Invariant spanning trees for quadratic rational maps
IINVARIANT SPANNING TREES FOR QUADRATICRATIONAL MAPS
ANASTASIA SHEPELEVTSEVA AND VLADLEN TIMORIN ‹ Abstract.
We study Thurston equivalence classes of quadraticpost-critically finite branched coverings. For these maps, we intro-duce and study invariant spanning trees. We give a computationalprocedure for searching for invariant spanning trees. This proce-dure uses bisets over the fundamental group of a punctured sphere.We also introduce a new combinatorial invariant of Thurston classes— the ivy graph. Introduction
Rational maps acting on the Riemann sphere are among central ob-jects in complex dynamics.
Thurston’s characterization theorem al-lows to study these algebraic objects by topological tools. It viewsrational maps within a much wider class of topological branched cov-erings. (Branched self-coverings of the sphere whose critical pointshave finite orbits are called
Thurston maps .) There is a natural equiv-alence relation on Thurston maps such that different rational func-tions are almost never equivalent. (All exceptions are known and well-understood.) Thurston’s theorem provides a topological criterion for aThurston map being equivalent to a rational function. Thus, classifi-cation of Thurston maps up to equivalence is an important problem.This fundamental problem has applications beyond complex dynam-ics, e.g., in group theory; it is a focus of recent developments, see e.g.[BN06, BD17, CG+15, KL18, Hlu17]. We approach the problem viaanalogs of Hubbard trees for quadratic rational maps: invariant span-ning trees .We will write S for the oriented topological 2-sphere. By a graph inthe sphere, we here mean a 1-dimensional cell complex embedded into S . By vertices and edges , we mean 0-cells and 1-cells, respectively. Mathematics Subject Classification.
Primary 37F20; Secondary 37F10.
Key words and phrases.
Complex dynamics; invariant tree, iterated monodromygroup.The study has been funded within the framework of the HSE University BasicResearch Program and the Russian Academic Excellence Project ’5-100. a r X i v : . [ m a t h . D S ] A p r A. SHEPELEVTSEVA AND V. TIMORIN
For a graph G , we will write V p G q for the set of vertices of G and E p G q for the set of edges of G . A tree is a simply connected graph. Avertex x of a tree T is called a branch point if T ´ t a u has more than2 components. Suppose that P Ă S is some finite subset. A tree T in S such that P Ă V p T q is called a spanning tree for P if V p T q ´ P consists of branch points.Let f : S Ñ S be an orientation preserving branched covering ofdegree 2. The map f has two critical points c p f q and c p f q . Let v p f q and v p f q be the corresponding critical values. The post-critical set of f is defined as the smallest closed f -stable set including t v p f q , v p f qu .The post-critical set of f will be denoted by P p f q . If P p f q is finite,then f is said to be post-critically finite . Recall that a Thurston map is a post-critically finite orientation preserving branched covering. Inthis paper, we will only consider degree two Thurston maps.
Definition 1.1 (Invariant spanning tree) . Let f : S Ñ S be aThurston map. A spanning tree T for P p f q is called an invariantspanning tree for f if:(1) we have f p T q Ă T ;(2) vertices of T map to vertices of T .This notion is close to what is called “invariant trees” in [Hlu17].Note that the restriction of f to an edge of T is injective unless theedge contains a critical point of f . Consideration of invariant spanningtrees is justified by the following examples (we describe those of them,which are quadratic maps, in more detail later, see Section 2):(1) Hubbard trees [DH85, BFH92, Poi93] can be connected to infin-ity to form invariant spanning trees.(2) Invariant spanning trees can be constructed for formal matings by joining the two Hubbard trees. (However, the tree structuresometimes does not survive in the corresponding topologicalmatings).(3) Classical captures in the sense of [Wit88, Ree92] often comewith invariant spanning trees. In fact, the original approach ofWittner used invariant trees.(4) Sufficiently high iterates of expanding Thurston maps possessinvariant spanning trees by [Hlu17]. This result has also beenextended in [Hlu17] to rational maps with Sierpinski carpet Ju-lia set.(5) Extended Newton graphs constructed in [LMS15] for post-criticallyfinite Newton maps are often invariant trees. Invariant span-ning trees can be obtained from them by erasing some of thevertices.
NVARIANT SPANNING TREES 3 (6) With each critically fixed rational map f , a certain bipartitegraph is associated in [CG+15]. Every edge if this graph isinvariant, thus every spanning tree of this graph is an invarianttree for f . Again, an invariant spanning tree can be obtainedby erasing some of the vertices.(7) Invariant spiders in the sense of [HS94] are invariant spanningtrees (possibly after removal of the critical leg in case of astrictly preperiodic critical point).Section 2 deals with examples of invariant spanning trees. However,we restrict our attention to the case of degree 2 Thurston maps.Let x be a vertex of a tree T Ă S , and e be an edge of T . If x is in theclosure of e , then we say that x is incident to e . We also say that e isincident to x . The following result shows how to recover the Thurstonequivalence class of f from an invariant spanning tree of f . Recallthat a ribbon graph (also known as a fat graph, or a cyclic graph)is an abstract graph in which the edges incident to each particularvertex are cyclically ordered. By [MA41], ribbon trees are the sameas isomorphism classes of embedded trees in S . Here an isomorphismof embedded trees is an orientation preserving self-homeomorphism of S that takes one tree to another. Recall that the orientation of S isassumed to be fixed. For a spanning tree T for P p f q , we write C p T q for the set of critical points of f in T . Theorem A.
Suppose that f , g : S Ñ S are two Thurston mapsof degree 2. Let T f and T g be invariant spanning trees for f and g ,respectively. Suppose that there is a cellular homeomorphism τ : T f Ñ T g with the following properties: (1) The map τ is an isomorphism of ribbon graphs. (2) We have τ ˝ f “ g ˝ τ on V p T f q Y C p T f q . (3) The critical values of f are mapped to critical values of g by τ .Then f and g are Thurston equivalent. In other words, to know the Thurston equivalence class of f , it suf-fices to know the following data:(1) the ribbon graph structure of T f ;(2) the restriction of the map f to the set V p T f q Y C p T f q .These data are discrete and can be encoded symbolically.Note that higher degree analogs of Theorem A will require an addi-tional structure on an invariant spanning tree (something like a struc-ture of an angled tree), as is shown by the following example. A. SHEPELEVTSEVA AND V. TIMORIN
Example 1.2.
This is a slightly modified example of [Mil09, Figure21]. Consider the following invariant tree T of a cubic polynomial f . x ˝ B (cid:47) (cid:47) (cid:79) (cid:79) A x ‚ C (cid:47) (cid:47) x ‚ x ˝ (cid:79) (cid:79) D ‚ Here we have x ÞÑ x ÞÑ x ÞÑ x ý . The points x and x are simplecritical points. The corresponding critical values (marked as solid) are x and x . The edges, oriented as shown in the picture, are mapped asfollows: A ÞÑ A Y B, B ÞÑ C ÞÑ C ´ Y B ´ Y A ´ , D ÞÑ D. The inverses here indicate the change of orientation. The part of thetree T that is obtained by removing the edge D and the vertex is theHubbard tree for f . This Hubbard tree is shown as an angled tree . Forexample, the angle between B and A ´ (computed counterclockwise)is . We measure angles so that the full turn corresponds to angle 1.In order to recover the Thurston class of f by its Hubbard tree, weneed to know the angles. For example, if we change the angle between B and A ´ to , then we obtain a different polynomial and a differentThurston class. However, there is still a correspondence τ between thetwo trees that satisfies the assumptions of Theorem A.An important algebraic invariant of a Thurston map is its biset overthe fundamental group of S ´ P p f q . A biset is a convenient algebraicstructure that carries a complete information about the Thurston classof f . The (perhaps better known) iterated monodromy group of f canbe immediately recovered from the biset. A formal definition of a bisetwill be given in Section 5. For now, we just emphasize that bisets admitcompact symbolic descriptions somewhat similar to presentations ofgroups by generators and relations or presentation of linear maps bymatrices. Theorem B.
Suppose that f is a Thurston map of degree 2, and T isan invariant spanning tree for f . There is an explicit presentation ofthe biset of f based only on the following data (1) the ribbon graph structure on T , (2) the restriction of f to V p T q Y C p T q . NVARIANT SPANNING TREES 5
In Section 5.5, we make the statement of Theorem B more precise.We provide an automaton representing the biset of f in Theorem 5.6.The construction is algorithmic. Theorem B solves, in a particular case,the problem of combinatorial encoding of Thurston maps by meansof invariant graphs. Different contexts of this problem are addressedin [CFP01, BM17, LMS15] for specific families of rational maps. Arelationship between the properties of invariant trees and the propertiesof the iterated monodromy group has been studied in [Hlu17].1.1. Dynamical tree pairs.
It is not always easy to find an invariantspanning tree for a Thurston map f . However, for any spanning tree T for P p f q , it is easy to find another spanning tree T ˚ that mapsonto T . The tree T ˚ is not uniquely defined; there are several waysof choosing suitable subtrees in the graph f ´ p T q . Suppose that wewant to find an invariant spanning tree for f . It is natural to look atan iterative process, a single step of which is the transition from T to T ˚ . Such an iterative process will be described below under the nameof ivy iteration .We now proceed with a more formal exposition. Let f : S Ñ S bea Thurston map of degree two. Consider two spanning trees T ˚ and T for P p f q such that f p T ˚ q Ă T . Moreover, we assume that(1) the vertices of T ˚ are mapped to vertices of T under f ;(2) all critical values of f are vertices of T ;If these assumptions are fulfilled, then p T ˚ , T q is called a dynamicaltree pair for f . Clearly, an invariant spanning tree T for f gives riseto a dynamical tree pair p T, T q . Thus, dynamical tree pairs generalizeinvariant spanning trees. Observe also that the restriction of f to everyedge of T ˚ is injective unless the edge contains a critical point of f .A spanning tree T for P p f q gives rise to a distinguished generatingset E T of the fundamental group π p S ´ P p f q , y q with y P S ´ T .Namely, E T consists of the identity element and the homotopy classesof smooth loops based at y intersecting T only once and transversely(we will make this more precise later).In Section 5, we state a theorem (Theorem 5.6) generalizing TheoremB. It follows from Theorem 5.6 that the biset of f is determined by adynamical tree pair p T ˚ , T q . More precisely, the biset can be explicitlypresented knowing the following discrete data:(1) the ribbon graph structures on T ˚ , T ;(2) the map f : V p T ˚ q Y C p T ˚ q Ñ V p T q ; A. SHEPELEVTSEVA AND V. TIMORIN (3) how elements of E T ˚ are expressed through elements of E T (orhow both E T ˚ , E T are expressed through some other generatingset of π p S ´ P p f q , y q ).Vice versa, given T and a presentation of the biset of f in the basisassociated with T , these data can be recovered.1.2. The ivy iteration.
The principal objective of this paper is tointroduce a computational procedure for finding invariant (or, moregenerally, periodic) spanning trees of degree 2 Thurston maps.Let f : S Ñ S be a degree 2 Thurston map. Ivy iteration operateson isotopy classes (rel. P p f q ) of spanning trees for P p f q , which we call ivy objects . Let Ivy p f q denote the set of all ivy objects for f . Let T bea spanning tree, and r T s be the corresponding ivy object. A symbolicpresentation of the biset of f plus a symbolic encoding of the ribbontree structure on T give rise to several choices of a spanning tree T ˚ such that p T ˚ , T q is a dynamical tree pair for f . Roughly speaking,several choices for T ˚ are related with different ways of choosing aspanning subtree in f ´ p T q . Consider the pullback relation r T s (cid:40) r T ˚ s on Ivy p f q . It equips Ivy p f q with a structure of an abstract directedgraph. A subset C Ă Ivy p f q is said to be pullback invariant if r T s P C and r T s (cid:40) r T ˚ s imply r T ˚ s P C .With the help of a computer, we found finite pullback invariant sub-sets in Ivy p f q for several simplest quadratic Thurston maps f . Withinthese pullback invariant subsets, we found all invariant ivy objects.Invariant ivy objects, obviously, correspond to invariant (up to homo-topy) spanning trees for f . Having a picture for a pullback invariantsubset, we can also see many periodic ivy objects of various periods.Some of these examples will be described in Section 2. Note that, if wefound a spanning tree for P p f q that is f -invariant up to homotopy, thenthis tree is a genuine invariant spanning tree for some map homotopicto f . This is good enough since we are interested in classification ofThurston maps up to Thurston equivalence (in particular, homotopicmaps are in the same class). How the ivy iteration compares with known combinatorial algorithms.
The ivy iteration can be considered in the following general context.The biset associated with f is a way of compactly representing the“combinatorics” of f . Unfortunately, the same biset may have very dif-ferent presentations in different bases. (A linear algebra analog is thatthe same linear map has different matrices in different bases.) Usually, On a somewhat similar note, the pullback relation on isotopy classes of simpleclosed curves in S ´ P p f q is discussed in [Pil03, KPS16]. NVARIANT SPANNING TREES 7 a combinatorial description of f yields a presentation of its biset. How-ever, given different combinatorial descriptions of a Thurston map, theproblem is whether they describe the same thing. This problem trans-lates into comparison of bisets: do different presentations correspondto the same biset?Up to date, there are several algorithmic approaches to the com-parison of bisets. We restrict our attention to bisets associated withrational, i.e., unobstructed, Thurston maps. A natural idea is to lookfor the “best” presentation of a biset. (This idea is somewhat similarto finding a Jordan normal form — or some other normal form — of alinear map.) This general idea works well for polynomials. In fact, the combinatorial spider algorithm of Nekrashevich [Nek09] aims at solv-ing the problem. Given a presentation of a biset by a twisted kneadingautomaton , the algorithm searches for the best presentation, which isassociated with a kneading automaton . The algorithm of [Nek09] isa combinatorial implementation of the spider algorithm originally de-veloped by Hubbard and Schleicher [HS94] for quadratic polynomials.A version of the combinatorial spider algorithm was used in [BN06] toidentify twisted rabbits with the rabbit, co-rabbit, or the airplane poly-nomials, as well as to classify the twists of z ` i . In fact, the authorsconsider not only bisets over the fundamental group but also bisetsover the pure mapping class group, which turns out to be useful fordistinguishing twisted rabbits. In [BN06], Bartholdi and Nekrashevichdevelop another approach to the same problem based on inspecting thecorrespondence on the moduli space associated with the Thurston pull-back map. This second approach relies on the fact that, in examplesunder consideration, there are few (namely, four) points in P p f q (if P p f q consists of only four points, then the moduli space has complexdimension one).The second approach of [BN06] has been further developed in [KL18],where all non-Euclidean Thurston maps with 4 or fewer post-criticalpoints are classified. The authors also provide an algorithm for iden-tifying the twists of all such maps. An extension of these results toThurston maps with bigger post-critical sets is currently unavailable,not only because there are too many objects to classify but also be-cause the technique is not easy to adapt. Thus, to the best of ourknowledge, purely combinatorial tools available up to date for compar-ing bisets are restricted either to specific types of Thurston maps (say,topological polynomials, expanding maps, etc.) or to maps with fewpost-critical points.The process of finding invariant trees described in [Hlu17] is in prin-ciple algorithmic. However, it applies under additional assumptions on A. SHEPELEVTSEVA AND V. TIMORIN the dynamics of the map (sphere or Sierpinski carpet Julia set) andproduces an invariant tree only for a sufficiently high iterate of themap.On the other hand, there are also “floating-point” algorithms, seee.g. [BD17, Section V.2]. Most of these algorithms aim at turningthe Thurston iteration into an efficient computation. For example,given two bisets without obstructions, one can compare them as follows(see Corollary V.9 of [BD17]). For each of the two bisets, computethe coefficients of the corresponding rational map using a version ofThurston’s algorithm. Then the two rational maps can be comparedby comparing the corresponding coefficients. This approach works asan efficient computation but fails to provide good combinatorial tagsto rational maps.The ivy iteration may be regarded as an attempt to generalize thespider algorithm. In fact, for quadratic polynomials, the spider algo-rithm (applied to a not necessarily invariant spider of an actual poly-nomial) is the same as the ivy iteration, except that, the issue of arbi-trary choices is resolved by specifying a kneading sequence. A spideris a tree of a very specific shape; it is a star. The ivy iteration may bein principle applied to arbitrary spanning trees, and to non-polynomialThurston maps. In this sense it is more general. However, it is notas good as the spider algorithm because it is not an algorithm at all.An important ingredient (an analog of a kneading sequence that wouldallow to make specific choices) is missing so far. On the positive side,the ivy iteration can be implemented as a purely combinatorial pro-cedure. At any step of the iteration, we obtain a presentation of thebiset associated with a Thurston map. If the iteration converges, thenwe obtain a good presentation, and the hope is that there are only fewgood ones. Moreover, the result is a nice visual tag associated witha rational map. To the best of our knowledge, the ivy iteration doesnot coincide with other known computational procedures, although itis conceptually unsophisticated and is based on the same general idea:that of taking pullbacks.
