ι ( γ, α ). The global attractor A consists of those γ forwhich ι ( γ, α ) ≤ α ∈ G .(c) If the virtual endomorphism φ f on the mapping class group is contracting, then f ← has a finite global attractor [KPS]. Nekrashevych [Nek, Thm. 7.2] establishes thiscontraction property in the case for hyperbolic polynomials.(d) Belk, Lannier, Margalit, and Winarski [BLMW] associate to a topological polynomial f (a Thurston map with a fixed point at infinity which is fully ramified) a simplicialcomplex T whose vertices are planar trees, and a simplicial map λ f : T → T inducedby lifting. For general complex polynomials, the associated Hubbard trees are fixedvertices, and the uniquness of the Hubbard tree for iterates of f leads to a contractionproperty of λ f that implies the existence of a finite global attractor for the pullbackrelation on curves. Question 4.2
For a general Thurston map, is there an associated natural action ona contractible simplicial complex? (e) If the correspondence on moduli space (in the direction of σ f ) has a nonempty in-variant compact subset, then φ f is contracting, so there is a finite global attractor.If moduli space admits an incomplete metric which is (i) uniformly contracted by σ f ,and (ii) whose completion is homeomorphic to that of the WP metric, then the trivialcurve is a finite global attractor [KPS, Thm. 7.2]. The latter occurs for f ( z ) = z + i ;the correspondence on moduli space is the inverse of a Latt`es map with three post-critical points and Julia set the whole sphere, which expands the Euclidean orbifoldmetric.(f) Intersection theory provides some insight. Consider the linear map λ f : R [ C ] → R [ C ]defined on basis vectors by γ (cid:55)→ (cid:80) γ f ← δ d δ · δ where deg( γ f ← δ ) = d δ . Then one canshow that the Dehn-Thurston coordinates of any orbit of λ f starting at any curve γ tends to zero. For otherwise, one has either an obstruction, ruled out by rationality,or a wandering curve with a uniform lower bound in the corresponding coordinatesand hence on the corresponding moduli of path families. But if α, β are any twocurves in (cid:98) C − P then mod( α )mod( β ) (cid:46) /ι ( α, β ) and so this is impossible. .2 Bounds on the size of the attractor Since up to conjugacy there are only finitely many non-flexible Latt`es rational maps with agiven degree d and size P of postcritical set, the cardinality of the finite global attractor A ,if one exists, must be bounded by some constant depending on d and P . I know very littleabout the behavior of this function.1. Certainly A can be large if P is large: for n ≥ /n -rabbit” quadratic polynomialwill have an n -cycle of curves. Other examples can be constructed by perturbing flexibleLatt`es examples. One can find hyperbolic sets consisting of invariant curves which arestable under perturbation; a result of X. Buff and T. Gauthier [BG, Cor. 3] impliesthat such maps are limits of sequences of hyperbolic Thurston maps with the maximumnumber 2 d − A can be small (say zero), by taking e.g. examples with σ f constant. Using McMullen’s compositional trick and Belyi functions one can easily buildboth hyperbolic rational maps and rational maps with Julia set the whole sphere havingthe property that σ f constant and P f arbitrarily large.3. Results of G. Kelsey and R. Lodge [KL] show that for quadratic rational maps f with P f = 4, we have A ≤ f corresponds (not quite bijectively)to a repelling fixed-point p of a correspondence g = Y ◦ X − on moduli space. In thenonexceptional cases, this is actually a rational map g : P → P . There appears to be anatural bijection between invariant (multi)curves for f and periodic internal rays joiningpoints in periodic superattracting cycles of g (these lie at infinity in moduli space) to p .I’ve confirmed this also for critically fixed polynomials with three finite critical points. Question 4.3 If f is rational and Γ is an invariant or periodic multicurve, does thereexist a stable manifold connecting the unique fixed-point of σ f in Teich( (cid:98) C , P ) to a fixed-point or periodic point in a boundary stratum corresponding to Γ ? For quadratics with four postcritical points, the analysis of [KL] seems to confirm theintuition that periodic curves in the dynamical plane are related to accesses landing atthe associated fixed-point that are periodic under the correspondence, in this case theinverse of a critically finite rational map with three postcritical points. But in higherdegrees with P f = 4, the correspondence need not the inverse of such a map, and thesituation is more complicated; see e.g. [Lod].In higher dimensions, another first natural example to try is the case of f a critically fixedpolynomial with four finite simple critical points. One would need to show the existenceof internal rays in two complex dimensions. This example is beautifully symmetric andpossesses many invariant lines that might make the problem more tractable. See [BEK,section 3]. Maps with nontrivial symmetries provide a source of non-rational examples without a finiteglobal attractor. We denote by Mod( f ) = { h : hf (cid:39) fh rel P f } ; here (cid:39) denotes isotopy. Werecall four facts:1. The pure mapping class group P Mod( S , P ) has no elements of finite order, so neitherdoes P Mod( f ). . If f is rational, P Mod( f ) is trivial, unless f is a flexible Latt`es example, in which case itis isomorphic to the free group on two generators.3. Suppose f has an obstruction Γ = ( γ , . . . , γ m ) with the property that the correspondinglinear map f Γ : R [Γ] → R [Γ] has 1 among its eigenvalues, with a corresponding nonnega-tive integer eigenvector ( a , . . . , a m ). Let T i denote the Dehn twist about γ i . Then somepower of T a · · · T a m m gives an element of Mod( f ) [Pil2].4. Thurston maps are like mapping classes. If f is obstructed, there is a canonical decom-position by cutting along a certain invariant multicurve [Pil1], [Sel]. The “pieces” mightcontain cycles of degree one: mapping class elements, each with its own centralizer. Thefact that the decomposition is canonical means that the centralizers of the pieces willembed into Mod( f ). Using this idea one can create examples of Thurston maps with avariety of prescribed behaviors. For example, if f is the identity on some sufficiently largepiece, then clearly A contains infinitely many fixed curves. If f is a pseudo-Anosov mapon some other sufficiently large piece, then there are infinitely many distinct orbits ofwandering curves.5. L. Bartholdi and D. Dudko give an explicit example of f with Mod( f ) infinitely generated[BD2]. As motivation, recall that if L is a flexible Latt`es example with postcritical set P , then σ L acts as the identity, and the pullback relation on curves is the identity function. So if f is nowan arbitrary Thurston map with with the same postcritical set P , then f ◦ L (cid:39) L ◦ f , and σ L ◦ f = σ f and the pullback relation on curves for f and L ◦ f are the same. Question 4.4
When do two Thurston maps have the same pullback relation on curves?
The examples in § f ; see Figure 2. The “rim” of the square pillowcase–the common boundary ofthe two squares at left– is an invariant Jordan curve containing P f : that is, f is the subdivisionmap of a finite subdivision rule. This map satisfies the combinatorial expansion propertiesof both Bonk-Meyer [BM] and Cannon-Floyd-Parry [CFP]. Appealing to either one of theseworks, we conclude there is a map isotopic to this example in which the diameters of the tilesgoes to zero upon iterated subdivision. This implies that this example has no Levy cycles.Appealing to [BD1], we conclude there is such an example that is expanding with respect to acomplete length metric.The vertical curve is an obstruction with multiplier 1, and the horizontal curve is invariant.Let T be the Dehn twist about this vertical curve, so that fT = T f . This immediately implies(i) A f is infinite, since the orbit of the horizontal curve under T will consist of f -invariantcurves, and (ii) if we put g = T f , then the horizontal curve wanders. To see this, note that g − n ( γ ) = ( T f ) − n ( γ ) = T − n f − n ( γ ) = T − n γ as required. I do not know if T f is isotopic to anexpanding map. Letting L denote the flexible Latt`es map induced by doubling on the torus,however, I would expect that T fL N is expanding for sufficiently large N . This should showthe existence of expanding maps with wandering curves. f goes in the opposite direction to the indicated arrow and defines a cellulardegree 5 map from the pillowcase to itself. The four corners of the pillowcase form the postcriticalset. Figure by W. Floyd.
Question 4.5
Suppose
Mod( f ) is trivial. Could there exist infinitely many periodic curves?Could there exist wandering curves? References [BD1] Laurent Bartholdi and Dzmitry Dudko. Algorithmic aspects of branched coveringsIV/V. Expanding maps.
Trans. Amer. Math. Soc. (2018), 7679–7714.[BD2] Laurent Bartholdi and Dzmitry Dudko. Algorithmic aspects of branched coveringsII/V: sphere bisets and their decompositions. https://arxiv.org/pdf/1603.04059.pdf .[BLMW] James Belk, Justin Lanier, Dan Margalit, and Rebecca R. Winarski. Recognizingtopological polynomials by lifting trees, https://arxiv.org/pdf/1906.07680.pdf ,06 2019.[BE] Israel Berstein and Allan L. Edmonds. On the construction of branched coverings oflow-dimensional manifolds.
Trans. Amer. Math. Soc. (1979), 87–124.[BM] Mario Bonk and Daniel Meyer.