Terminological conventions.
In this paper, we talk about graphsin the sphere as well as abstract graphs. The former notion belongsto topology, and the latter — to combinatorics. We try to clearlydistinguish these notions. A graph (without a specification) usuallymeans a graph in the sphere. When referring to an abstract graph, wealways say “abstract”. We sometimes consider oriented edges of graphsin the sphere. These are edges, for which some orientation (=direction)is specified. On the other hand, we talk of directed edges in abstract
NVARIANT SPANNING TREES 9 directed graphs. In this sense, a directed edge is a fundamental notion,which can be defined as an ordered pair of vertices. It is not “anedge equipped with a direction”. Thus our terminological discrepancybetween “oriented edges” and “directed edges” is intentional.2.
Examples of invariant spanning trees
In this section, we describe some examples of invariant spanningtrees. We confine ourselves with degree two rational maps.2.1.
Quadratic polynomials.
Let p p z q “ z ` c be a post-criticallyfinite quadratic polynomial. We will write J p p q for the Julia set of p and K p p q for the filled Julia set of p . Set X to be the forward p -orbit of c , i.e., the set t p ˝ n p c q | n ě u . The landing point of the dynamicalexternal ray R p p q of p with argument 0 is denoted by x β . This pointis usually called the β -fixed point . We will use the terminology of[Mil09, Poi93, Poi10], in particular, the notion of a regulated hull.Define T as the union of t8uY R p p q and the regulated hull of X Yt x β u .Then T is an invariant spanning tree for p .Recall that the regulated hull of X is called the Hubbard tree of p .Thus T is strictly bigger than the Hubbard tree of p . It is importantthat the tree T contains both critical values of p . In our terminology,the Hubbard tree itself is not a spanning tree for p . To specify a graphstructure on T , we need to define vertices. By definition, the vertices of T are post-critical points and branch points of T . On the other hand, x β is never a vertex of T .As an example, an invariant spanning tree T for the basilica polyno-mial p p z q “ z ´ ´ ‚ ˝ ‚ Critical values are shown as solid, and other vertices of T as circles.For the rabbit polynomial z ` c , where c « ´ . ` . i ,the tree T looks as follows: v ‚ w ˝ x α ˝ ˝ ‚ Here x α is the α -fixed point , i.e., the landing point of external rays witharguments , and . The edge 0 contains the β -fixed point x β andthe points ´ x α , ´ w . The latter three points are not in V p T q since theyare neither post-critical nor branch points.2.2. Matings.
Let p and q be two post-critically finite quadratic poly-nomials. Consider a compactification C of C obtained by adding acircle of infinity. More precisely, the circle at infinity is parameterizedby the arguments of external rays. For an angle θ P R { Z , we let R p p θ q be the corresponding external ray in the dynamical plane of p . We willwrite E p p θ q for the corresponding point in the circle at infinity. Let p and q act on different copies of C , say, p acts on C p and q on C q .Then E p p θ q and E q p θ q will refer to points in C p and C q , respectively.Consider the disjoint union Y “ C p \ C q . Let „ be an equivalencerelation on Y defined as follows. We have x „ y and x ‰ y if and onlyif one of the two points, say, x , has the form E p p θ q , and the other point y has the form E q p´ θ q . The quotient space S p > q “ Y { „ is called the formal mating space of p and q . It is easy to see that S p > q is home-omorphic to S . The map F : Y Ñ Y defined as p on C p and q on C q descends to the quotient space. Thus we have a naturally definedmap f : S p > q Ñ S p > q . We write f “ p > q and call f the formal mating of p and q . To construct an invariant spanning tree for f , it sufficesto construct invariant spanning trees for p and q as above, and thentake the union of the two trees. Below, the thus constructed invariantspanning tree is shown for p > q , where p is the rabbit polynomial, and q is the basilica polynomial. v ‚ w ˝ x α ˝ ˝ ˝ ´ ‚ Here, 0 and ´ ´ C q (more precisely, in the image of this plane in the space S p > q ). Thepoint (more precisely, the image of E p p q and E q p q in S p > q ) isnot a critical value anymore. Moreover, this point is not a vertex of theinvariant spanning tree shown above. It belongs to the edge connecting0 with 0. NVARIANT SPANNING TREES 11
Captures.
The following definition of a capture is equivalent tothe one from [Ree92]. However, we phrase the definition somewhatdifferently. Introduce a smooth structure on S . We also fix a smoothspherical metric on S . Given a vector v x at some point x P S and ε ą
0, there is a vector field D p v x , ε q such that(1) outside of the ε -neighborhood of x with respect to the sphericalmetric, D p v x , ε q “ x , the vector D p v x , ε q x coincides with v x .We may consistently choose vector fields D p v x , ε q for all x , v x and ε sothat they depend continuously (or even smoothly) on all parameters.Consider a smooth path β : r , s Ñ S and choose a small ε ą
0. De-fine the map σ β : S Ñ S as the time r , s flow of the non-autonomousvector field D p β p t q , ε q . Here β p t q is the velocity vector of β at the point β p t q . The map σ β is a self-homeomorphism of S with the followingproperties:(1) we have σ β p β p qq “ β p q ;(2) the map σ β is the identity outside of the ε -neighborhood U ε p β q of β r , s ;(3) the map σ β is homotopic to the identity modulo S ´ U ε p β q .The homeomorphism σ β depends on β , ε and on a particular choice of D p v x , ε q . However, if the path β is fixed, then any two such homeo-morphisms σ β and ˜ σ β are homotopic relative to S ´ U ε p β q .We can consider a composition σ β ˝ p , where p is a post-criticallyfinite quadratic polynomial, and the choice of β depends on p . Set β p q “ 8 , and place β p q at some strictly preperiodic point that isnot postcritical. If U ε p β q does not contain finite post-critical points ofthe map p and iterated images of β p q , then all such maps σ β ˝ p withfixed β are equivalent. In other words, the Thurston equivalence classof f “ σ β ˝ p depends only on β and p . The post-critical set of f is theunion of P p p q and the forward orbit of β p q , including β p q . Note that β p q is a critical value of f , the image of the critical point . In fact,the homotopy class of f does not change if we deform β within thesame homotopy class relative to P p f q . When talking about σ β ˝ p , wewill always assume that the set β r , q is disjoint from P p p q and fromthe forward orbit of β p q . The path β is called a capture path for p . Definition 2.1.
The map σ β ˝ p defined as above is called the (gener-alized) capture of p associated with β . The capture σ β ˝ p is said to be simple if there is only one t P r , s with β p t q P J p p q . In the lattercase, the corresponding capture path is called a simple capture path . Suppose that β p q is eventually mapped to a periodic critical point of p , i.e., to 0 if p p z q “ z ` c . Then a simple capture path β : r , s Ñ S looks as follows. There is a parameter t P p , q such that β r , t q isin the basin of infinity, β p t , s is in the Fatou component eventuallymapped to a super-attracting periodic basin, and β p t q is a point ofthe Julia set. We may arrange β | r ,t q to go along an external ray,and β | p t , s to go along an internal ray. If β p q P J p p q , then β r , s can be chosen as the union of an external ray and its landing point.Different simple capture paths lead to at most two different Thurstonequivalence classes of captures provided that p and β p q are fixed, cf.[Ree10, Section 2.8].Generalized captures were first defined by M.Rees in [Ree92]. Simplecaptures go back to B.Wittner [Wit88]. Both Wittner and Rees usedthe word “capture” to mean simple capture. We, on the contrary, usethe word “capture” to mean a generalized capture. It is worth notingthat the original approach of Wittner also used invariant trees. Thestudy of captures is motivated by the following theorem of M.Rees: Theorem 2.2 (Polynomial-and-Path Theorem, Section 1.8 of [Ree92]) . Suppose that R is a rational function of degree two with a periodiccritical point c . Suppose also that the other critical point c of R is not periodic but is eventually mapped to c . Then R is equivalentto some capture σ β ˝ p . Moreover, the quadratic polynomial p has aperiodic critical point of the same period as c . Suppose that β is a simple capture path for p , and f “ σ β ˝ p is thecorresponding capture. Let T be the minimal subtree of the extendedHubbard tree of p that includes P p f q . Then T satisfies the property p p T q Ă T . Note that it may happen that f p T q Ć T , so that T is notan invariant spanning tree for f . For example, let p be the airplanepolynomial. Choose β p q to be an iterated p -preimage of 0 on an edgeof the Hubbard tree of p . Then T coincides with the Hubbard treeset-theoretically but has more vertices. Some edge e of T maps under p so that the β p q P p p e q but β p q is not an endpoint of p p e q . The latteris a consequence of the fact that there are no vertices of T mapping to β p q . The homeomorphism σ β displaces p p e q so that σ β p p p e qq no longercontains β p q . Thus T is not forward invariant under f “ σ β ˝ p .It may seem plausible that T can be deformed slightly into a genuineinvariant spanning tree. Unfortunately, this is not always true. It isknown that different simple captures (even those for which β p q is thesame) may yield different Thurston equivalence classes, see e.g. [Ree10,Section 2.8]. If T were deformable into an invariant spanning tree, then, NVARIANT SPANNING TREES 13 by Theorem A, all simple captures with given β p q would be Thurstonequivalent, a contradiction.In the following lemma, by a support of a homeomorphism σ : S Ñ S we mean the closure of the set of points x P S with σ p x q ‰ x . Lemma 2.3.
Let p , β and T be as above. Assume that the supportof σ β is a sufficiently narrow neighborhood of β r , s , i.e., a subset ofthe ε -neighborhood of β r , s for sufficiently small ε ą . Then T is aninvariant spanning tree for f “ σ β ˝ p whenever β r , s X p p T q “ ∅ . Recall our assumption that the capture path β is simple. Proof.
Suppose that β r , s X p p T q “ ∅ . Then the support of σ β canbe made disjoint from p p T q . It follows that σ β “ id on p p T q , therefore, f p T q “ σ β p p p T qq “ p p T q Ă T . (cid:3) Proof of Theorem A
Let f : S Ñ S be a Thurston map of degree two. It will beconvenient to mark the critical points of f , i.e., to distinguish between c p f q and c p f q . Definition 3.1 (Marked Thurston maps) . A (critically) marked Thurstonmap of degree two is an ordered triple p f, c , c q , where f is a Thurstonmap of degree two, and t c , c u is the set of all critical points of f .Thus, if c ‰ c , then p f, c , c q and p f, c , c q are different markedThurston maps. To lighten the notation, we will sometimes write f fora Thurston map p f, c , c q . In this case, we will write c p f q , c p f q toemphasize the dependence on f .We now recall the definition of Thurston equivalence. Definition 3.2 (Thurston equivalence) . Let f and g be two Thurstonmaps. They are said to be Thurston equivalent if there are two orien-tation preserving homeomorphisms φ , ψ : S Ñ S with the followingproperties:(1) We have φ “ ψ on P p f q , and φ p P p f qq “ P p g q .(2) The maps φ and ψ are isotopic modulo P p f q .(3) We have ψ ˝ f “ g ˝ φ .If f and g are marked Thurston maps of degree two, then we addition-ally require that φ p v i p f qq “ v i p g q for i “ , Lemma 3.3.
Let f t , t P r , s be a continuous family of Thurston mapswith P p f t q “ P p f q . Then all f t are Thurston equivalent. Thurston maps f and f from Lemma 3.3 are said to be homotopic .This lemma is known but we will sketch a proof for completeness. Sketch of a proof.
By the covering homotopy theorem, there is a ho-motopy φ t with φ “ id and f t ˝ φ t “ f . It is easy to see that φ t are orientation preserving homeomorphisms. Setting g “ f t , f “ f , φ “ φ t , ψ “ id , we see that the requirements of Definition 3.2 arefulfilled. (cid:3) Cyclic sets and pseudoaccesses.
Recall Theorem A. We aregiven two quadratic Thurston maps f , g with invariant spanning trees T f , T g . There is an isomorphism τ : T f Ñ T g of ribbon graphs thatconjugates f with g on V p T f qY C p T f q and maps critical values to criticalvalues. We want to prove that τ extends to a Thurston equivalencebetween f and g .Modify the trees T f , T g by adding to their vertices the critical pointsof f belonging to T f , T g , respectively. We will write T f , T g for themodified trees. These are also ribbon graphs. Note that τ takes thevertices of T f to the vertices of T g . Moreover, τ induces an isomorphismof ribbon graphs.The proof will consist of two steps. The first step is to define a ribbongraph isomorphism between f ´ p T f q and g ´ p T g q . In other words, ifjust T f is given (in which the critical values are marked), then f ´ p T f q can be recovered as a ribbon graph, even without knowing f . In orderto recover f ´ p T f q , a classical construction of the Riemann surface for f ´ helps. (This construction is essentially the same as for the Riemannsurface of z ÞÑ ? z .) We make a cut between two critical values of f , and then glue two copies of the slitted sphere along the slits. Ifthe cut is disjoint from T f (except the endpoints), then it suffices tosee how copies of T f in the two slitted spheres are glued together.To translate this process to combinatorics, we need some terminologyrelated to cyclic sets and pseudoaccesses. The next definition followsthe terminology of [Poi93]. Definition 3.4 (Pseudoaccess) . Let A be a cyclic set , i.e., a set witha distinguished cyclic order of elements. A pseudoaccess of A is an(ordered) pair p a, b q of elements of A such that b is the immediatesuccessor of a in the cyclic order. The terminology is motivated by thefollowing picture. Suppose that A consists of Jordan arcs in the planethat share an endpoint and are otherwise disjoint (the cyclic order on A follows the counterclockwise direction around the endpoint). A Jordan NVARIANT SPANNING TREES 15
Figure 1.