Expanding Thurston maps , volume 225 of
Mathe-matical Surveys and Monographs . American Mathematical Society, Providence, RI,2017.[BEKP] Xavier Buff, Adam Epstein, Sarah Koch, and Kevin Pilgrim. On Thurston’s pullbackmap. In
Complex dynamics , pages 561–583. A K Peters, Wellesley, MA, 2009. BEK] Xavier Buff, Adam L. Epstein, and Sarah Koch. B¨ottcher coordinates.
Indiana Univ.Math. J. (2012), 1765–1799.[BG] Xavier Buff and Thomas Gauthier. Perturbations of flexible Latt`es maps. Bull. Soc.Math. France (2013), 603–614.[CFP] J. W. Cannon, W. J. Floyd, and W. R. Parry. Finite subdivision rules.
Conform.Geom. Dyn. (2001), 153–196 (electronic).[CFPP] J. W. Cannon, W. J. Floyd, W. R. Parry, and K. M. Pilgrim. Nearly EuclideanThurston maps. Conform. Geom. Dyn. (2012), 209–255.[CGN + ] Kristin Cordwell, Selina Gilbertson, Nicholas Nuechterlein, Kevin M. Pilgrim, andSamantha Pinella. On the classification of critically fixed rational maps. Conform.Geom. Dyn. (2015), 51–94.[Cui] Guizhen Cui. Rational maps with (constant) pullback map. .[DH] A. Douady and John Hubbard. A Proof of Thurston’s Topological Characterizationof Rational Functions. Acta. Math. (1993), 263–297.[FLP] Albert Fathi, Fran¸cois Laudenbach, and Valentin Po´enaru.
Thurston’s work on sur-faces , volume 48 of
Mathematical Notes . Princeton University Press, Princeton, NJ,2012. Translated from the 1979 French original by Djun M. Kim and Dan Margalit.[FPP] William Floyd, Walter Parry, and Kevin M. Pilgrim. Rationality is decidable fornearly Euclidean Thurston maps. to appear, Geom. Dedicata (2020).[Har] W. J. Harvey. Boundary structure of the modular group. In
Riemann surfaces andrelated topics: Proceedings of the 1978 Stony Brook Conference (State Univ. NewYork, Stony Brook, N.Y., 1978) , volume 97 of
Ann. of Math. Stud. , pages 245–251.Princeton Univ. Press, Princeton, N.J., 1981.[Hlu] Mikhail Hlushchanka. Tischler graphs of critically fixed rational maps and theirapplications, https://arxiv.org/pdf/1904.04759.pdf , 04 2019.[KL] Gregory Kelsey and Russell Lodge. Quadratic Thurston maps with few postcriticalpoints, https://arxiv.org/abs/1704.03929 . (04 2017).[KPS] Sarah Koch, Kevin M. Pilgrim, and Nikita Selinger. Pullback invariants of Thurstonmaps.
Trans. Amer. Math. Soc. (2016), 4621–4655.[Lod] Russell Lodge. Boundary values of the Thurston pullback map.
Conform. Geom.Dyn. (2013), 77–118.[MBI] M. Hlushchanka M. Bonk and A. Iseli. Thurston maps and the dynamics of curves.research announcement, Quasiworld seminar, 2020.[Mal] Edgar Arturo Saenz Maldonado. On Nearly Euclidean Thurston Maps. PhD Thesis,Virginia Polytechnic University (2012).[Mas] Howard Masur. Extension of the Weil-Petersson metric to the boundary of Teich-muller space.
Duke Math. J. (1976), 623–635.[Nek] Volodymyr Nekrashevych. Combinatorial models of expanding dynamical systems. Ergodic Theory Dynam. Systems (2014), 938–985.[Par] Walter Parry. NET map slope functions. September 2018.[Pil1] Kevin M. Pilgrim. Canonical Thurston obstructions. Adv. Math. (2001), 154–168.[Pil2] Kevin M. Pilgrim.
Combinations of complex dynamical systems , volume 1827 of
Lecture Notes in Mathematics . Springer-Verlag, Berlin, 2003. Pil3] Kevin M. Pilgrim. An algebraic formulation of Thurston’s characterization of rationalfunctions.
Ann. Fac. Sci. Toulouse Math. (6) (2012), 1033–1068.[PT] Kevin M. Pilgrim and Lei Tan. Combining rational maps and controlling obstruc-tions. Ergodic Theory Dynam. Systems (1998), 221–245.[Sel] Nikita Selinger. Thurston’s pullback map on the augmented Teichm¨uller space andapplications. Invent. Math. (2012), 111–142.(2012), 111–142.