Recovering G f “ f ´ p T f q from T f . Top left: westart with T f and make a cut along a simple curve C f connect-ing the critical values outside of T f . As T f , we took an invariantspanning tree for the rabbit polynomial p ; we set v “ p p q and w “ p p v q . Top right: widening the cut, we obtain a hemisphere U f with a copy T f of T f . Vertices of T f are labeled as the correspond-ing vertices of T f . Bottom left: attach the opposite hemisphere U f with another copy T f of T f . Then G f is the union of T f and T f .This construction shows that G f is uniquely defined as a ribbongraph once T f is given and the critical values are distinguishedamong the vertices of T f . Vertices of G f are now labeled as theyappear in G f (new labels are preimages of the former labels). Bot-tom right: we now removed the edges of G f that do not appearin T f . What remains is a tree that identifies with T f (one shouldrotate the sphere and deform the trees to attain the coincidence). arc disjoint from all elements of A except for the same endpoint definesa pseudoaccess of A . This is illustrated by the figure below, in which the pseudoaccess p a, b q is represented by the dashed segment. c bd ‚ (cid:111) (cid:111) a The cyclic set A here is represented by the four arcs a , b , c , d in thiscyclic order.3.2. A homeomorphism between f ´ p T f q and g ´ p T g q . Considera Thurston map f of degree 2 with an invariant spanning tree T f , and aThurston map g of degree 2 with an invariant spanning tree T g . We willwork under the assumptions of Theorem A. In particular, we considera homeomorphism τ : T f Ñ T g with the properties listed there. Thefirst step in the proof of Theorem A is to extend τ to G f “ f ´ p T f q .Note that we view G f not only as a subset of S but also as a graph.Vertices of G f are defined as preimages of vertices of T f . Edges of G f are defined as components of f ´ p T f ´ V p T f qq . Definition 3.5 (Pseudoaccesses of graphs) . Let G be a graph in thesphere. A pseudoaccess of G at a vertex a is defined as a pseudoaccessof E p G, a q . Here E p G, a q is the cyclic set of all edges of G incident to a . Recall that the cyclic order of edges incident to a is induced by theorientation of S .Consider a Jordan arc C f that connects v p f q with v p f q and isotherwise disjoint from T f , see Figure 1, top left. Then C f definestwo pseudoaccesses of T f , one at each of the critical values. Since τ preserves the cyclic order of edges at every vertex, it defines a corre-spondence between pseudoaccesses of T f and pseudoaccesses of T g . Thetwo pseudoaccesses of T f defined by C f give rise to two distinguishedpseudoaccesses of T g . Since τ maps the critical values of f to the criti-cal values of g , the two distinguished pseudoaccesses of T g are at v p g q and v p g q . Clearly, there exists a Jordan arc C g connecting v p g q with v p g q , otherwise disjoint from T g and defining the two distinguishedpseudoaccesses of T g .The set U f “ S ´ C f is a disk. The restriction of f to f ´ p U f q is an unbranched covering since both critical values of f are in C f .Therefore, f ´ p U f q is a disjoint union of two open disks U f and U f .These disks are shown as hemispheres in Figure 1, bottom left. Thereis an ambiguity in labeling U f and U f . One of the two disks has to be NVARIANT SPANNING TREES 17 labeled U f , and the other U f . However, which disk gets which label isup to us. Similarly, g ´ p U g q is a disjoint union of two disks U g and U g .Again, the labeling of these disks should be specified somehow.The common boundary of the disks U f , U f is the Jordan curve f ´ p C f q . Consider the closure T if (in S ) of the f -pullback of T f ´ C f in U if , where i “ ,
1. Clearly, T if is a tree isomorphic to T f . Moreover, T if and T f have isomorphic ribbon graph structures. We may view T f and T f as two copies of T f . Observe that these two copies are glued atthe critical points of f to form the graph G f “ f ´ p T f q . Observe alsothat the critical points of f are the vertices of T if that correspond, un-der the natural isomorphism between T if and T f , to the critical values v p f q and v p f q . Thus there is an abstract description of the ribbongraph G f . It involves making two copies of T f and gluing them at thevertices corresponding to v p f q , v p f q . See again Figure 1. Note thatthe representation of G f as a union T f Y T f depends on the choice of C f . More precisely, it depends on the choice of the two pseudoaccessesof T f . A similar representation can be obtained for G g “ g ´ p G g q . Lemma 3.6.
Either τ p T f X T if q Ă T g X T ig for every i “ , or τ p T f X T if q Ă T g X T ´ ig for every i “ , . Lemma 3.6 is a manifestation of the fact that the construction shownin Figure 1 is essentially unique. Define the label (cid:96) p e q of an edge e P E p T f q so that e Ă T (cid:96) p e q f . Thus the label of an edge can take values 0 or1. Labels are defined on edges of T f and on edges of f ´ p T f q but maynot be well-defined on edges of T f . (Recall that T f was defined above asa subdivision of T f , in which critical points of f in T f become vertices.)Lemma 3.6 asserts that τ either preserves all labels or reverses all labels.We will choose the labeling of U g , U g so that all labels are preservedby τ . Proof of Lemma 3.6.
We have to show that, if τ p e r q Ă T (cid:96) p e r q g for some e r P E p T f q , then the same holds for every edge e of T f . In other words, τ preserves all labels. To this end, we compare every edge e with e r .The latter will be called the reference edge . Consider critical points of f in T f . Note, however, that T f does not have to contain all criticalpoints of f . Suppose that some critical point c (which is necessarily c p f q or c p f q ) lies in T f . Let v “ f p c q be the corresponding criticalvalue. The curve f ´ p C f q defines two pseudoaccesses of T f at c , notnecessarily different. We will call these distinguished pseudoaccesses critical pseudoaccesses . Clearly, critical pseudoaccesses depend onlyon the ribbon graph structure of T f and on the choice of C f , more precisely, on the two pseudoaccesses defined by C f . The latter twopseudoaccesses will be referred to as post-critical pseudoaccesses . Thetwo critical pseudoaccesses of T f at c may separate some pairs of edgesincident to c .The values of the function (cid:96) can be computed step by step, startingat e r and passing from edges to adjacent edges. Suppose that (cid:96) p e q is known, and e shares a vertex a with e . If a is not critical, then (cid:96) p e q “ (cid:96) p e q . If a is critical, then (cid:96) p e q ‰ (cid:96) p e q if and only if e and e are separated by the critical pseudoaccesses at a . The just describedcomputational description of (cid:96) follows from the observation that edges e , e P E p T f , a q are separated by the critical pseudoaccesses at a if andonly if they are separated by f ´ p C f q , i.e., lie in different componentsof f ´ p U f q .We now need to prove that (cid:96) p τ p e qq “ (cid:96) p e q . To this end, it is enoughto observe that τ maps C p T f q to C p T g q and that τ maps critical pseu-doaccesses of T f to critical pseudoaccesses of T g . Indeed, suppose that c is a critical point in T f . Then v “ f p c q is a critical value, and τ p v q is also a critical value by property p q of τ listed in the state-ment of Theorem A. On the other hand, by property p q of τ , we have τ p v q “ τ ˝ f p c q “ g p τ p c qq . Since g p τ p c qq is a critical value, and g hasdegree two, τ p c q is a critical point. It is also clear that critical pseu-doaccesses at c map under τ to critical pseudoaccesses at τ p c q . Weconclude that τ preserves the labels, as desired. (cid:3) Recall that G f “ f ´ p T f q and G g “ g ´ p T g q . Recall also that V p G f q Ą V p T f q Ą V p T f q , and similarly for g . Proposition 3.7.
There is a ribbon graph isomorphism τ ˚ : G f Ñ G g with the following properties: (1) We have τ ˚ “ τ on V p T f q . (2) We have τ ˚ ˝ f “ g ˝ τ ˚ on all vertices of G f .Proof. Recall that, by Lemma 3.6, the map τ : T f Ñ T g preserveslabels. This map lifts to T if , where i “ ,
1, by the homeomorphisms f : T if Ñ T f and g : T ig Ñ T g . In other words, we can define a map τ i : T if Ñ T ig by the formula τ i “ g ´ i ˝ τ ˝ f , where g ´ i is the inverse of g : T ig Ñ T g . We set τ ˚ to be the map from G f to G g , whose restrictionto T if is τ i . Then we need to prove that properties p q ´ p q hold for τ ˚ .Let us first prove that τ ˚ “ τ on V p T f q . On V p T f q X T if , the map τ satisfies the property τ ˝ f “ g ˝ τ by the assumptions of Theorem A. ByLemma 3.6, under τ the set T f X T if maps to T g X T ig (note that T f “ T f NVARIANT SPANNING TREES 19 as sets, and hence T f X T if “ T f X T if set-theoretically). Therefore, wehave τ “ g ´ i ˝ τ ˝ f on V p T f q X T if . It remains to note that the righthand side coincides with the definition of τ i .We now prove that the cyclic order of edges incident to a vertex a ˚ P V p G f q is preserved by τ ˚ . Let f ´ i be the inverse of f : T if Ñ T f .If a ˚ “ f ´ i p a q , where a is not a critical value, then the statementis obvious since both f : U if Ñ S and g : U ig Ñ S preserve theorientation. Now, if a is a critical value, then the restriction of τ ˚ tothe union of edges incident to a ˚ is glued from the two maps τ and τ . The cyclic order of edges of G f at a ˚ is as follows. First come alledges of T f incident to a ˚ that are mapped by τ in an order preservingfashion. Then come all edges of T f incident to a ˚ that are mapped by τ in an order preserving fashion. It follows that τ ˚ preserves the cyclicorder on edges of G f incident to a ˚ .It remains to prove that τ ˚ ˝ f “ g ˝ τ ˚ on all vertices of G f . Indeed,let a ˚ be a vertex of G f . Then a ˚ “ f ´ i p a q for i “ τ ˚ ˝ f p a ˚ q “ τ p a q “ g ˝ τ i ˝ f ´ i p a q “ g ˝ τ ˚ p a ˚ q . In the first equality, we used that τ ˚ “ τ on V p T f q . In the secondequality, we used the definition of τ i . (cid:3) An extension of τ to the sphere. We keep the notation of The-orem A. Consider the homeomorphism τ ˚ : G f Ñ G g constructed inProposition 3.7. The restriction of τ ˚ to T f is in general different from τ . However, these two maps match on V p T f q “ V p T f q Y C p T f q . More-over, τ ˚ restricted to T f also satisfies assumptions p q – p q of TheoremA. Thus we may consider τ ˚ in place of τ .We will now extend τ ˚ to the entire sphere. Such an extension ispossible due to the following result. Theorem 3.8 (Corollary 6.6 of [BFH92]) . Let G and G be two con-nected graphs embedded into S . Consider a homeomorphism h : G Ñ G that induces an isomorphism of ribbon graphs. Then there is anorientation preserving homeomorphism h ˚ : S Ñ S whose restrictionto G is h . Applying Theorem 3.8 to our specific situation, we obtain the fol-lowing corollary.
Corollary 3.9.
Suppose that τ ˚ : G f Ñ G g satisfies the propertieslisted in Proposition 3.7. Then τ ˚ extends to an orientation preservinghomeomorphism τ ˚ : S Ñ S . The homeomorphism τ ˚ maps complementary components of G f tocomplementary components of G g . The following notion helps to saywhich components are mapped to which components in combinatorialterms: Definition 3.10 (Boundary Circuits) . Let G be a graph in S . If anorientation of an edge e P E p G q is fixed, then e is called an orientededge of G . The endpoints of e form an ordered pair p a, b q , where a isthe initial endpoint and b is the terminal endpoint of e . We also saythat e originates at a and terminates at b . The same edge equippedwith different orientations gives rise to two different oriented edges.A boundary circuit of G (also known as a left-turn path in G ) is acyclically ordered sequence r e , . . . , e n ´ s of oriented edges of G withthe following property: if e i terminates at a vertex a , then e i ` p mod n q originates at a , and e i ` p mod n q is the immediate predecessor of e i inthe cyclic order on E p G, a q . Clearly, any oriented edge belongs to someboundary circuit.As above, let W be some complementary component of G . There isa boundary circuit Σ W “ r e , . . . , e n ´ s associated with W . Informally,it is obtained by tracing the boundary of W counterclockwise. Thecorrespondence W ÞÑ Σ W between components of S ´ G and boundarycircuits of G is one-to-one. Observe that the same edge may enter Σ W twice with different orientations. Observe also that the rotation from e k to e k ` around the terminal point of e k is clockwise.3.4. Homotopy.
Theorem A will be deduced from Theorem 3.11 statedbelow. Theorem 3.11 is not new: Proposition 3.4.3 of [Hlu17] containsa more general fact; it is based in turn on a similar statement from[BM17]. However, since notation and terminology in [BM17, Hlu17]are somewhat different, we sketch a proof here. The proof will bebased on a technical lemma from [BFH92].
Theorem 3.11.
Suppose that two Thurston maps f and g of degree twoshare an invariant spanning tree T . Moreover, suppose that f ´ p T q “ g ´ p T q “ G , that f “ g on V p G q , and that the critical values of f coincide with the critical values of g . Then there is an orientationpreserving homeomorphism ψ isotopic to the identity relative to V p T q and such that f “ g ˝ ψ . Note that the equality f ´ p T q “ g ´ p T q means the equality of graphsrather than just sets. In particular, we assume that the two graphs havethe same vertices. Note also that all critical points of f are among ver-tices of these graphs. Theorem 3.11 implies that f and g are Thurstonequivalent. In fact, they are even homotopic. NVARIANT SPANNING TREES 21
Proof of Theorem A using Theorem 3.11.
Let f , g and τ : T f Ñ T g beas in Theorem A. As before, set G f “ f ´ p T f q and G g “ g ´ p T g q . ByProposition 3.7, there is a homeomorphism τ ˚ : G f Ñ G g that inducesan isomorphism of ribbon trees and is such that(1) we have τ ˚ “ τ on V p T f q ;(2) we have g ˝ τ ˚ “ τ ˚ ˝ f on all vertices of G f .Replacing τ with τ ˚ if necessary, we may assume that τ satisfies theseproperties. In particular, τ maps G f to G g .By Corollary 3.9, the map τ extends to an orientation preservinghomeomorphism τ : S Ñ S . Set g ˚ “ τ ´ ˝ g ˝ τ . Clearly, this is aThurston map of degree two. Then T f is an invariant spanning tree for g ˚ . Since τ maps the critical values of f to the critical values of g , themaps f and g ˚ share the critical values. Finally, g ´ ˚ p T f q “ τ ´ ˝ g ´ ˝ τ p T f q “ τ ´ p G g q “ G f “ f ´ p T f q . Thus all assumptions of Theorem 3.11 hold for f and g ˚ . By Theorem3.11, the map g ˚ is homotopic to f . Since g is topologically conjugateto g ˚ , we conclude that g is Thurston equivalent to f . (cid:3) Consider two graphs G and T in the sphere. Let h : G Ñ T be acontinuous map that is injective on the edges of G and is such that theforward and inverse images of the vertices are vertices. Such a mapis called a graph map in [BFH92]. Suppose that a graph map h hasan extension h : S Ñ S . If h is an orientation preserving branchedcovering injective on every complementary component of G , then h iscalled a regular extension of h . This terminology also follows [BFH92]. Theorem 3.12 (Corollary 6.3 of [BFH92]) . Consider two graph maps h , h : G Ñ T admitting regular extensions h , h . Suppose that h “ h on V p G q and h p e q “ h p e q for every e P E p G q . Then there is ahomeomorphism ψ : S Ñ S such that h “ h ˝ ψ , and ψ is isotopic tothe identity relative to V p G q . We are now ready to deduce Theorem 3.11 from Theorem 3.12.
Proof of Theorem 3.11.
Apply Theorem 3.12 to h “ f : G Ñ T and h “ g : G Ñ T . These are clearly graph maps admitting regularextensions. All assumptions of Theorem 3.12 are satisfied. It followsthat there exists a homeomorphism ψ : S Ñ S such that f “ g ˝ ψ on S , and ψ is isotopic to the identity relative to V p G q . (cid:3) Thus we proved Theorem 3.11, and the latter implies Theorem A. No dynamics: spanning trees
In this section, we associate certain combinatorial objects with aspanning tree. Recall that, given a finite set P of (marked) pointsin S , a spanning tree for P is a tree T Ă S with the property that V p T q “ P Y B p T q , where B p T q is the set of branch points of T . Thusthe notion of a spanning tree is an non-dynamical notion. Supposethat the sphere S is glued of a polygon ∆ by identifying some edgesof it. Then the boundary of ∆ becomes a spanning tree for the set P of all vertices of ∆. Alternatively, some vertices of ∆ can be droppedfrom P if these give rise to branch points of the tree.4.1. The generating set E T of π p S ´ P q . Let T be a spanning treefor a finite marked set P . Assume that the base point y P S ´ T isfixed once and for all. We now define a certain generating set E “ E T of π p S ´ P q “ π p S ´ P, y q . (This is the same generating set as in[Hlu17]; Hlushchanka refers to its elements as edge generators ).Endow S with some smooth structure. (It will be clear howeverthat our construction is independent of this structure). Consider anoriented smooth Jordan arc A . Let γ be a smooth path that crosses A only once and transversely. By a transverse intersection we meanthat the tangent lines to A and γ at the intersection point are different,and that the intersection point is not an endpoint of A . We say that γ approaches A from the left if, at the intersection point, the velocityvectors to γ and to A (in this order) form a positively oriented basisin the tangent plane to the sphere. With every oriented edge e of T , we associate an element g e P π as follows. The homotopy class g e is represented by a smooth loop γ e that crosses e just once andtransversely, approaches it from the left, and has no other intersectionpoints with T . (We assume of course that the loop γ e is based at y ).A smooth loop γ e with the indicated properties is said to be adapted to T at e . Consider the subset E “ E T Ă π p S ´ P q consisting of id , the neutral element, and elements g e , where e ranges through alloriented edges of T . Note that the same edge equipped with differentorientations gives rise to two different elements of E . These elementsare inverse to each other.Thus E is a generating set of π p S ´ P q that is symmetric ( E ´ “ E )and such that id P E . Lemma 4.1.
Different oriented edges of T give rise to different ele-ments of E .Proof. Consider two different oriented edges e , e of T with g e “ g e .Let γ e i : r , s Ñ S be smooth simple loops as above so that g e i “ r γ e i s . NVARIANT SPANNING TREES 23
We can also arrange that γ e p , q is disjoint from γ e p , q . Let D bethe croissant shaped region bounded by γ e r , s and γ e r , s .Since g e “ g e , the loops γ e i are homotopic rel. P . Therefore,there are no points of P in D . We claim that there are also no branchpoints of T in D . Indeed, if x is such point, then there is a componentof T ´ t x u disjoint from both e and e . This component must endsomewhere in D . On the other hand, by definition of a spanning tree, allendpoints of T are in P . A contradiction with the fact that P X D “ ∅ .Since e ‰ e , there is at least one vertex x of T in D . However, thisis impossible since all vertices of T are in P Y B p T q . (cid:3) Consider a set E of smooth loops based at y with the followingproperties. Firstly, we assume that every γ P E is adapted to T at someoriented edge e of T . Secondly, there is exactly one loop γ e P E adaptedto T at e , and, as we change the orientation of e , the correspondingloop also changes orientation but otherwise remains the same. Thirdly,we assume that the constant loop belongs to E , and that different loopsfrom E are either disjoint (except the common basepoint y ) or the same(up to the change of direction). If these assumptions are satisfied, thenwe say that E is an adapted set of loops for T . Clearly, any spanningtree admits an adapted set of loops. The set E T equals r E s , the set ofclasses in π p S ´ P q of all elements from E .The following lemma will help us translate the pullback operations onspanning trees into a combinatorial language. We will assume that thebasepoint y is chosen outside of all spanning trees under consideration. Lemma 4.2.
Let H be an inner automorphism of π p S ´ P q . Supposethat two spanning trees T and T are such that E T “ H p E T q in π p S ´ P q . Then T and T are homotopic rel. P .Proof. Let h be an element of π p S ´ P q such that H is the conjugationby h . We will write PMod p S , P q for the pure mapping class group of S with marked point set P . Consider the homomorphism P ush : π p S ´ P, y q Ñ
PMod p S , P Y t y uq from the Birman exact sequence(cf. Section 4.2.1 of [FM12]). It is easy to see that ψ “ P ush p h q actson π p S ´ P, y q as H . Moreover, the Birman exact sequence impliesthat ψ is isotopic to the identity rel. P but not rel. P Y t y u . Replacing T with ψ ´ p T q , we can arrange that E T and E T coincide. Thus wewill assume from now on that E T “ E T .We may assume that both T and T are composed of smooth arcs.Suppose that sets E , E of smooth loops based at y are adapted to T , T , respectively. The sets E , E form embedded graphs Γ, Γ , respec-tively, in S with the single vertex y . Every complementary component (“face”) of Γ contains a single vertex of T , and similarly for Γ . Thereis a homeomorphism φ : Γ Ñ Γ that is simultaneously a graph map.Moreover, for every edge of Γ, there is an isotopy transforming thisedge to its φ -image. (Indeed, two loops are homotopic rel. P if andonly if they are isotopic rel. P .) Then φ can be extended as an ori-entation preserving homeomorphism φ : S Ñ S fixing P pointwise.This follows from Lemma 2.9 of [FM12]. Moreover, it follows from thesame lemma that φ is isotopic to the identity rel. P . Applying φ ´ to T and E , we may now assume that E “ E . The correspondingedges of T and T connect the same complementary components of Γand cross the same edge of Γ. It follows that the corresponding edgesof T and T are homotopic rel. P , as desired. (cid:3) The converse of Lemma 4.2 is also true. We say that two subsets E and E of a group π are conjugate if there is u P π such that E coincideswith the set of all elements of the form uvu ´ , where v runs through E . Proposition 4.3.
Let T and T be spanning trees for P . The trees T and T are homotopic rel. P if and only if the corresponding generatingsets E T and E T are conjugate.Proof of Proposition 4.3. To lighten the notation, we will write E and E instead of E T and E T . We silently assumed that both E and E aresubsets of the same group π “ π p S ´ P, y q corresponding to a certainbasepoint y . Thus the basepoint is fixed.Suppose first that E and E are conjugate. Then T and T are homo-topic rel. P , by Lemma 4.2. Suppose now that T and T are homotopic.We may assume that both T , T are formed by smooth arcs. Let E bea set of smooth loops adapted to T .Now consider a homotopy T t of T (so that T “ T , T “ T , and t runsthrough r , s ). We may assume that this homotopy is smooth. Thenthere is a homotopy φ t : S Ñ S consisting of orientation preservingdiffeomorphisms such that φ “ id and φ t p T q “ T t . Clearly, E t “ φ t p E q is adapted to T t . In particular, E t represents the symmetric generatingset E t “ E T t in π p S ´ P, φ t p y qq .Recall that any homotopy class c of paths connecting two givenpoints y , y P S ´ P gives rise to an isomorphism H c : π p S ´ P, y q Ñ π p S ´ P, y q . Two different isomorphisms of this type differ by an in-ner automorphism of the target group. All groups π p S ´ P, φ t p y qq canbe identified along the path t ÞÑ φ t p y q . In particular, π p S ´ P, φ p y qq identifies with π . NVARIANT SPANNING TREES 25
Modifying the homotopy if necessary, we may arrange that φ p y q “ y .Thus, E and E lie in the same group, and, by definition of E , we musthave E “ E . On the other hand, E identifies with E “ E under theautomorphism H c , where c is the homotopy class of the loop t ÞÑ φ t p y q .Since H c is an inner automorphism, E and E are conjugate. Thus theproposition is proved. (cid:3) Vertex structures.
Below, we will introduce some formal al-gebraic/combinatorial notions. The purpose of these is to translatetopological objects, namely, spanning trees, into a symbolic language.For any finite set E , we write FS p E q for the free semi-group generatedby E . The semi-group FS p E q can also be thought of as the set of allfinite words in the alphabet E . The empty word is allowed as an elementof FS p E q ; it is the neutral element of the semi-group. For g , h P E , theproduct of g and h in FS p E q will be written as g ¨ h .Suppose now that π is a group and that E Ă π . We also supposethat id P E . Here id means the identity element of π . It is not to beconfused with the neutral element of FS p E q , which is not an element of E or of π . We set E ‹ to be the quotient of FS p E q modulo the relations id ¨ g “ g ¨ id “ g for all g P E . Now assume that E Q id is symmetric,i.e., that g P E implies g ´ P E . Here g ´ is the inverse of g in thegroup π . Then there is a natural map Π : E ‹ Ñ π that takes everyword in the alphabet E to the product of its symbols. (The latterproduct is with respect to the group operation in π .) We will referto Π as the evaluation map . For example, an element g ¨ g P E ‹ ismapped to g g P π . Intuitively, an element u P E ‹ is a way of writingthe element Π p u q of the subgroup of π generated by E as a product ofgenerators. Different ways of writing the same element may differ bya sequence of cancellations. However, we disregard all appearances of id . For example, g ¨ h ¨ g ¨ g ´ is different from g ¨ h as an element of E ‹ .However, it is the same as g ¨ id ¨ h ¨ g ¨ id ¨ g ´ ¨ id ¨ id , for example.A vertex structure on E is a subset V Ă E ‹ with the following prop-erty: for every g P E , there is a unique element of V of the form u ¨ g ¨ u for some u , u P E ‹ . Any vertex structure gives rise to an abstract di-rected graph G p V q as follows. The vertices of G p V q are identified withelements of V . The oriented edges of G p V q are labeled by elements of E . Two vertices v , w P V are connected with an oriented edge g (from v to w ) if v “ v ¨ g ¨ v , w “ w ¨ g ´ ¨ w for some elements v , v , w , w of E ‹ . Since E is symmetric, the edgesof G p V q always come in pairs so that paired edges connect the samevertices but go in different directions. These pairs of edges correspond to pairs of the form t g, g ´ u in E . Thus G p V q can also be regardedas an undirected graph, by identifying each pair of oppositely directededges with an undirected edge. A vertex structure V on E is called a tree structure if G p V q is a tree.Observe that the graph G p V q also carries a natural ribbon graphstructure. Indeed, directed edges of G p V q originating at a given vertex v P E ‹ are linearly ordered. (We refer to the linear order of symbols inwords from E ‹ ). Relaxing this linear ordering to a cyclic ordering, weobtain a ribbon graph structure on G p V q .4.3. Vertex words.
In this section, we explain how a spanning tree T for a finite marked set P defines a tree structure on E “ E T . Tothis end, we need to equip T with a bit of extra structure. Namely, weassume that some pseudoaccess is fixed at every vertex of T .Recall that any oriented edge e of T gives rise to a group element(edge generator) g e P E . Moreover, by Lemma 4.1, different edgescorrespond to different edge generators. Thus we may think of E as acombinatorial analog for the set of oriented edges of T . We now definea combinatorial analog of a vertex. Definition 4.4 (Vertex word) . Let x be a vertex of T . Consider alledges e , . . . , e k ´ incident to x and oriented outwards. The linearorder of these edges is well defined if we impose that(1) it follows the natural clockwise order around x ;(2) the chosen pseudoaccess at x coincides with p e k ´ , e q .Then we define the vertex word of x as the product g e ¨ ¨ ¨ ¨ ¨ g e k ´ P E ‹ .For example, if k “
3, then x “ g e ¨ g e ¨ g e (the product is in E ‹ , notin π p S ´ P q !). Let V be the set of all vertex words associated withthe vertices of T . Then V is clearly a tree structure on E such that G p V q is isomorphic to T as a ribbon graph.The construction presented above may seem artificial. In order toshed some light on it, let us consider an example. The following is aspanning tree T for a set of three marked points: ˝ (cid:79) (cid:79) B ˝ (cid:111) (cid:111) A ¨ C (cid:47) (cid:47) ˝ (The marked points, shown as circles, are precisely the endpoints ofthe tree.) We write A , B , C for the oriented edges of T originating atthe branch point. Set a “ g A , b “ g B , c “ g C . Then the generatingset E T consists of 7 elements id , a , a ´ , b , b ´ , c , c ´ . The vertexword corresponding to the branch point of the tree is a ¨ b ¨ c . Note NVARIANT SPANNING TREES 27 that this word is different from the neutral element of E ‹ even throughΠ p a ¨ b ¨ c q “ abc “ id in π . This example explains why we need toconsider E ‹ . The vertex structure associated with T is V “ t a ¨ b ¨ c, a ´ , b ´ , c ´ u . Clearly, the combinatorial structure of G p V q represents that of T .5. Dynamics: computation of the biset
In this section, we consider a Thurston map f of degree two with aninvariant spanning tree T . We will find a presentation for the biset of f using only the combinatorics of the map f : T Ñ T . We start withrecalling the terminology.5.1. Bisets and automata.
A biset is a convenient algebraic invari-ant of a Thurston map, which fully encodes the Thurston equivalenceclass.Fix some basepoint y P S ´ P p f q . Define the set X f p y q as theset of all homotopy classes of paths from y to f ´ p y q in S ´ P p f q .To lighten the notation, we will write π f for the fundamental group π p S ´ P p f q , y q . There are natural left and right actions of π f on X f p y q . For this reason, the set X f p y q is referred to as a π f -biset .The left action of π f on X f p y q is the usual composition of paths. Let γ be a representative of an element r γ s P π f , and let α be a representativeof an element r α s P X f p y q . Then r γ sr α s , the left action of the element r γ s P π f on an element r α s P X f p y q , is defined as the element r γα s of X f p y q represented by the composition γα of γ and α : we first traverse γ , and then α . According to our convention, paths are composed fromleft to right. The right action of π f on X f p y q is defined as follows.For r γ s P π f and r α s P X f p y q as above, let β be the composition of α and the pullback of γ originating at the terminal point of α . Then theelement r α s . r γ s P X f p y q , the right action of r γ s on r α s , is defined as r β s .We will refer to X f p y q as the biset of f . Now that we have a particularexample at hand, we give a general algebraic definition of a biset. Definition 5.1 (Biset) . Let π be a group. A set X is called a bisetover π , or a π -biset , if commuting left and right actions of π on X aregiven. The biset X is said to be left free if there exists a subset B Ă X such that every element a P X can be uniquely represented as gb , where g P π and b P B . The subset B is then called a basis of X . Let π beanother group, and X be a π -biset. A group isomorphism ρ : π Ñ π is said to conjugate X with X if there is a bijection σ : X Ñ X withthe property that σ p g a.g q “ ρ p g q σ p a q ρ p g q for all g , g P π and a P X . If ρ and σ with these properties exist, then X and X are said to be conjugate . If moreover π “ π and ρ “ id , we say that X and X are isomorphic . For more details on these formal notions, we referthe reader to [Nek05, BD17] (note that bisets are called bimodules in[Nek05], see Chapter 2).Clearly, the biset of a Thurston map is well defined up to conjuga-tion. Recall the following theorem of Nekrashevich (Theorem 6.5.2 of[Nek05], see also [Kam01, Pil03]), which says that, reversely, the con-jugacy class of the biset determines the Thurston equivalence class ofthe map: Theorem 5.2.
Let f and f be Thurston maps, and X f i be the cor-responding π f i -bisets, i “ , . Here π f i is the fundamental group of S ´ P p f i q . (1) The maps f and f are Thurston equivalent if and only if thereexists an orientation preserving homeomorphism h : S Ñ S such that h p P p f qq “ P p f q and the induced isomorphism h ˚ : π f Ñ π f conjugates X f with X f . (2) Suppose that P p f q “ P p f q “ P and the base points chosen for X f , X f coincide. The maps f and f are homotopic rel. P ifand only if X f and X f are isomorphic. Let us go back to a degree 2 Thurston map f . A basis of X f p y q consists of two elements. These are homotopy classes of two pathsconnecting y with its preimages y , y . Thus, to choose a basis of X f p y q is the same as to choose two paths α , α , up to homotopy rel. P p f q , so that α ε connects y with y ε , for ε “
0, 1. Once some basis of X f p y q is chosen, we can associate an automaton with X f p y q . Definition 5.3 (Automaton) . Let A and S be some sets. In practicallyimportant cases both A and S are finite. The set A is called an alphabet ,and its elements are called symbols . The set S is called the set of states ,and its elements are called states . An automaton can be defined as amap Σ : A ˆ S Ñ S ˆ A , or rather as a triple p A, S, Σ q . Let FS p A q bethe set of finite words in the alphabet A , including the empty word.This is a f ree s emi-group generated by A , thus the notation. If we fixsome initial state s P S , then we obtain a self-map of FS p A q as follows.Imagine that a machine reads a word w P FS p A q symbol by symbol, right to left . Suppose, at some point, it reads a symbol a P A and itsstate is s . Set p t, b q “ Σ p a, s q . Then the machine writes b in place of a , changes the state to t , and moves one step left. In other words, anautomaton p A, S, Σ q gives rise to a right action of S on FS p A q . If Σ isfixed, then it is common to write Σ p a, s q simply as as . NVARIANT SPANNING TREES 29
Consider an abstract left free π -biset X . Assume that some basis B of X is chosen. Then, for every a P B and every g P π , there areelements a ˚ P B and g ˚ P π with ag “ g ˚ a ˚ . Thus, we have a well-defined map Σ B : B ˆ π Ñ π ˆ B taking p a, g q to p g ˚ , a ˚ q . By definition,this is an automaton with π being the set of states. We will refer tothis automaton as the full automaton of X in the basis B . Clearly, thefull automaton defines X up to isomorphism. On the other hand, thefull automaton carries excessive information. It is enough to know thevalues Σ B p a, g q for all g in some generating set of π . If B is finite and π is generated by a finite set S , then the image of B ˆ S under Σ B is finite. In particular, this image lies in S ˚ ˆ B , where S ˚ is also afinite subset of π . Thus, in order to describe the biset, it suffices toindicate the map Σ B : B ˆ S Ñ S ˚ ˆ B between finite sets. This map iscalled a (finite) presentation of X . We see that finitely presented bisetscan be efficiently described, and computations with them are easy toimplement. However, the isomorphism problem for bisets is not easy,cf. [BD17].We now go back to the biset X f p y q of a quadratic Thurston map f . Ina number of important situations, there is a finite generating set E Ă π f with the following property. For ε P t , u and any element a P E , wehave r α ε s .a “ a ˚ r α ε ˚ s for some ε ˚ P t , u and a ˚ P E depending on a and ε . Define an automaton Σ : t , u ˆ E Ñ E ˆ t , u taking p ε, a q to p a ˚ , ε ˚ q . This automaton has then a finite set of states. Such automataare practically important and are called finite state automata . Observethat Σ defines a finite presentation of X f p y q . We will see that a simplepresentation of X f p y q by a finite state automaton can be associated withevery invariant spanning tree of f . This observation was also made in[Hlu17] in a more general context but with a less explicit descriptionof the automaton.5.2. A base edge and labels.
We now assume that p T ˚ , T q is a dy-namical tree pair for f . Let Z be the smallest subarc of T containingboth v and v . (In Figure 1, top left, this is the union of the arcs x α , and x α v .) Then f ´ p Z q is a Jordan curve containing the criticalpoints c and c . (In Figure 1, bottom left, this is the only simple cyclein the graph.) We will regard both Z and f ´ p Z q as graphs in thesphere whose vertices are the vertices of T and f ´ p T q , respectively,contained in Z and f ´ p Z q , respectively. Since the tree T ˚ cannot con-tain the Jordan curve f ´ p Z q , there is at least one edge e b of f ´ p Z q not contained in T ˚ . (In Figure 1, we removed an edge of f ´ p Z q whenpassing from the bottom left to the bottom right picture. We may set e b to be this removed edge.) Choose one such edge, and call e b “ f p e b q the base edge of T . There may be several ways of choosing a base edge.The two arcs with endpoints c , c mapping onto Z will be denotedby Z and Z . Here Z is chosen to include e b . Then Z includes theother pullback of e b .Set G “ f ´ p T q . Suppose now that some post-critical pseudoac-cesses (i.e., pseudoaccesses at critical values) are chosen for T . In-termediate steps in the computation of an automaton for X f p y q , butnot the final result, will depend on this choice. The choice of the post-critical pseudoaccesses gives rise to a representation G “ T Y T . Here T , T are two trees mapping homeomorphically onto T under f . InSection 3.2, we defined T and T using a Jordan arc C connecting v with v outside of T . However, it is easy to see that T i depend onlyon the pseudoaccesses defined by C . To fix the labeling, we assumethat Z i Ă T i for i “ ,
1. In fact, Z i is an “invariant” part of T i ,independent of the choice of the pseudoaccesses.Modify T ˚ so that the critical points of f lying in T ˚ become vertices.To distinguished the new (modified) tree from T ˚ , we denote it by T ˚ .Let e be an edge of T ˚ . Then e lies in T (cid:96) p e q , where (cid:96) p e q “ (cid:96) p e q is called the label of e , cf. the proof of Lemma 3.6. Wenow reproduce the combinatorial definition of labels. Definition 5.4 (The label of an edge) . Define the critical pseudoac-cesses of G “ f ´ p T q as the preimages of the post-critical pseudoac-cesses of T . There is a unique function (cid:96) : E p G q Ñ t , u with thefollowing properties:(1) we have (cid:96) p e b q “ e , e share a vertex; then (cid:96) p e q “ (cid:96) p e q ifand only if e , e are not separated by the critical pseudoac-cesses.The function (cid:96) with these properties is called the labeling . For e P E p G q ,the value (cid:96) p e q is called the label of the edge e . An edge of T ˚ may consistof several edges of G . These edges have the same label since the criticalpoints of f in T ˚ are vertices of T ˚ . The label of an edge of T ˚ is definedas the label of any edge of G contained in it. Thus the labeling is alsodefined on E p T ˚ q .Note that the labeling may not be well defined on E p T ˚ q if there areedges of T ˚ subdivided by critical points of f . This was the reason forpassing from T ˚ to T ˚ . NVARIANT SPANNING TREES 31
Figure 2.
The sequences S p T q and S p T q . In this example,the sequence S p T q consists of the oriented edges v x , x v , v y , y x , x y , y v , taken in this order. The sequence S p T q consistsof the edges v y , y v , v x , x v , taken in this order. The sig-nature of v x and y x is p , q . The signature of v y and y v is p , q . The opposite edges y v and v y have signature p , q .The signature of v x is p , q . The dashed line is the curve C corresponding to the chosen pseudoaccesses at the critical values. Signatures.
As before, T is a spanning tree for P p f q with spec-ified pseudoaccesses at the critical values. We also need a function onthe edges of T . Definition 5.5 (Signatures of edges) . Let C p T q be the only boundarycircuit of T . Informally: if a particle x loops around T in a smallneighbourhood of T so that T is kept on the right , then the cyclicallyordered sequence of oriented edges, along which x moves, coincides with C p T q . Even more informally: C p T q corresponds to walking around T clockwise . The choice of the direction is explained as follows: as wewalk around T clockwise, we walk around S ´ T counterclockwise. Thepostcritical pseudoaccesses divide all oriented edges from C p T q intotwo groups (segments) S p T q and S p T q . The labeling of S p T q and S p T q is chosen as follows. By definition, S p T q originates at v andterminates at v . Then S p T q originates at v and terminates at v .See Figure 2 for an illustration. We can now assign signatures to alledges of T . We say that an oriented edge e of T is of signature p i, j q if e appears in S i p T q , and e ´ appears in S j p T q . Here, for an orientededge e , we let e ´ denote the same edge with the opposite orientation.Thus there are four possible signatures: p , q , p , q , p , q , and p , q .For i, j “ ,
1, we write S i p T j q for the pullback of S i p T q in T j . Thecomplement of G in S consists of two disks Ω and Ω . These disksare bounded by S p T q Y S p T q and S p T q Y S p T q . We assumethat Ω to be the disk bounded by S p T q Y S p T q . See Figure 3 foran illustration. More precisely, the oriented boundary of Ω , regarded Figure 3.
The graph G and the disks Ω and Ω . Here G “ f ´ p T q , where T is the tree from Figure 2. The preimages of v , v are the critical points c , c , respectively. The preimages of othervertices of T are denoted as their images followed by a label 0 or 1in the parentheses. All preimages in U (the upper half-plane) arelabeled 0 and all preimages in U (the lower half-plane) are labeled1. Then T and T are the copies of T in U and U , respectively.These copies are deformed but topologically the same as T . Thesequence S p T q goes through the vertices c , x p q , c , y p q , x p q , y p q , c . The sequence S p T q goes through the vertices c , y p q , c , x p q , c . The disk Ω bounded by S p T q Y S p T q is the exterior of the quadrilateral c y p q c y p q with the arcs c x p q , c x p q and y p q x p q removed. as a chain of oriented edges of G , is the concatenation of S p T q and S p T q . Then Ω is bounded by S p T q Y S p T q in a similar sense.We will assume that Ω Q y . The problem, however, is that the twoassumptions(1) that y P Ω , and(2) that Ω is bounded by the concatenation of S p T q and S p T q may not be compatible. There are two ways of making them both hold.On the one hand, we can choose y differently. Although this is easyin theory, we will not do this in practice. A basepoint will be fixedonce and for all (see Assumption 6.3 for the principle of choosing thebasepoint). On the other hand, we may relabel T and T by choosing e b differently. The edge e b of G is one of the two pullbacks of e b ; theone not in T ˚ . If we change T ˚ , then we can also replace e b with theother pullback of e b . In this way, we can satisfy both assumptions. Thisis how we will act in practice. At each step of our iterative process,we will define T ˚ (and e b ) so that both assumptions hold. The exact NVARIANT SPANNING TREES 33 procedure will be described later. For now, we just assume that bothassumptions are satisfied.5.4.
The choice of paths α , α . We assume that p T ˚ , T q is a dy-namical tree pair for f . As before, T comes with a specific choice ofpseudoaccesses at the critical values. Recall that f ´ p y q “ t y , y u . Welabel the preimages y , y of y so that y i P Ω i for i “ ,
1. Choose apath α connecting y with y outside of T ˚ . Similarly, choose a path α connecting y with y outside of T ˚ . Then B “ tr α s , r α su is a basisof X f p y q . The basis B is well defined and depends only on T ˚ and y .This description of B is sufficient for now. However, for later use, wewill need a more accurate description of α and α . We describe themup to a homotopy rel f ´ p V p T qq rather than rel V p T q . Choose a path α connecting y to y so that it is disjoint from G . This is possible. In-deed, according to assumption p q made in Section 5.6, we have y P Ω .Recall also that Ω is a topological disk, and that y P Ω by definitionof y . Therefore, y can be connected with y P Ω by a path in Ω .This path is automatically disjoint from G ; and we take this path as α .The path α should be chosen so that it crosses G only once in a pointof e b . We may arrange that α is smooth and that the intersection istransverse. Since e b is not included into T ˚ , this description of α isconsistent with the earlier description.For example, in Figure 3, the path α goes from the outside of thequadrilateral c y p q c y p q to the inside. It may cross either c y p q or y p q c ; thus there are two possible choices for e b .5.5. A more precise statement of Theorem B.
In this section, werestate Theorem B more precisely and in a greater generality. Recallthat f : S Ñ S is a Thurston map of degree two. We assume that f has a dynamical tree pair p T ˚ , T q . Let E T and E T ˚ be the generating setsof π f defined as in Section 4.1. We will describe a map Σ : t , uˆ E T Ñ E T ˚ ˆ t , u . By definition, Σ p ε, r γ sq is pr α ε γ ˚ α ´ ε ˚ s , ε ˚ q , where γ ˚ is an f -pullback of γ originating at y ε and terminating at y ε ˚ . Note that ε ˚ and γ ˚ are determined by ε and γ . The map Σ defines a presentationof X f p y q .Recall our assumption on the basepoint y : the complementary com-ponent Ω of f ´ p T q containing y is bounded by S p T q and S p T q . Theorem 5.6.
We use the terminology and notation introduced above.Suppose that ε P t , u and g P E T . Then we have Σ p ε, g q “ p g ˚ , ε ˚ q , where ε ˚ and g ˚ are defined as follows. If g “ id , then ε ˚ “ ε and g ˚ “ id . Suppose now that g “ g e , where e is an oriented edge of T ofsignature p ε ` δ, ε ˚ ` δ q . Here the addition is mod 2; observe that δ and ε ˚ are determined by ε and the signature of e . If there is an orientededge e ˚ of T ˚ labeled δ that maps over e preserving the orientation,then g ˚ “ g e ˚ . If there is no such edge, then g ˚ “ id . An edge e ˚ mapping over e preserving the orientation means that f p e ˚ q is an oriented Jordan arc containing e , and that the orientationof f p e ˚ q is consistent with that of e . Note that an element g e ˚ for e ˚ P E p T ˚ q is also an element of E T ˚ . If, say, a critical point dividesan edge of T ˚ into two edges of T ˚ , then these two edges give riseto the same pair of mutually inverse elements of E T ˚ . Suppose that T ˚ “ T , then T is an invariant spanning tree for f . In this case, weobtain a finite state automaton pt , u , E T , Σ q . Theorem 5.6 gives anexplicit description of this automaton. Thus it provides a specificationof Theorem B. Corollary 5.7.
A dynamical tree pair p T ˚ , T q for f determines themap Σ : t , u ˆ E T Ñ E T ˚ ˆ t , u that provides a presentation for thebiset of f . In particular, the isomorphism class of the biset X f p y q andhence the homotopy class of f are determined by p T ˚ , T q . Note that the description provided in Theorem 5.6 depends on thechoice of pseudoaccesses. However, the end result, i.e., the map Σ : t , u ˆ E T Ñ E T ˚ ˆ t , u , is obviously independent of these choices. Example 5.8 (An automaton for the basilica polynomial) . Recall thatthe basilica polynomial is p p z q “ z ´
1. It is easy to find a presentationfor the biset of p directly (cf. [Nek05, Section 5.2.2]). However, we willuse Theorem 5.6 in order to illustrate its statement. Let T be theinvariant spanning tree for p defined in Example 2.1: ´ ‚ A (cid:47) (cid:47) ˝ B (cid:47) (cid:47) ‚ Since T is invariant, we may take T ˚ “ T . The tree T has three vertices ´
1, 0, and two edges: A “ r´ , s and B “ r , . Note that A is the Hubbard tree for p . Orient these two edges from left to right(in the picture, the orientations are represented by arrows). Observethat A maps onto A reversing the orientation, and B maps onto A Y B preserving the orientation. We may represent this symbolically as A Ñ A ´ , B Ñ A, B.
NVARIANT SPANNING TREES 35
Observe also that B contains the β -fixed point x β of p , i.e., the landingpoint of the invariant external ray (the latter ray is also a part of B ).Set a “ g A and b “ g B . Thus we have E “ E T “ t id, a, b, a ´ , b ´ u .Theoretically, we have to make some choices. Observe that the lo-cation of the basepoint is irrelevant since the complement of T in thesphere is simply connected. The only possible choice for a base edge is B since f ´ p A q Ă T . Also, we need to choose two pseudoaccesses atthe critical values v “ ´ v “ 8 . However, since both criticalvalues are endpoints of T , these pseudoaccesses are unique. In orderto implement the algorithm described in Theorem 5.6, we need to findlabels and signatures. Since B is the base edge, and B maps over B ,we have (cid:96) p B q “
0. Indeed, the edge e b of f ´ p G q not in T but mappingalso over B has label 1 by definition. The two critical pseudoaccessesat 0 separate A from B , hence we have (cid:96) p A q “
1. The boundary circuit C p T q is r A, B, B ´ , A ´ s (square brackets denote a cyclically orderedset). Here A ´ , B ´ stand for the edges A , B equipped with the oppo-site orientation. The two post-critical pseudoaccesses divide C p T q into S p T q “ p A, B q and S p T q “ p B ´ , A ´ q . By Definition 5.5, both A and B have signature p , q . The oriented edges A ´ , B ´ have signa-ture p , q . We can now compute p g ˚ , ε ˚ q for each pair p ε, g q P t , u ˆ E according to Theorem 5.6.The computations can be organized as follows. Draw the followingtable: a p , q b p , q a ´ p , q b ´ p , q a ´ p q b p q a p q b ´ p q b p q b ´ p q In the top row, we list all elements of E ´ t id u . After each element,we indicate its signature. Thus, columns of the table (except for theleftmost one) are marked by oriented edges of T . These are the A -column, then the B -column, etc. The last row is temporarily filled asfollows. In the A -column, we write all elements of the form g e , where e is mapped over A preserving the orientation. In our case, theseelements are a ´ “ g A ´ and b “ g B . We proceed similarly with othercolumns. After each element of E ´ t id u in the last row, we indicate inthe parentheses the label of the corresponding edge. Now we can fill the second and the third rows of the table. Forexample, look at the A -column. Take one of the entries in the lastrow, say, a ´ p q . Here a ´ is an element of E and 1 is the label. Addthe label to both components of the signature written in the samecolumn. In our case, we obtain p ` , ` q “ p , q . This meansthat .a “ a ´ by Theorem 5.6. We write a ´ A -column with the row marked 1. Now take the remaining entryin the last row, b p q . Adding the label to the signature, we obtain p ` , ` q “ p , q . This means that .a “ b . We write b A -column with the row marked 0.More generally, consider the column marked by an oriented edge e of T . In the first row, we indicated the signature p ε ` δ, ε ˚ ` δ q of this edge,right after the corresponding element g e . In the last row, we indicatedan edge e ˚ mapping over e in an orientation preserving fashion, and thelabel δ of e ˚ . We add δ to both components of p ε ` δ, ε ˚ ` δ q to obtain p ε, ε ˚ q . Then we have ε.g e “ g e ˚ ε ˚ by Theorem 5.6. We write g e ˚ ε ˚ atthe intersection of the e -column with the row marked ε . If there is noedge e ˚ of label δ mapping over e preserving orientation, then we write ε ˚ .Acting in this way, we obtain the following table (from which weremoved the last row as it was not needed anymore). a b a ´ b ´ b b a a ´ b ´ b ´ p ε, g e q , one has to look at the intersectionof the row marked ε P t , u with the column marked g e P E . The cell ofthe table at the given position contains g e ε or ε . In the former case, wehave Σ p ε, g e q “ p g e , ε q . In the latter case, we have Σ p ε, g e q “ p id, ε q .The following is the Moore diagram for the obtained automaton. NVARIANT SPANNING TREES 37 a p , q (cid:47) (cid:47) p , q (cid:18) (cid:18) b p , q (cid:32) (cid:32) p , q (cid:7) (cid:7) ida ´ p , q (cid:82) (cid:82) p , q (cid:47) (cid:47) b ´ p , q (cid:62) (cid:62) p , q (cid:84) (cid:84) Proof of Theorem 5.6.
Set G “ f ´ p T q . Since f p T ˚ q Ă T ,we have T ˚ Ă G set-theoretically. Moreover, all vertices of T ˚ arealso vertices of G but, in general, not the other way around. Recallthat complementary components of G correspond to boundary circuitsof G . We will write C i p G q for the boundary circuit corresponding toΩ i . Recall that the labeling of Ω i was defined so that C p G q is theconcatenation of S p T q and S p T q . The boundary circuit C p G q isthen the concatenation of S p T q and S p T q .Let e be an edge of T . Then f ´ p e q can be represented as a union e Y e , where e P E p T q and e P E p T q . The following propositionis an alternative description of the boundary circuits C i p G q , where i “ , Proposition 5.9.
Let e be an oriented edge of T of signature p i, j q .Then e belongs the boundary circuit C i p G q and e belongs to the bound-ary circuit C ´ i p G q . In other words, e δ belongs to C i ` δ p G q , where δ “ , and the addition is mod 2.Proof. Suppose that the signature of e is p i, j q . It follows by definitionof a signature that e P S i p T q . The edge e of G is a part of T hencealso of S i p T q . By definition, this means that that e P C i p G q . Theproof of the claim that e P C ´ i p G q is similar. (cid:3) We are now ready to prove Theorem 5.6.
Proof of Theorem 5.6.
Suppose that we are given ε P t , u and g P E T .If g “ id , then the conclusion is obvious. Thus we may assume that g “ g e for some e P E p T q . Let g ˚ and ε ˚ be as in the statement ofTheorem 5.6. Namely, let p i, j q be the signature of e . Set δ “ ε ` i mod 2. Then we also have i “ ε ` δ mod 2. Define ε ˚ as j ` δ mod 2.Then we also have j “ ε ˚ ` δ . Thus the signature of e can be writtenas p ε ` δ, ε ˚ ` δ q . If there is an edge e ˚ of T ˚ labeled δ that maps over e preserving the orientation, then we set g ˚ “ g e ˚ . Note that, if anedge e ˚ exists, then it is unique. Indeed, there is only one edge e δ of T δ mapping to e . This edge e δ may or may not be a subset of T ˚ . If itis, then it is contained in a unique edge e ˚ of T ˚ . We equip e ˚ with theorientation induced from the orientation of e by the map f : e δ Ñ e .Thus e ˚ is uniquely determined as an oriented edge of T ˚ . If e δ is nota subset of T ˚ , then we set g ˚ “ id .We now need to prove that Σ p ε, g q “ p g ˚ , ε ˚ q , i.e., that r α ε s .g “ g ˚ r α ε ˚ s . Let γ e be a smooth loop based at y , crossing T just oncetransversely and approaching it from the left. Thus r γ e s “ g . Bydefinition r α ε s .g is (the homotopy class of) the concatenation of α ε anda pullback γ ˚ e of γ e . The pullback γ ˚ e should start at y ε , where α ε ends.The path γ ˚ e approaches some boundary edge e of Ω ε . Thus e is an edgeof G . Equip e with an orientation such that γ ˚ e approaches e from theleft. Then e is an element of the boundary circuit C ε p G q correspondingto the boundary of Ω ε . Observe that e must be a pullback of e , henceit must coincide with e or with e . We need to find which one. ByProposition 5.9, the edge e δ belongs to C i ` δ p G q “ C ε p G q . Therefore,we have e “ e δ . Since e is of signature p ε ` δ, ε ˚ ` δ q , the two sidesof the arc e belong to Ω ε and Ω ε ˚ . Indeed, the left side of e is Ω ε , aswe already know. On the other hand, by Proposition 5.9, the oppositeedge p e q ´ belongs to the boundary circuit C j ` δ p G q “ C ε ˚ p G q . Itfollows that the right side of e is Ω ε ˚ . When crossing e , the path γ ˚ e leaves Ω ε and enters Ω ε ˚ (it may be that ε “ ε ˚ ).It follows that the path α ε .γ e terminates in Ω ε ˚ . We must have then r α ε s . r γ s “ r α ε γ ˚ e α ´ ε ˚ s r α ε ˚ s , and it remains to show that r α ε γ ˚ α ´ ε ˚ s “ g ˚ .Suppose first that e is not a subset of T ˚ (then g ˚ “ id ). Then γ ˚ e is disjoint from T ˚ . Since, by our assumption, α , α are also disjointfrom T ˚ , the loop α ε γ ˚ e α ´ ε ˚ lies entirely in S ´ T ˚ . The set S ´ T ˚ issimply connected, therefore, this loop is contractible in S ´ T ˚ and in S ´ P p f q Ą S ´ T ˚ . Thus both sides of the equality r α ε γ ˚ e α ´ ε ˚ s “ g ˚ equal id , and the equality holds.Finally, suppose that e is a subset of T ˚ . Then, since the edge e ˚ of T ˚ has label δ , we have e ˚ Ą e . The path γ ˚ e intersects G once, andapproaches e ˚ from the left. Therefore, r α ε γ ˚ e α ´ ε s “ g e ˚ , which provesthe desired. (cid:3) NVARIANT SPANNING TREES 39 The ivy iteration
We start with a geometric explanation of the process, after which weprovide a formal combinatorial implementation.6.1.
A geometric description of the iterative process.
Considera marked Thurston map f : S Ñ S of degree 2 with critical values v and v .We now describe a procedure that, given a spanning tree T for f ,allows to recover a dynamical tree pair p T ˚ , T q . Take the full preimage G “ f ´ p T q . The basic idea is to select a spanning tree T ˚ in G . Moreprecisely, we select some subtree of G containing P p f q and then erasesome of its vertices. Thus the choice of T ˚ is in general not unique.Below, we will give more precise comments on what it involves to makethis choice, in terms of combinatorics.Recall that the basepoint y is assumed to be outside of T Y G . Wealso assume that y and y are in the same component of S ´ G . Choosea base edge e b of T . As above, we assume that e b separates v from v . Having chosen a base edge e b , we can recover T ˚ . There are twopullbacks of e b in G “ f ´ p T q . We choose one of the two pullbacks e b so that the following properties hold: ‚ The edge e b of G is oriented so that f : e b Ñ e b preserves theorientation. ‚ Consider a path in S ´ G originating at y and approaching e b .This path approaches e b from the left.Recall that Z is the smallest arc in T connecting v with v . Then theJordan curve f ´ p Z q consists of two pullbacks of Z . Both pullbacks of e b are in f ´ p Z q . They are oriented both from c to c or both from c to c . Thus, one of them, e b , is oriented as the boundary of thecomponent of S ´ f ´ p Z q containing y . This shows that e b is welldefined.We can now define a spanning tree T ˚ . Clearly, f ´ p Z q is the onlysimple loop in G . Thus, removing e b from G leads to a tree. We set p T tobe the smallest subtree of this tree containing P p f q . (In particular, allendpoints of p T must be in P p f q ). Finally, define T ˚ as the tree obtainedfrom p T by erasing all vertices of p T that are not in P p f q and are notbranch points of p T . The erased vertices become points in the edges of T ˚ . Then p T ˚ , T q is a dynamical tree pair for f . By definition of labelsgiven in Section 5.2, we have (cid:96) p e b q “
1, equivalently, e b P T . By theproperties of e b listed above, the orientation of e b corresponds to theorientation of S p T q . Hence the boundary circuit C p G q corresponding to Ω is the concatenation of S p T q and S p T q . Thus our assumptionmade in Section 5.6 is fulfilled, and Theorem 5.6 applies to p T ˚ , T q .The topological ivy iteration is aimed at finding an invariant spanningtree for f , up to homotopy, or, more generally, at finding periodic (alsoup to homotopy, to be made precise later) spanning trees. Note thatan invariant (or periodic), up to homotopy, spanning tree for f yields agenuine invariant (or periodic) spanning tree for some map homotopicto f . Consider a dynamical tree pair p T ˚ , T q as above. Since there arefinitely many choices for e b , there are also finitely many choices for T ˚ . Definition 6.1 (Topological ivy object) . A (topological) ivy object isdefined as a homotopy class of spanning trees for P p f q . We will writeIvy p f q for the set of all ivy objects for f . For a fixed f with | P p f q| ě p S , P p f qq on the set of ivy objects of f . Ingeneral, this action is not transitive as, for example, spanning treesmay have different combinatorics.Note that, as a set, Ivy p f q depends only on P p f q , not on f . However,we will introduce a relation on Ivy p f q that will depend on the dynamicsof f . If T is a spanning tree for f , then r T s will denote the correspond-ing ivy object. Suppose that p T ˚ , T q is a dynamical tree pair as above.Then we will write r T s (cid:40) r T ˚ s , and call the thus defined relation onIvy p f q the pullback relation . The pullback relation on Ivy p f q can berepresented by a structure of an abstract directed graph. There arefinitely many arrows originating at each element of Ivy p f q . The setIvy p f q equipped with the graph structure just described is called the ivy graph of f .Recall that, a subset C Ă Ivy p f q is pullback invariant if the follow-ing property holds: whenever r T s P C and r T s (cid:40) r T ˚ s , we also have r T ˚ s P C . In other words, in the associated directed graph, there areno edges originating in C and terminating outside of C . Finding pull-back invariant subsets of Ivy p f q is obviously related to finding periodicspanning trees. By the way, we can now rigorously define a periodicobject τ P Ivy p f q as an element of some directed cycle in Ivy p f q . Thelength of any simple cycle containing τ is called a period of τ . (Notethat τ may have several periods according to this definition. Moreover,as a rule, periodic ivy objects do have several different periods.) If T isa spanning tree such that r T s is periodic of period p , then we say thatthe spanning tree T is periodic of period p , up to homotopy.There is no way of defining the forward image of an ivy object r T s under f simply because homotopies rel. P p f q are not preserved by f . NVARIANT SPANNING TREES 41
This is also confirmed by the existence of a periodic ivy object withseveral different periods.6.2.
A formal description of the ivy iteration.
The purpose ofthis section is to give a compact and precise description of the com-putational scheme. The scheme has been implemented (as a WolframMathematica code) according to this description. Motivations and geo-metric explanations are given above. However, we will still need somework to relate the geometric story to the combinatorial story.
Push forward of a generating set.
Let X be an abstract left free bisetover a group π . Consider a basis B of X and a symmetric generating set E of π . (Recall that symmetric means that g ´ P E for every g P E ). Wewill assume that 1 P E . Consider the full automaton Σ : B ˆ π Ñ π ˆ B of X . Set Σ “ p σ, ι q , so that Σ p a, g q “ p σ p a, g q , ι p a, g qq for all a P B , g P π . Define the push forward P E of E as the set σ p B , E q . In otherwords, P E consists of all elements of the form σ p a, g q , where a P B and g P E . Note also that the set P E coincides with the set of all restrictions(sections) of generators from E under the associated wreath recursion,in Nekrashevych’s [Nek05] terminology. It is easy to see that P E isalso a symmetric set containing 1. Note that a push forward of E willcorrespond to a pullback of a tree and, in our case, it will also be agenerating set. For our purposes, we need to combine the push forwardoperation with a change of a basis. Push forwards with simultaneous basis changes.
As before, X is anabstract left free biset over a group π . Let B be a basis of X . Anyfunction λ : B Ñ π defines a basis change: the new basis consists of λ p a q a , where a P B . Denote this basis change by C λ . We will alsowrite a λ for λ p a q a . The new basis obtained from B through C λ will bedenoted by C λ B . Let C λ Σ be the full automaton of X in the new basis C λ B . We will now see how the maps Σ and C λ Σ are related to eachother. The following simple computation solves the problem. Take any a P B and any g P π , and set p g ˚ , a ˚ q “ Σ p a, g q . Then the followingequalities hold in X : a λ .g “ λ p a q a.g “ λ p a q g ˚ a ˚ “ λ p a q g ˚ λ p a ˚ q ´ a ˚ λ . This computation shows that C λ Σ “ p C λ σ, C λ ι q is given by C λ σ p a λ , g q “ λ p a q g ˚ λ p a ˚ q ´ , C λ ι p a λ , g q “ a ˚ λ . Given a symmetric generating set E of π containing 1, set P λ E to bethe generating set obtained as the push forward of E under C λ Σ. Push forwards of tree structures.
In contrast to the above, we nowexplicitly assume that B consists of two elements. As always, we assumethat E is symmetric and contains 1. We say that E is tree-like if thereexists a tree structure V on E , cf. Section 4.2. Fix a tree-like generatingset E . Below, we will describe a certain set of basis changes λ for which P λ E are also tree-like. For each such λ , the corresponding tree structure P λ V on P λ E is defined below. The map Σ : B ˆ E Ñ P E ˆ B can beextended to a mapΣ ‹ “ p σ ‹ , ι ‹ q : B ˆ E ‹ Ñ p P E q ‹ ˆ B . The map Σ ‹ is defined inductively as follows. If ∅ denotes the emptyword in E ‹ , then we set Σ ‹ p a, ∅ q “ p ∅ , a q . Suppose now that anelement of E ‹ has the form g ¨ w , where g P E and w P E ‹ . Set p g ˚ , a ˚ q “ Σ p a, g q . Then we set σ ‹ p a, g ¨ w q “ g ˚ σ ‹ p a ˚ , w q , ι ‹ p a, g ¨ w q “ ι ‹ p a ˚ , w q . Replacing B with C λ B , we may assume that λ ”
1. Suppose that B “ t a, b u . In order to define the new vertex set P V Ă p P E q ‹ , we firstconsider the following three sets: V p a q “t σ ‹ p a, v q | v P V , ι ‹ p a, v q “ a u , V p b q “t σ ‹ p b, v q | v P V , ι ‹ p b, v q “ b u , V p a, b q “t σ ‹ p a, v q ¨ σ ‹ p b, v q | v P V , ι ‹ p a, v q “ b u . Now take the union of these three sets, remove the trivial element1 P E ‹ form it as well as all elements of the form g ¨ g ´ , where g P E .The remaining set is P V . The combinatorial ivy iteration.
Thus, for every tree-like generatingset E of π , there are several tree-like generating sets of the form P λ E .Set B “ t a, b u . We will only consider basis changes associated withfunctions λ : B Ñ π such that λ p a q “
1. This is equivalent to sayingthat the first basis element a will be fixed once and for all. In ourimplementation, this will be the class of a constant loop.Suppose that g P E is an element with the property that ι p , g q “ g is called a base element . With any base element g , we associate the function λ g : B Ñ π such that λ g p b q “ σ p , g q .(Recall that λ p a q “ combinatorial ivy iteration is the processof passing from E to P λ g E .Define a combinatorial ivy object as a tree-like generating set, up toconjugacy. More precisely, a combinatorial ivy object is a conjugacyclass of tree-like generating sets. If E is such a generating set, thenits conjugacy class will be denoted by r E s . Let Ivy c p f q be the set of NVARIANT SPANNING TREES 43 all combinatorial ivy objects with π “ π f . For a pair of generatingsets E and E “ P λ g E as above, connect r E s with r E s by a directededge. Each pair pr E s , r E sq yields only one directed edge, no matter inhow many ways r E s can be represented in the form r P λ g E s . In this way,Ivy c p f q becomes a directed graph. We will show that the combinatorialivy iteration represents the topological ivy iteration. In particular, thegraph Ivy p f q is isomorphic to a subgraph of Ivy c p f q . A more precisestatement is given in the following theorem: Theorem 6.2.
There is an isomorphic embedding of
Ivy p f q into Ivy c p f q .This embedding takes a class r T s of a spanning tree T to the conjugacyclass of the corresponding generating set E T . Let T and a base edge e b of T define a dynamical tree pair p T ˚ , T q as in Section 6.1. If g is theelement of E T corresponding to e b , then T ˚ corresponds to P λ g E T . Thereis a canonical tree structure V on E T such that G p V q is isomorphic to T . The tree structure V ˚ on E T ˚ corresponding to T ˚ is obtained as P λ g V . Translation from geometry to combinatorics.
Proposition4.3 implies that there is at least a set-theoretic embedding of Ivy p f q into Ivy c p f q . Our symbolic implementation of the ivy iteration willrely on the following assumption. Assumption 6.3.
Whenever we consider a spanning tree T for P p f q ,we assume that there is a fixed point y of f outside of T . Moreover, aslong as we deal with spanning trees that eventually map to T , the point y is kept the same. The point y will be used as the basepoint for π f . Assumption 6.3 can always be fulfilled if we replace f with a homo-topic Thurston map. Indeed, choose a point y R T with f p y q R T anda path β connecting f p y q to y outside of T . Then the assumption issatisfied if we replace f with σ β ˝ f . Here σ β is a path homeomorphismintroduced in Section 2.3. Now, if y is fixed under f , then y is disjointfrom f ´ n p T q for all n ě
0. In particular, if p T ˚ , T q is a dynamical treepair for f , then y P S ´ T ˚ . Thus we can keep the same f -fixed basepoint during the ivy iteration. The basis change associated with a dynamical tree pair.
Consider aspanning tree T for f , and choose a base edge e b of T . The choice of e b defines a dynamical tree pair p T ˚ , T q as in Section 6.1. Considerthe basis B “ tr α s , r α su of X f p y q associated with T . According tothe convention introduced in Section 5.4, the element r α s is the classof the constant path. The path α connects y with another preimage y of y outside of T . This property defines B uniquely. Let now B ˚ “ tr α s , r α ˚ su be the basis associated with T ˚ . We need to showthat r α ˚ s “ σ p , g qr α s , where g is the element of E T corresponding to e b . Then we will have B ˚ “ C λ g B . This is a part of the correspondencebetween the geometric and the combinatorial ivy iterations.Set h “ σ p , g q . Let e b be the edge of f ´ p T q of label 1 that is notin T ˚ . Recall that α ˚ is defined as a path from y to y crossing e b and disjoint from f ´ p T q otherwise. In Section 6.1, we have chosen e b so that α ˚ approaches it from the left. It follows that r α s .g “ r α ˚ s in X f p y q . Thus we can define α ˚ as α .γ e b . On the other hand, we haveΣ p , g q “ p h, q , therefore, r α ˚ s “ r α s .g “ h r α s , as desired.6.4. Pullback and vertex words.
In this section, we study the effectof the pullback relation on vertex words.
Proposition 6.4.
Let x be a vertex of T , and v P E ‹ be the corre-sponding vertex word. The vertex x is a critical value of f if and onlyif ι ‹ p , v q “ . In this case, we also have ι ‹ p , v q “ .Proof. Let Π : E ‹ Ñ π f be the evaluation map. Note that ι ‹ p ε, v q “ ι p ε, Π p v qq for each ε P t , u . The element Π p v q P π f is represented bya loop around x that crosses T only in a small neighborhood of x . Wemay assume that this loop γ is smooth and simple. Then it bounds adisk D such that D X V p T q “ t x u . The two f -pullbacks of γ are loopsor not loops depending on whether x is a critical value or not. On theother hand, these pullbacks are loops if and only if ι p ε, Π p v qq “ ε forall ε “ , (cid:3) Suppose now that v “ a ¨ ¨ ¨ a k ´ . Consider elements σ ‹ p , v q “ b ¨ ¨ ¨ b k ´ and σ ‹ p , v q “ c ¨ ¨ ¨ c k ´ of p P E q ‹ . Here b i and c i are ele-ments of π f , for i “ . . . , k ´
1. Set ε “ ε i “ ι ‹ p , a ¨ ¨ ¨ a i ´ q for i “ . . . , k . Similarly, we set δ “ δ i “ ι ‹ p , a ¨ ¨ ¨ a i ´ q for i “ . . . , k . Suppose first that ε k “ δ k “ w p , v q “ w p , v q P p P E q ‹ as σ ‹ p , v q σ ‹ p , v q “ b ¨ ¨ ¨ b k ´ ¨ c ¨ ¨ ¨ c k ´ . Suppose now that ε k “ δ k “ w p , v q “ σ ‹ p , v q “ b ¨ ¨ ¨ b k ´ , w p , v q “ σ ‹ p , v q “ c ¨ ¨ ¨ c k ´ . Proposition 6.5.
Suppose that the basis B of X f p y q corresponds to T ˚ . Then the tree structure V ˚ on P E corresponding to T ˚ coincideswith the set of w p ε, v q P p P E q ‹ , where ε runs through t , u , and v runsthrough V , except that we omit w p ε, v q if it is empty or it has the form a ¨ a ´ for some a P E . In other words, we have V ˚ “ P V . NVARIANT SPANNING TREES 45
Proof.
Let x be a vertex of T , and v be the corresponding vertex word.Let A , . . . , A k ´ be the oriented edges of T such that a i “ g A i for i “ . . . , k ´
1. Define β i as a smooth path disjoint from T , startingat y , and ending in a small neighborhood of x . Choose β i so that itapproaches x between A i ´ and A i . Here i ´ k so that for i “ A i ´ “ A k ´ . Choose the paths γ A i with a i “ r γ A i s as β i γ i β ´ i ` , where γ i is a short path in a small neighborhoodof x crossing A i just once and transversely. We may assume that γ i isdisjoint from all other edges of T . For ε P t , u , consider the pullback β εi of β i originating at y ε .Suppose first that x is not a critical value of f . Then there are twopreimages x and x of x . We will prove that the vertex words of x , x are w p , v q , w p , v q (not necessarily in this order), provided that x and x are vertices of T ˚ . Without loss of generality, we may assume that β ends near x . (Otherwise, simply swap x and x .) Set γ εi to be thepullback of γ i near x ε . An important observation is that a pullback of ashort path is short. Therefore, γ εi must indeed stay near one preimage of x rather than wander between the two preimages. Then the inductionon i shows that b i is represented by α ε i β ε i i γ i p β ε i ` i ` q ´ α ´ ε i ` and that all β ε i i end near x . Note also that b i “ id if and only if γ i does not crossany edge of T ˚ . Otherwise it crosses exactly one edge. The compositionof all γ i is a small loop around x . Moreover, this loop is a pullback ofthe small loop γ around x , which is the composition of all γ i . It followsthat the oriented edges of T ˚ coming out of x correspond precisely tonon-identity elements b i . This means that w p , v q is the vertex wordfor x ; the proof of w p , v q being the vertex word for x is similar.Suppose now that x is a critical value of f . Then there is just onepreimage x “ x of x . We will prove that w p , v q “ w p , v q is thevertex word for x “ x provided that x is a vertex of T ˚ . Set γ i be thepullback of γ i originating where β ε i i ends, and γ i be the pullback of γ i originating where β δ i i ends. Since δ i ‰ ε i , the paths γ i and γ i are alwaysdifferent pullbacks of γ i . Similarly to the above, b i is represented by α ε i β ε i i γ i p β ε i ` i ` q ´ α ´ ε i ` and c j is represented by α δ j β δ j j γ j p β δ j ` j ` q ´ α ´ δ j ` .The composition of all γ i is a pullback of γ but it is not a loop; it isonly a “half” of a loop. The other half is the composition of all γ i ,which is also the other pullback of γ . It follows that oriented edges of T ˚ coming out of x correspond precisely to non-identity elements b i or non-identity elements c j . This means that w p , v q “ w p , v q is thevertex word for x .To conclude the proof, we observe that any vertex of T ˚ is mappedto a vertex of T . Thus any vertex of T ˚ can be obtained as described above. If w p ε, v q is empty, then obviously, the corresponding point x ε of f ´ p T q does not belong to T ˚ . If w p ε, v q has the form a ¨ a ´ , then x ε belongs to an edge of T ˚ corresponding to a (thus, in particular, x ε is not a vertex of T ˚ ). Conversely, if x ε is not a vertex of T ˚ , thenthis may be due to one of the following reasons. Firstly, we may have x ε R T ˚ , then w p ε, v q is empty. Secondly, x ε may belong to some edgeof T ˚ . In this case, w p ε, v q must have the form a ¨ a ´ , where a P P E corresponds to this edge. (cid:3) Proposition 6.5 concludes the proof of Theorem 6.2.7.
Examples of the ivy iteration
In this section, we describe some particular computations of the ivygraphs made according to the ivy iteration. We consider only thesimplest examples, for which other, sometimes more efficient, compu-tational approaches to distinguishing Thurston equivalence classes areavailable. In particular, in most examples, particular invariant span-ning trees are known. We find (conjecturally) all periodic spanningtrees in these examples. Also, we find some pullback invariant sets ofivy objects and show their combinatorial structure.In [KL18], all non-Euclidean Thurston maps with at most 4 post-critical points are classified, and an algorithm is suggested for solvingthe twisting problem for such maps. However, invariant spanning treesfor rational maps from [KL18] are not immediate from the provideddescription.More complicated examples will be worked out in a separate publi-cation.7.1.
The basilica polynomial.
Let us go back to Example 5.8. Thisexample deals with the basilica polynomial f p z q “ z ´
1. We startedwith an invariant spanning tree ´ ‚ A (cid:47) (cid:47) ˝ B (cid:47) (cid:47) ‚ and deduced the corresponding presentation of the biset X f p y q , seeFigure 4. This is enough to start the combinatorial ivy iteration. Thisprocess leads to a pullback invariant subset of Ivy c p f q consisting of 3combinatorial ivy objects. Two of these objects correspond to invariantspanning trees.The two (up to homotopy) invariant spanning trees of f are easy todescribe. The first one is the Hubbard tree connected to infinity asdescribed in Section 2.1. The second one is a spider in the sense ofHubbard–Schleicher [HS94]. There is only one remaining ivy object in NVARIANT SPANNING TREES 47
Figure 4.
A pullback invariant subset of 3 elements in Ivy p f q ,where f p z q “ z ´ T . Vertex2 corresponds to an invariant spider for f . Vertex 3 corresponds toa spanning tree that is not invariant up to homotopy; it is periodicof period 2. Note that vertex 1 has periods 1 and 2. the given pullback invariant subset of Ivy p f q . It does not correspondto an invariant spanning tree for f .Note that Ivy p f q consists of 5 objects. These objects are the samefor all f with a given post-critical set P p f q of 3 elements. However,the pullback relations (in particular, pullback invariant subsets) de-fined by f are different. Three of the elements of Ivy p f q correspondto the unions of two sides of the triangle with vertices in P p f q . Theremaining two elements are stars with endpoints in P p f q . The twostars differ by the cyclic order of edges at the only branch point. Re-call that Teichm¨uller theory provides powerful invariants of Thurstonequivalence classes in form of certain spaces, groups, correspondencesbetween spaces, and virtual homomorphisms between groups. How-ever, all these invariants are trivial in the case | P p f q| “ The rabbit polynomial.
The rabbit polynomial p p z q “ z ` c is such that 0 is periodic of period 3, and Im p c q ą
0. These conditionsdetermine c uniquely. Indeed, the period 3 assumption leads to a cubicequation on c , which has one real and two complex conjugate roots.An invariant spanning tree T for f constructed as in Section 2.1 looksas follows. v ‚ (cid:79) (cid:79) Bw ˝ (cid:111) (cid:111) C x α ˝ A (cid:29) (cid:29) ˝ D (cid:47) (cid:47) ‚ The choice of the basepoint y for the fundamental group π f “ π p S ´ P p f q , y q is irrelevant. Indeed, the complement of T is simply connected.The elements of π f associated with the edges of T will be denoted by a , b , c , d , so that a small letter denotes g e , where e is the edge denotedby the corresponding capital letter. Thus E T consists of id , a , b , c , andtheir inverses. The edges of T map forward as follows: A Ñ B, B Ñ C, C Ñ A, D Ñ B ´ AD.
Let us compute the map Σ : t , u ˆ E T Ñ E T ˆ t , u from Theorem5.6. We choose D as the base edge. Then we have the following labels: (cid:96) p A q “ (cid:96) p B q “ (cid:96) p C q “ , (cid:96) p D q “ . The two pseudoaccesses of T at the critical values v “ v and v “ 8 are unique. We have S p T q “ p B ´ , A, D q , S p T q “ p D ´ , A ´ , C, C ´ , B q . By Definition 5.5, the edges A and D have signature p , q , the edge B has signature p , q , the edge C has signature p , q .By Theorem 5.6, a presentation for X f p y q looks as follows a b c d a ´ b ´ c ´ d ´ d a b d c ´ d b ´ c d ´ d ´ a ´ d ´ a “ b ´ c ´ , so that it is enough to use only b , c and d as generators of π f . We will identify π f with the free group generatedby b , c , and d . Then the tree structure on E T consists of b ´ c ´ ¨ c ¨ b , cb ¨ d , b ´ , c ´ , and d ´ . Using the combinatorial ivy iteration, we founda pullback invariant subset of Ivy p p q consisting of 10 ivy objects, seeFigure 5. This is the pullback invariant subset containing the class ofthe tree T .We see that, similarly to the basilica, there are two invariant span-ning trees for f , up to homotopy, among the trees representing objectsin the found pullback invariant subset. One tree corresponds to theHubbard tree connected to . The other tree is an invariant spider.7.3. Simple capture of the basilica at ? . Let p p z q “ z ´ ? p of preperiod2: it maps to 1, and 1 maps to ´
1. It follows that the simple capture f of p at ? T : ´ ‚ A (cid:47) (cid:47) ˝ B (cid:47) (cid:47) ˝ C (cid:47) (cid:47) ? ‚ NVARIANT SPANNING TREES 49
Figure 5.
The pullback invariant subset of Ivy p f q containing r T s , where f is the rabbit polynomial, and T is the invariant span-ning tree for f obtained by connecting the Hubbard tree to .This subset consists of 10 elements. Vertex 5 represents an invari-ant spider, and vertex 3 represents T . We oriented the edges of T from left to right, and labeled them A , B , C . The corresponding symmetric generating set of π f is E T “t , a ˘ , b ˘ , c ˘ u , where a “ g A , b “ g B , c “ g C .Since ´ ? T , there areunique pseudoaccesses at ´ ?
2. As the base edge of T , we take C . Note that 0 is the only critical point in T , thus it separates edgesof different labels. We may assume that (cid:96) p A q “ , (cid:96) p C q “ (cid:96) p B q “ G “ f ´ p T q mapping onto C . They are separatedby a critical point mapping to ?
2. One or the other assignment of labelsdepends on which of the two edges is chosen as e b . The latter, in turn,depends on which complementary component of G contains the point Figure 6.
The pullback invariant subset of 40 elements inIvy p f q containing r T s . Here f is a simple capture of the basil-ica at ?
2, and T is the invariant spanning tree for f introducedin Section 7.3. Vertices 3, 35 and 38 represent invariant spanningtrees for f . y . The oriented edges A , B , C have signature p , q . The oppositeoriented edges A ´ , B ´ , C ´ have signature p , q .By Theorem 5.6, the biset of f is represented as follows: a b c a ´ b ´ c ´ a ´ b ´ c ´ b c a p f q of order 40 containing r T s . Within this subset, thereare three invariant spanning trees for f , up to homotopy, see Figure 6.Vertex 3 corresponds to the invariant spanning tree ´ ‚ ˝ ˝ ? ‚ Vertex 35 corresponds to the invariant spanning tree ´ ‚ ˝ ˝ ? ‚˝ NVARIANT SPANNING TREES 51
Finally, vertex 38 corresponds to the invariant spanning tree ˝ ´ ‚ ˝ ˝ ? ‚ A capture of the Chebyshev polynomial.
Finally, we con-sider an example, where an invariant spanning tree is not known apriori. Namely, we take a simple capture of the Chebyshev polynomial p p z q “ z ´ p , namely, ˘?
2. We restrict our attention to asimple capture of p at ?
2. There are two simple captures of p at ? β u and β d (“u” and “d” are from “ u p”and “ d own”). We may define β u as a path along the external ray ofargument , and β d as a path along the external ray of argument .Clearly, any other simple capture path for p ending at ? β u or β d relative to the set t? , , ´ , u (which is the post-criticalset of the captures). The extended Hubbard tree T with vertices in t? , , ´ , u is the following: ´ ‚ ˝ ? ‚ ˝ (We have marked post-critical points of the capture rather than of p .)Consider f u “ σ β u ˝ p . To lighten the notation, we will write σ u instead of σ β u . The full preimage G u “ f ´ u p T q can be obtained as thefull preimage under p of σ ´ u p T q . The tree σ ´ u p T q can be represented(up to homotopy) as follows. Comparing σ ´ u p T q to T : the edge r´ , s is preserved in σ ´ u p T q . The edges r , ? s and r? , s are replaced withthe external rays of arguments and 0, respectively. See Figure 7, topright, for an illustration of σ ´ u p T q .Next, G u is obtained as the full p -preimage of σ ´ u p T q , see Figure7, bottom left. The graph G u consists of the line segments r´? , s , r , ? s , and the external rays R p p q , R p p q , R p p q , R p p q . Here R p p θ q stands for the external ray of argument θ in the dynamical plane of p .We want to find a spanning tree T ˚ Ă G u so that to make p T ˚ , T q intoa dynamical tree pair.Denote the edges of T as A , B , C and orient them as in Figure 7, topleft. Then Z “ A Y B is a simple arc in T connecting the two criticalvalues (this is consistent with the meaning of the symbol Z in Sections5.2 and 6.1). The full preimage f ´ u p Z q is the simple closed curveconsisting of the segments r´? , s , r , ? s , the rays R p p q , R p p q ,and the point . We need to chose an edge e b in f ´ u p Z q that will not Figure 7.
The simple capture f u “ σ u ˝ p of p p z q “ z ´
2. Copies of the sphere are represented as disks, in which theboundary circles are assumed to be collapsed. Thus, in each of thefour pictures, one should think of the entire boundary circle as onepoint. be included into T ˚ . Take e b to be the edge that goes along the ray R p p q . Then the segment r´? , s should also be removed from T ˚ since ´? P p f u q . Thus T ˚ is as shown in Figure 7, bottomright. Denote the edges of T ˚ by A ˚ , B ˚ , C ˚ , D ˚ and orient them asin Figure 7, bottom right. The edges of T ˚ map over the edges of T asfollows: A ˚ Ñ A, B ˚ Ñ B, C ˚ Ñ C ´ , D ˚ Ñ C. Thus, in this example, every edge of T ˚ maps over just one edge of T ,which is not the case in general.We will write a , b , c for the elements of π f u corresponding to the edges A , B , C , respectively. Thus the generating set E “ E T consists of id , a ˘ , b ˘ , c ˘ . Similar convention will apply to T ˚ , so that E ˚ “ E T ˚ consists of id , a ˚ , b ˚ , c ˚ , d ˚ , and their inverses. Elements of E T ˚ areshown through their representatives in Figure 7, bottom right (dashed NVARIANT SPANNING TREES 53
Figure 8.
The pullback invariant subset of Ivy p f u q containing r T s . There are 81 objects in this subset. Vertex 44 represents aninvariant ivy object. loops). Inspecting how the dashed loops cross the edges of T , we canexpress elements of E ˚ through those of E : a ˚ “ ba ´ , b ˚ “ ca ´ , c ˚ “ ac ´ a ´ , d ˚ “ a ´ . We now compute the presentation of the biset of f u associated with p T ˚ , T q as in Theorem 5.6. To this end, we first need to choose post-critical pseudoaccesses for T . The critical values of f u are v “ ´ v “ ?
2. Since v is an endpoint of T , there is only one pseudoaccessat v . However, v admits to pseudoaccesses: one is from above, andthe other is from below. We choose the one from above. According tothis choice of pseudoaccesses, we have S p T q “ p A, B q and S p T q “p C, C ´ , B ´ , A ´ q . Therefore, the edges A and B have signature p , q ,and C has signature p , q . Next, we need to compute the labels for alledges of T ˚ . Since e b by definition has label 1, we have (cid:96) p A ˚ q “ (cid:96) p B ˚ q “ C ˚ and D ˚ since weneed to look at the critical pseudoaccesses at . The pseudoaccesses at separate B ˚ and C ˚ from D ˚ . Therefore, we have (cid:96) p C ˚ q “ (cid:96) p D ˚ q “ X f u p y q associ-ated with p T ˚ , T q : a b c a ´ b ´ c ´ a ˚ b ˚ d ˚ d ˚´
01 0 0 c ˚´ a ˚´ b ˚´ c ˚ p f q containing r T s , see Figure 8. This subset containsan invariant ivy object. Thus, we found an invariant (up to homotopy)spanning tree for f . This invariant tree is a star.7.5. Some open questions.
The following are open questions aboutthe pullback relation on spanning trees that seem important: ‚ For a quadratic rational Thurston map f , can there be an in-finite sequence of pairwise different ivy objects r T n s such that r T n s (cid:40) r T n ` s ? ‚ Is there a uniform upper bound on the number of invariantspanning trees, up to isotopy, for a quadratic rational Thurstonmap? Is there a post-critically finite quadratic polynomial withmore than two invariant ivy objects? ‚ Can there be two different pullback invariant subsets of Ivy p f q ,for a quadratic rational Thurston map f ?7.6. Acknowledgements.
The authors are grateful to D. Dudko andM. Hlushchanka for useful discussions, to D. Schleicher and Jacobs Uni-versity Bremen for hospitality and inspiring working conditions duringthe workshop “Dynamics, Geometry and Groups” in May 2017, wherethese and other enlightening discussions took place. We are also grate-ful to the referee for valuable remarks and suggestions.
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Scuola Normale Superiore, 7 Piazza dei Cavalieri,56126 Pisa, Italy
E-mail address , Anastasia Shepelevtseva: [email protected]
E-mail address , Vladlen Timorin:, Vladlen Timorin: