The Sullivan dictionary and Bowen-Series maps

The Sullivan dictionary between Kleinian groups and rational dynamics describes striking similarities between the fields, both in terms of the objects of study as well as the techniques used. We give an expository account of a recent bridge between the two sides of the dictionary by describing a framework for combining a Fuchsian group with a complex polynomial into a single dynamical system on the Riemann sphere.


Introduction
In this expository article, we draw heavily from and build upon two strands of Dennis Sullivan's work: (1) The Sullivan dictionary between Kleinian groups and rational dynamics [Sul85].
(2) The Patterson-Sullivan measure [Sul79,Sul84].We shall survey these two themes in the light of a recent combination theorem or a bridge between Kleinian groups and polynomial maps discovered by the authors [MM21].An essential ingredient in the building of this bridge is the Bowen-Series map [Bow79,BS79].
Sullivan's dictionary [Sul85, p. 405] was based on the empirical insight that Kleinian groups and rational dynamics share many common features.For instance, the limit set (resp. the domain of discontinuity) of a Kleinian group corresponds to the Julia set (resp. the Fatou set) of a rational map.Sullivan extended these similarities to a deeper similarity between techniques by introducing quasiconformal methods into the field of rational dynamics.This led to the proof of his celebrated no wandering domains theorem.In fact, using these techniques, Sullivan gave a new proof of Ahlfors' finiteness theorem along the lines of the no wandering domains theorem.
Our focus here is on a line in the Sullivan dictionary that observes the similarity between the following: (1) The Bers simultaneous uniformization theorem in Kleinian groups (2) Polynomial mating in rational dynamics, introduced by Douady and Hubbard [Dou83].The first step is to replace the Kleinian group by a single map that captures its dynamics.This brings us to the notion of a mateable map (see Section 2 below for details).With the context of mateable maps in place, we address the following question: Question 1.1.Which mateable maps and polynomials can be mated in the spirit of Douady and Hubbard?
It turns out that Bowen-Series maps [Bow79,BS79] for punctured sphere groups provide such examples.Surprisingly, there exists a new class of related maps which we call higher Bowen-Series maps that also fit the bill and give rise to combination theorems as well as 'dynamically natural' homeomorphisms between limit and Julia sets.As the name suggests, higher Bowen-Series maps are closely related to Bowen-Series maps.Indeed, higher Bowen-Series maps appear as second iterates of suitable Bowen-Series maps.Higher Bowen-Series maps can also be characterized as 'amalgams' of several Bowen-Series maps of the same Fuchsian group with overlapping fundamental domains.This part of the story is complex analytic in flavor and is taken largely from [MM21].
It is worth mentioning that examples of dynamically natural homeomorphisms between limit sets of Kleinian reflection groups (i.e., discrete subgroups of Aut( C) generated by reflections in finitely many Euclidean circles) including the classical Apollonian gasket limit set and Julia sets of anti-holomorphic rational maps were first constructed in [LLMM19], and this phenomenon was studied systematically in a general framework in [LMM22,LLM22].To the best of our knowledge, [MM21,Theorem 7.16] gives the first example of such an explicit connection between limit sets and Julia sets in the holomorphic setting.
In the last section of this survey, we turn to the measurable dynamics of mateable maps and the resulting matings.From the point of view of group theory, the measure-theoretic framework, naturally and rather appropriately, turns out to be that of Patterson-Sullivan measures.On the other hand, since mateable maps share features of rational maps, the limit set of a mateable map supports a natural dynamically defined measure: the measure of maximal entropy (the existence of a unique maximal entropy measure for a rational map was proved in [Lyu83] and independently in [Mn83,FLMn83]).The fact that a mateable map is an object halfway between groups and polynomials is reflected in close connections between maximal entropy measures of mateable maps and suitable Patterson-Sullivan measures.We conclude the article with some estimates of Hausdorff dimensions of maximal entropy measures of (higher) Bowen-Series maps and related open questions.
The phenomenon of "mating" of rational maps with Fuchsian groups was discovered in the 1990s by Bullett and Penrose in the context of iterated algebraic correspondences [BP94] and was studied comprehensively in [BL20,BL21].Specifically, they constructed a family of algebraic correspondences of bi-degree (2, 2), and showed that the members of this family can be interpreted in an appropriate sense as matings of the modular group with quadratic rational maps.This is quite different from our mating framework as we extract a non-invertible map (a mateable map) from a Kleinian group (i.e., a semi-group dynamics from the dynamics of a non-commutative group) and then combine this map with the dynamics of a polynomial thereby producing a hybrid dynamical system in one complex variable.It would be quite interesting to know if our mating framework has deeper connections with that of Bullett-Penrose-Lomonaco.

Mateability
Let Aut(D) denote the group of all conformal automorphisms of the unit disk D. A Fuchsian group Γ is a discrete subgroup of Aut(D).The aim of this section is to spell out what it means to mate a Fuchsian group with a polynomial.We provide the definition of mateability at the outset.The definition below will imply that Γ is a lattice (Lemma 2.9).Definition 2.1.A continuous map A : S 1 → S 1 is a mateable map associated with a Fuchsian group Γ if the following are satisfied: (1) A is orbit equivalent to Γ.
(2) A is piecewise analytic on S 1 .
(3) A is an expansive covering map of degree greater than one.(4) A is Markov.
(5) No periodic break-point of A is asymmetrically hyperbolic.
The failure of any of the conditions in Definition 2.1 provides an obstruction to mateability.Somewhat surprisingly, it turns out that these necessary conditions are also sufficient (see Proposition 2.10).
We elaborate now on the terms used in Definition 2.1.Let A : S 1 → S 1 be a (not necessarily continuous) map.The grand orbit of a point x ∈ S 1 under A is defined as GO A (x) := {x ∈ S 1 : A m (x) = A n (x ), for some m, n ≥ 0}.
Let Γ be a Fuchsian group with limit set equal to Λ ⊂ S 1 .We say that a (not necessarily continuous) map A : S 1 → S 1 is orbit equivalent to Γ on Λ if for every x ∈ Λ, Γ • x = GO A (x).
A (not necessarily continuous) map A : S 1 → S 1 is piecewise Möbius if there exist k ∈ N, closed arcs I j ⊂ S 1 , and g j ∈ Aut(D) for j ∈ {1, • • • , k}, such that (2) int I m ∩ int I n = ∅ for m = n, and (3) A| Ij = g j .
A piecewise Möbius map A as above is called piecewise Fuchsian if g 1 , • • • , g k generate a Fuchsian group, which we denote by Γ A .If the maps g j are assumed only to be complex-analytic in some small neighborhoods of I j (without requiring them to be Möbius), then f is said to be piecewise analytic.
The maps g j will be called the pieces of A. We shall occasionally refer to the domains I j of g j also as pieces of A when there is no scope for confusion.
Remark 2.2.We think of the partition of S 1 into the closed arcs {I j } as a part of the data of the piecewise Möbius/analytic map A. This can be formalized by defining a piecewise Möbius/analytic map A as a pair {g j } k j=1 , {I j } k j=1 .Lemma 2.3 below upgrades the regularity of A considerably.
Lemma 2.3.[MM21, Lemma 2.8] Let A : S 1 → S 1 be a (not necessarily continuous) piecewise analytic map that is orbit equivalent to a finitely generated Fuchsian group Γ.Then, A is piecewise Fuchsian, and the pieces of A form a generating set for Γ.
Suppose that x 1 , • • • , x k are a cyclically ordered collection of k points on S 1 defining the pieces I j = [x j , x j+1 ] of A (j + 1 taken modulo k).We shall say that A is minimal, if the decomposition of S 1 given by x 1 , • • • , x k is minimal; i.e., there does not exist i and h . Thus, a minimal A has no superfluous break-points.
Let A be a continuous piecewise Möbius map on the circle.Let D denote the unit disk.Let I 1 , • • • , I k be a circularly ordered family of intervals with disjoint interiors such that (1) I j ∩ I j+1 = {x j+1 } (the indices being taken mod k).
Let γ j be the semi-circular arc in D between x j , x j+1 meeting S 1 at right angles at x j , x j+1 , and let D j ⊂ D be the closed region bounded by I j and γ j .Then A, the canonical extension of A to a piecewise Möbius map in D is defined on ∪ j D j as A = g j on D j .Set D := ∪ j D j and call D the canonical domain of definition of A. Let R = D\D.We shall call R the fundamental domain of A, as well as the fundamental domain of A. Each bi-infinite hyperbolic geodesic contained in the boundary ∂R will be called an edge of R. The ideal vertices of R will be called the vertices of R. Let S be the set of vertices of R. A pair of non-adjacent points in S, or equivalently the bi-infinite geodesic joining them in R will be called a diagonal of R.
Remark 2.4.We note that the fundamental domain of a piecewise Fuchsian map A may not be a fundamental domain for the Fuchsian group Γ A generated by the pieces of A (see Subsection 4.1).
Remark 2.5.Let A : S 1 → S 1 be a continuous piecewise Möbius map with pieces A continuous map f : S 1 → S 1 is said to be expansive if there exists δ > 0 such that for any a = b ∈ S 1 , there exists n ∈ N such that d(f n (a), f n (b)) > δ.
We endow S 1 with the counter-clockwise orientation.For a, b ∈ S 1 , we denote the counter-clockwise arc of S 1 connecting a, b by > ab.Suppose that y 0 is a periodic point of period n of a piecewise Möbius covering map A : S 1 → S 1 .Then, A n is orientation-preserving, and it maps an arc of the form > y 1 y 0 to an arc of the form > y 2 y 0 .We define the one-sided multipliers of A at y 0 to be the one-sided derivatives of A n : where y = y 0 is any point on S 1 .See [MM21, Section 2] for properties of one-sided multipliers of A.
Let x be a periodic point (of period n) of a piecewise Möbius, expansive circle covering A. Then x is said to be parabolic on the right (resp., on the left) if x is said to be a periodic point of mixed type if it is hyperbolic on one side, but parabolic on the other.
Lemma 2.6.[MM21, Lemma 2.15] Let A : S 1 → S 1 be a piecewise Fuchsian expansive covering map having x 1 , • • • , x k as the break-points of its piecewise definition.Further, let x j be a periodic point of A. Then, x j is not of mixed type.
Definition 2.7.Let X be a topological space and f : X → X be a continuous map.
It is well-known that continuous, open and distance expanding self-maps of compact metric spaces admit Markov partitions (see [PU10,§3]).In particular, the polynomial map z → z d , restricted to the unit circle S 1 , admits a Markov partition (in fact, explicit Markov partitions for z d can be easily constructed).
We call A : S 1 → S 1 a piecewise Fuchsian Markov map if it is a piecewise Fuchsian expansive covering map (of degree d at least two) such that the pieces I j (intervals of definition) of A in S 1 give a Markov partition for A : S 1 → S 1 .The restrictions A| Ij = g j (∈ Γ A ) of A to I j will be referred to as pieces of A.
By the Markov property of A, each interval I j has exactly d pre-images under A. This gives us a natural transition matrix for A −1 given by a jl = 1 if there exists a point in the interior of I l mapped to I j under A, and a jl = 0 otherwise.Further, there is a naturally associated topological Markov chain, which we now describe (compare [Ser86,ECH+92]).We construct a d−regular directed graph G with k vertices (one for each I j ) and a directed edge from vertex j to vertex l if and only if a jl = 1.Further, we label such a directed edge from j to l by g −1 l (since the piece of A on I l is g l , the inverse branch from I j to I l is g −1 l ).Note that there are exactly d branches of A −1 at each interior point of an I j and any such branch is given by the inverse of one of the pieces of A; i.e., for each piece g i of A, g −1 i is a label of some edge of G and each label of an edge of G is of this form.
We now follow a point z ∈ S 1 under backward iteration of A. Let {z = z 0 , z 1 , • • • } be a (finite or infinite) sequence of points in S 1 such that A(z i+1 ) = z i .Then any such sequence encodes a geodesic in G; i.e., an isometric immersion of an interval [0, a], or [0, ∞) into G such that [i, i + 1] maps isometrically to an edge of G labeled by (the unique) g satisfying by the following: (1) z i ∈ I j(i) .
(3) A restricted to The labeled directed graph G (also known as a topological Markov chain) imposes a structure akin to that of an automatic group [ECH+92] on backward orbits of points via backward orbits of intervals I j .Thus, a sequence of backward orbits of an interval I j may be given by ).This sequence is also encoded by the same geodesic in G described above, since the pair {I j(i) , I j(i+1) } corresponds to a unique edge in G, and the label on the edge is the unique g ∈ Γ such that g −1 is a piece of A satisfying I j(i) ⊂ g −1 (I j(i+1) ).
A more concise version of Definition 2.1 can now be furnished by saying that Definition 2.8.A piecewise Fuchsian Markov map A : S 1 → S 1 is mateable if A is orbit equivalent to the Fuchsian group Γ A generated by its pieces, and none of the periodic break-points of A are asymmetrically hyperbolic.
We finally provide the lemma promised before Definition 2.1 guaranteeing that Γ A is a lattice.Lemma 2.9.[MM21, Lemma 2.18] If A is mateable, then Γ A is a lattice (or equivalently, Γ A is a finitely generated Fuchsian group such that D/Γ A has finite hyperbolic area).
For a complex polynomial P , its filled Julia set K(P ) is the completely invariant set of all points whose forward orbits (under P ) stay bounded.A polynomial is said to be hyperbolic if each of its critical points converges to an attracting cycle under forward iteration.The set of all hyperbolic polynomials (of a given degree) is open in the parameter space.A connected component of degree d hyperbolic polynomials is called a hyperbolic component in the parameter space of degree d polynomials.The hyperbolic component of degree d polynomials containing the map z d is called the principal hyperbolic component, and is denoted by H d .The filled Julia set of each map in H d is a quasidisk, and the dynamics of such a map on its Julia set is quasisymmetrically conjugate to the action of z d on S 1 .
The next proposition says that the conditions of Definition 2.8 are sufficient to guarantee conformal mateability of piecewise Fuchsian Markov maps and polynomials in principal hyperbolic components.
For a Jordan curve J on the Riemann sphere, we denote its complementary components by D in and D out .The canonical extension A : D → D of a mateable map is said to be conformally mateable with a polynomial P in a principal hyperbolic component if there exist a holomorphic map F defined on a subset of C, a Jordan curve J ⊂ Dom(F ), and a pair of conformal maps φ in : D → D in and φ out : K(P ) → D out that conjugate A and P (respectively) to F .The following is the first main result of [MM21].
Remark 2.11.A mateable map may have parabolic fixed points on S 1 , and hence the topological conjugacy between z d and A is not necessarily quasisymmetric.This renders classical quasiconformal tools (such as the ones used in the proof of Bers simultaneous uniformization theorem) insufficient for the purpose of conformally mating polynomials with mateable maps associated with Fuchsian groups.However, an appropriate class of 'generalized quasiconformal maps', called David homeomorphisms (maps with suitable Sobolev regularity satisfying a quantitative control on the area of the region where the dilatation blows up), allows one to perform the conformal mating construction.Two results that lie at the analytic heart of the proof of Proposition 2.10 are the David integrability theorem (this can be seen as a generalization of the measurable Riemann mapping theorem, see [Dav88], [AIM09, Theorem 20.6.2]) and a David extension theorem for certain circle homeomorphisms (which plays the role of the Ahlfors-Beurling extension theorem in the current setting, see [LMMN20, Theorem 4.9]).In fact, Item 5 in Definition 2.1 is required to guarantee the existence of a David extension of a circle homeomorphism conjugating z d to a mateable map.
3. Bowen-Series maps of Fuchsian punctured sphere groups 3.1.Bowen-Series maps for General Fuchsian groups.Archetypal examples of piecewise Fuchsian Markov maps of the circle that are orbit equivalent to finitely generated Fuchsian groups are given by Bowen-Series maps.These first appeared in the work of Bowen and Series [Bow79,BS79].
A finitely generated Fuchsian group Γ (of the first kind) admits a fundamental domain R (⊂ D) that is a (possibly ideal) hyperbolic polygon.Denote the edges of R by {s i } n i=1 (labeled in counter-clockwise order around the circle).Each edge s i of R is identified with another edge s j by a corresponding element h(s i ) ∈ Γ.The set {h(s i )} n i=1 forms a generating set for Γ.Let C(s i ) be the Euclidean circular arc in D containing s i and meeting S 1 orthogonally.Further, let N be the net in D consisting of all images of edges of R under elements of Γ.The fundamental domain R is said to satisfy the even corners property if C(s i ) lies completely in N , for i ∈ {1, • • • , n}.Definition 3.1 (Bowen-Series map).Suppose that a fundamental domain R of Γ satisfies the even corners property.Label (following [BS79]) the endpoints of C(s i ) on S 1 , P i , Q i+1 (with Q n+1 = Q 1 ) with P i occurring before Q i+1 in the counterclockwise order.These points occur along the circle in the order 1).The Bowen-Series map A Γ,BS : S 1 → S 1 of Γ (associated with the fundamental domain R) is defined piecewise as A Γ,BS ≡ h(s i ), on the sub-arc [P i , P i+1 ) of S 1 (traversed in the counter-clockwise order).Proposition 3.2.[BS79, Lemma 2.4] The map A Γ,BS is orbit equivalent to Γ, except (possibly) at finitely many points modulo the action of Γ.
We shall simply denote A Γ,BS by A Γ .The Bowen-Series maps corresponding to Fuchsian groups uniformizing positive genus surfaces (possibly with punctures) are discontinuous.Let us illustrate this with two examples.In the left diagram in Figure 1, R is a fundamental domain for a (closed) genus two surface where the color coding determines the side-pairings.Note that h(s 1 )(x) = y, and h(s 2 )(P 2 ) = Q 5 .Thus, for continuity of the corresponding Bowen-Series map at P 2 , the map h(s 1 ) must send the geodesic ray from x to P 2 to the geodesic ray from y to Q 5 .But the former ray lies in the net N (by the even corners property), while the latter ray passes through int R.This is absurd as R is a fundamental domain, proving discontinuity of the Bowen-Series map at P 2 .In the right diagram in the above figure, R is a fundamental domain for a once punctured torus where the sides are paired according to their colors.The side-pairing transformations h(s 1 ) maps P 2 to P 3 , while h(s 2 ) carries P 2 to P 1 .This causes discontinuity of the associated Bowen-Series map at P 2 .
Thus, to get continuous Bowen-Series maps, we need to restrict our attention to punctured sphere groups (possibly with orbifold points) equipped with special fundamental domains.In fact, it turns out that the Bowen-Series maps of Fuchsian punctured sphere groups constructed below are coverings of S 1 with degree at least two.3.2.Bowen-Series maps for punctured spheres.We mention at the outset that we always associate Bowen-Series maps with Fuchsian groups decorated with preferred fundamental domains and side-pairing transformations.
We first construct a specific Fuchsian group G d uniformizing a (d + 1)−times punctured sphere equipped with a preferred fundamental domain.The group G d (equipped with the preferred fundamental domain) will serve as a base-point in the Teichmüller space of (d + 1)−times punctured spheres.Since any (marked) group Γ ∈ Teich(G d ) is conjugate to G d via a quasiconformal homeomorphism of C, the Bowen-Series map of Γ equipped with a marked fundamental domain determined by the quasiconformal conjugacy is easily seen to be a quasiconformal conjugate of the Bowen-Series map of G d .
Fix d ≥ 2. For j ∈ {1, • • • , d}, let C j be the hyperbolic geodesic of D connecting p j := e πi(j−1)/d and p j+1 := e πij/d , and C −j be the image of C j under reflection in the real axis.We further denote the complex conjugate of p j by p −j , j ∈ {2, • • • , d}.Choose a Möbius automorphism g j of D defined as reflection in C j followed by complex conjugation.By construction, g j carries C j onto C −j (cf. Figure 2).Note that for j ∈ {1, • • • , d−1}, the Möbius map g j+1 g −1 j is the composition of reflections in the circular arcs C j+1 and C j .Since C j and C j+1 touch at p j+1 , a straightforward computation (using the formula of circular reflections) shows that g j+1 g −1 j fixes p j+1 and has derivative equal to one at this fixed point.Therefore, g j+1 g −1 j is parabolic with its unique fixed point at p j+1 .Likewise, the maps g 1 , g d fix p 1 , p d+1 (respectively), and have derivative equal to one there.Thus, g 1 , g d are also parabolic with their unique fixed points at p 1 , p d+1 , respectively.Let (1) For d ≥ 2, the Bowen-Series map We refer the reader to [MM21, Propositions 3.4, 3.5] for an orbifold variant of Proposition 3.3.Remark 3.4.In the above examples, the chosen fundamental domains of the groups coincide with those of the corresponding Bowen-Series maps.
3.3.Mateability of Bowen-Series maps.We note now that Bowen-Series maps for punctured spheres fit into our mating framework.Recall that H k stands for the principal hyperbolic component in the space of degree k polynomials.

Folding and higher Bowen-Series maps for Fuchsian groups
The aim of this section is to describe a new class of piecewise Fuchsian Markov maps (following [MM21]), beyond the Bowen-Series examples that are mateable with polynomials.We start with classes of maps that we shall be considering in this section.Recall that the fundamental domain of a piecewise Fuchsian Markov map A is denoted by R. The set D = D \ R is the canonical domain of definition of A in D, and a bi-infinite geodesic in R joining a pair of non-adjacent vertices of R is called a diagonal of R.
It is instructive to go through the following two definitions in conjunction with the two explicit examples of piecewise Fuchsian Markov maps given in Subsection 4.1 (cf. Figure 3).

Definition 4.1 (Completely folding map). A piecewise Fuchsian Markov map
A : S 1 → S 1 is said to be a completely folding map if there exist finitely many diagonals δ 1 , • • • , δ l of R such that the following hold: (1) For every edge (2) The ideal endpoints p i , q i of δ i are fixed points of A for all i; i.e., A(p i ) = p i and A(q i ) = q i whenever p i , q i are ideal endpoints of δ i .(3) For p i , q i as above, q i = p i+1 .(4) δ i ∩ δ j = ∅ for i = j.Further, p 1 = q l ; i.e., the sequence of diagonals δ i forms a chain of non-intersecting bi-infinite geodesics such that, after adjoining the ideal endpoints, one obtains a 'piecewise geodesic' embedding of the closed interval [0, 1] in the closed disk D.

Definition 4.2 (Higher degree map without folding). [MM21, Definition 4.2]
A piecewise Fuchsian Markov map A : ) are the endpoints of α 1 (resp.α 2 ) and p, q are the endpoints of δ, then A(a 1 ) = p = A(a 3 ) and A(a 2 ) = q by continuity of A on S 1 .A piecewise Fuchsian Markov map A : S 1 → S 1 is said to be a higher degree map without folding if (1) there exists an (open) ideal polygon D ⊂ R such that all the edges δ 1 , Cyclically ordering the edges α 1 , • • • , α k of R, it follows from Definition 4.2, that under a higher degree map without folding A, consecutive edges α i , α i+1 of R go to consecutive edges of D. Note however that an counter-clockwise cyclic ordering of edges of R may be taken to a clockwise cyclic ordering of edges of D under A. In any case we have a continuous map A : ∂R → ∂D.Adjoining the ideal endpoints of R and D, A has a well-defined degree d.Further, each edge of D has exactly |d| pre-images under A since there are no folds.Also, since each δ i is a diagonal of R, we have |d| > 1.We call |d| the polygonal degree of A. (Since |d| > 1, we call A a higher degree map without folding .)Remark 4.3.A piecewise Fuchsian Markov map with a diagonal fold need not be a completely folding map ; see Subsection 4.4.2 for an example.

4.1.
A completely folding map and a higher degree map without folding for the sphere with three punctures.We now give two simple examples: a completely folding map and a higher degree map without folding which are orbit equivalent to Γ 0 corresponding to a sphere with three punctures.Then Γ 0 is isomorphic to F 2 , the fundamental group of S 0,3 (see Figure 3).We will denote a bi-infinite hyperbolic geodesic in D having its (ideal) endpoints at a, b ∈ S 1 by ab.
Fix a (closed) fundamental domain W of Γ 0 , given by an ideal quadrilateral with its ideal vertices at the fourth roots of unity (the quadrilateral 1236 in the figure).The generators of Γ 0 are given by h, g, where h takes the edge 12 to 16, g takes 32 to 36, and g −1 h is parabolic.The combinatorics in this case is relatively simple and the case-by-case analysis for proving orbit equivalence in Proposition 4.4 is easy.4.1.1.A completely folding map for S 0,3 .We shall first construct a completely folding map , and then modify the construction slightly to obtain a higher degree map without folding .We define the fundamental domain R of the completely folding map A Γ0,cfm (to be constructed) as Thus, R is the interior of the octagon 12345678 in Figure 3.We define the pieces of A Γ0,cfm as follows.In the list below, an arc will be indicated by > ij where the pair of numbers i, j are its endpoints, provided there are no other break-points of A Γ0,cfm in the arc.Otherwise, we will denote the arc by all the break-points it contains.Further the label of the arrow will denote the piece of A Γ0,cfm that takes the domain arc to the range arc.
Fundamental domains for a completely folding map and a higher degree map without folding: 3 punctures 4.1.2.A higher degree map without folding for S 0,3 .There is a higher degree map without folding naturally associated with the completely folding map above.Note that the completely folding map A Γ0,cfm is not minimal.The pieces of A Γ0,cfm for the contiguous arcs > 34 and > 45 is g −1 .Similarly, the pieces of A Γ0,cfm for the contiguous arcs > 78 and > 81 is h −1 .We define A Γ0,hBS : S 1 → S 1 to be the minimal piecewise Fuchsian Markov map agreeing with A Γ0,cfm everywhere (here 'hBS' is an acronym for 'higher Bowen-Series', the reason behind this terminology will be explained in Remark 4.8).Although A Γ0,cfm and A Γ0,hBS agree pointwise, they are formally different piecewise Fuchsian Markov maps as A Γ0,cfm has more pieces (some of which are repeated).Consequently, their canonical extensions A Γ0,cfm and A Γ0,hBS have different domains of definition.
It is easy to see that A Γ0,hBS is a higher degree map without folding .The fundamental domain R for A Γ0,hBS is the interior of the ideal hexagon 123567 contained in R. The inner domain of A Γ0,hBS is given by the ideal triangle 136 (see Definition 4.2).The pieces of A Γ0,hBS are given by the following list (note that 4, 8 are not break-points of A Γ0,hBS and hence we omit them from the notation): The polygonal degree of A Γ0,hBS is 2.
Proof.Since A Γ0,cfm and A Γ0,hBS agree as maps on S 1 , it suffices to check this for A Γ0,hBS .It is easy to see that A Γ0,hBS −grand orbits are contained in Γ 0 −orbits simply because the pieces of A Γ0,hBS are elements of Γ 0 .It therefore suffices to show that if x, y are in the same Γ 0 −orbit then they lie in the same A Γ0,hBS −grand orbit.It suffices to check this for the generators g, h and their inverses.
Let y = g.x.We want to show that x, y lie in the same grand orbit under A Γ0,hBS .Case 1: y ∈ > 32176.Then x ∈ > 32 and the piece of A Γ0,hBS restricted to > 32 is g.
Note that the branch of A Γ0,hBS restricted to > 56 is h • g −1 , and the branch of A Γ0,hBS restricted to > 12 is h.Hence, This shows that x and y are grand orbit equivalent under A Γ0,hBS .Next, if y = g −1 .x,then x = g.y and exchanging the roles of x, y in the previous paragraph shows that x, y are grand orbit equivalent under A Γ0,hBS .Finally, by the symmetry of the setup, the same argument applies to h, h −1 .
As a circle covering, the degree of A Γ0,hBS is equal to 4. This can be easily seen from the actions of the pieces of A Γ0,hBS (along with their range) listed in Section 4.1.2.Thus, we have now exhibited two different examples of piecewise Fuchsian Markov maps that are orbit equivalent to a thrice punctured sphere Fuchsian group; namely, the Bowen-Series map (of degree 3) and the higher degree map without folding A Γ0,hBS defined above (of degree 4).Moreover, the polygonal degree of A Γ0,hBS is 2, while the Bowen-Series map induces a self-homeomorphism on the boundary of its fundamental domain.4.2.Folding and higher degree maps for general punctured spheres.We follow the scheme of Section 4.1 above and generalize it to the case of S 0,k -a sphere with k punctures, k > 3. We shall use Figure 4 below as an illustration for the general case.Fix a (closed) fundamental domain of Γ 0 = G k−1 (see Subsection 3.2 for the definition of G k−1 ), given by an ideal (2k − 2)−gon W (the figure illustrates the k = 4 case).For definiteness, let us assume that the ideal vertices of W are the (2k − 2)-th roots of unity.To make the book-keeping a little easier, we modify the notation as follows.
(1) The vertices of W on the bottom semi-circle are numbered 1 wise order.(3) Between vertices i, i + 1 (and including i, i + 1) on the top semi-circle, there are 2k − 2 vertices given by the vertices of g i .W (noting that g i .W ∩ W equals the bi-infinite geodesic i(i + 1)).We label the 2k − 4 vertices strictly between i, i + 1 as {i, 2}, {i, 3}, The generators of Γ 0 are given by g 1 , • • • , g k−1 , where g i takes the edge i − (i + 1) − to the bi-infinite geodesic i(i + 1).

A completely folding map for
Figure 4. Fundamental domains for a completely folding map and a higher degree map without folding: 4 punctures As in Section 4.1, we define A Γ0,cfm in terms of its pieces as follows.Recall that > ij stands for an arc with its endpoints at the break-points i, j such that there are no other break-points of A Γ0,cfm in the arc.
and on each of the k−1 short arcs > {i, j}{i, j + 1} for i ≤ j ≤ i + k − 2 between i, i + 1, define A Γ0,cfm to be g −1 i .Then A Γ0,cfm (∪ i+k−2 j=i > {i, j}{i, j + 1}) equals the upper semi-circle between 1 and k. (Here, for notational convenience, we identify {i, 1} with i and {i, i + 2k − 2} with i + 1.) Also, for i ≤ j ≤ i + k − 2, A Γ0,cfm maps the clockwise arc from {i, j} to {i, j + 1} onto the clockwise arc from j to j + 1.We refer to the clockwise arcs from {i, j} to {i, j + 1} (for i ≤ j ≤ i + k − 2) as short folding arcs under A Γ0,cfm .
We refer to the clockwise arcs from {i, j} to {i, j + 1} (for j ≤ i − 1 or i + k − 1 ≤ j) as long folding arcs under A.
It is easy to see from the above definition that A Γ0,cfm : S 1 → S 1 is a completely folding map .As any (marked) group Γ ∈ Teich(Γ 0 ) is conjugate to Γ 0 via a quasiconformal homeomorphism of C that preserves S 1 , D and respects the markings, we define the associated completely folding map A Γ,cfm to be the conjugate of A Γ0,cfm under such a quasiconformal homeomorphism.
Remark 4.5.It is not hard to cook up other examples of completely folding maps.However, we do not know of any other completely folding map that is orbit equivalent to the Fuchsian group generated by its pieces.4.2.2.A higher degree map without folding for S 0,k .Again, as in Section 4.1, define A Γ0,hBS to be the minimal piecewise Fuchsian Markov map coinciding with A Γ0,cfm on S 1 .Denote the canonical extension of A Γ0,hBS by A Γ0,hBS , its canonical domain of definition in D by D Γ0,hBS , and the fundamental domain of A Γ0,hBS by R Γ0,hBS .Further, let D be the open ideal polygon bounded by the bi-infinite geodesics 12, 23, • • • , (k − 1)k, k1.Evidently, all the edges of D are (non-intersecting) diagonals of R Γ0,hBS , each ideal vertex of D is fixed by A Γ0,hBS , each edge of R Γ0,hBS is mapped by A Γ0,hBS to an edge of D, and A Γ0,hBS has no diagonal folds.Therefore, A Γ0,hBS is a higher degree map without folding having D as its inner domain.
Definition 4.6 (Higher Bowen-Series map).We call the piecewise Fuchsian Markov map A Γ0,hBS the higher Bowen-Series map of Γ 0 (associated with the fundamental domain W ). For any (marked) group Γ ∈ Teich(Γ 0 ), we define the higher Bowen-Series map of (the marked group) Γ to be the conjugate of A Γ0,hBS under a quasiconformal homeomorphism of C that conjugates Γ 0 to Γ (and respects the marking), and denote it by A Γ,hBS .
Clearly, the higher Bowen-Series map of each Γ ∈ Teich(Γ 0 ) is a higher degree map without folding .We refer the reader to [MM21, Proposition 5.2] for a characterization of higher Bowen-Series maps among all higher degree maps without folding.4.2.3.Connections between Bowen-Series and higher Bowen-Series maps.The next two propositions are about the relationship between Bowen-Series maps and higher Bowen-Series maps (for Γ ∈ Teich(Γ 0 )).In fact, Proposition 4.7 will give an alternative, more direct construction of the higher Bowen-Series map of Γ in terms of the Bowen-Series maps of Γ associated with various overlapping fundamental domains.
Proposition 4.7 (Characterizing higher Bowen-Series maps as piecewise Bowen-Series maps).[MM21, Proposition 4.5] Let W be a (closed) fundamental domain for a Fuchsian group Γ ∈ Teich(Γ 0 ) (uniformizing a k-times punctured sphere) which is an ideal (2k − 2)-gon.We label the ideal vertices of W as counterclockwise order, and assume that the side-pairing transformations of W (generating Γ) are given by g 1 , • • • , g k−1 , where g i takes the edge i − (i + 1) − to the edge i(i + 1).
Further, let D be the interior of the ideal polygon bounded by the bi-infinite geodesics 12, 23, • • • , (k − 1)k, k1, and P the interior of the ideal polygon bounded by the bi-infinite geodesics Conversely, a map A : S 1 → S 1 defined as in condition (2) above is a higher Bowen-Series map orbit equivalent to Γ, and the fundamental domain of A is given by Remark 4.8.The preceding description of A Γ,hBS shows that A Γ,hBS is made up of Bowen-Series maps corresponding to various (overlapping) fundamental domains of Γ.This justifies the terminology 'higher Bowen-Series maps'.
Higher Bowen-Series maps also arise as second iterates of suitable Bowen-Series maps.
Proposition 4.9 (Higher Bowen-Series as second iterate of Bowen-Series).In all cases, the second iterate of the Bowen-Series map of Γ is orbit equivalent to an index-two subgroup of Γ.
The degree of the higher Bowen-Series map as a self-covering of S 1 is (χ − 1) 2 , where χ = 2 − k is the Euler characteristic of S 0,k .
The last statement may be found in [MM21, Section 4.3.2].We do not know if higher iterates of Bowen-Series maps produce further examples of mateable maps (see Question 4.16 below).With careful combinatorial book-keeping, the arguments of the proof of Proposition 4.4 can be adapted for the general case.
Proposition 4.10 (Orbit equivalence).[MM21, Proposition 4.7] Let Γ ∈ Teich(Γ 0 ), and A Γ,cfm , A Γ,hBS be as above.Then A Γ,cfm , A Γ,hBS are orbit equivalent to Γ. 4.3.Consequences.We now discuss some consequences.4.3.1.Interpolating between completely folding maps and higher degree maps without folding.The completely folding map A Γ,cfm and the higher Bowen-Series map A Γ,hBS described in Sections 4.1 and 4.2 agree on S 1 .We denote the interior of the polygon in Section 4.2 with vertices 1, • • • , k by D. Note that D is 'half' the (closed) fundamental domain W in the sense that doubling D along the bi-infinite geodesic 1k gives W . Choose 1 = i 1 < i 2 < • • • < i l+1 = k to be a selection of vertices in clockwise cyclic order along the upper semi-circle.Let ∪ 1≤j≤l (i j , i j+1 ) = L denote a finite union of edges and diagonals of W contained in D. Let W 0 denote the part of W contained above L and let Then R L is the fundamental domain of the piecewise Fuchsian Markov map A L whose canonical extension A L has domain Note that, for all L, the map A L equals A Γ,cfm on S 1 .The map A Γ,hBS is the unique minimal representative and corresponds to the case 1 = i 1 < i 2 = k.The map A Γ,cfm lies at the other end of the spectrum, with l + 1 = k, and i j = j for j = 1, • • • , k.The maps A L are non-minimal representatives whenever l > 1. 4.3.2.Mateability of completely folding maps and higher Bowen-Series maps.We now record the fact that higher Bowen-Series maps satisfy the conditions of Definition 2.1, and hence can be conformally mated with hyperbolic complex polynomials (of appropriate degree) with Jordan curve Julia sets.
In light of Proposition 2.10, we make the following definition.Definition 4.12 (Moduli space of matings).The moduli space of matings between a topological surface Σ and complex polynomials in principal hyperbolic components consists of triples (Γ, A Γ , P ), where (1) Γ is a Fuchsian group uniformizing Σ, (2) A Γ is a minimal mateable map orbit equivalent to Γ on S 1 , and (3) P is a polynomial in a principal hyperbolic component with deg(P ) = deg(A Γ : S 1 → S 1 ).
An immediate implication of Theorems 3.5 and 4.11 is that the moduli space of matings between the topological surface S 0,k (k ≥ 3) and complex polynomials in principal hyperbolic components is disconnected.Specifically, it contains at least two components corresponding to • Bowen-Series maps associated to groups in Teich(S 0,k ) and polynomials in H 2k−3 , and • higher Bowen-Series maps associated to groups in Teich(S 0,k ) and polynomials in H (k−1) 2 .
We refer the readers to [MM21, §6.4] for further details.Yet another application of orbit equivalence between higher Bowen-Series maps and Fuchsian punctured sphere groups is the failure of orbit equivalence rigidity for Fuchsian groups (see [FW03] for general background on orbit equivalence rigidity and positive results, and [MM21, §8] for a precise statement of its failure in the Fuchsian case).

Two non-examples.
In this subsection, we will consider two modifications of higher Bowen-Series maps and show that the resulting piecewise Fuchsian Markov maps are not orbit equivalent to the groups generated by their pieces.4.4.1.A non-example without folding.The following description of the higher Bowen-Series map A Γ0,hBS on S 1 is straightforward to check from its construction (see Subsection 4.2): Γ0,BS , otherwise, where A Γ0,BS denotes the Bowen-Series map of Γ 0 associated with the fundamental domain W .
In fact, the agreement of A Γ0,hBS and A Γ0,BS on the arcs ) played an important role in the proof of orbit equivalence of Γ 0 and A Γ0,hBS (see Proposition 4.4).However, if one replaces A Γ0,BS by A 2 Γ0,BS on these arcs as well, the resulting minimal piecewise Fuchsian Markov map Proposition 4.13.[MM21, Proposition 4.9] The map B : S 1 → S 1 is not orbit equivalent to the Fuchsian group Γ 0 generated by its pieces.
Remark 4.14.The map B has no diagonal fold, but it is not a higher degree map without folding .This is because some edges of the boundary of the fundamental domain of B are not mapped to diagonals, and hence Condition (3) of Definition 4.2 is violated (see Figure 5 (left)).

4.4.2.
A non-example with folding.Yet another example of a piecewise Fuchsian Markov map that is not orbit equivalent to the Fuchsian group generated by its pieces is given by the following symmetric version of higher Bowen-Series maps.For simplicity, we illustrate the thrice punctured sphere case.
Consider the Fuchsian group Γ 0 of Section 4.1, the (closed) fundamental domain W with vertices at the fourth roots of unity, and the generators g, h that pair the sides of W (as in Figure 3).We set The piecewise Fuchsian Markov map C : S 1 → S 1 depicted in Figure 5 (right) has degree 5.It has diagonal folds in the sense of Definition 4.2, but it is not a completely folding map since Condition (4) of Definition 4.1 fails.Moreover, Proposition 4.15.The piecewise Fuchsian Markov map C : S 1 → S 1 of Figure 5 (right) is not orbit equivalent to the Fuchsian group Γ 0 generated by its pieces.
Proof.Observe that g(−i) = i, and thus the points ±i lie in the same Γ 0 −orbit.But both these points are fixed by C, and hence they cannot lie in the same grand orbit of C.
The proofs of orbit equivalence for Bowen-Series and higher Bowen-Series maps only involve looking at the first iterates of the maps.On the other hand, the orbit equivalence property is ruled out for the above two piecewise Fuchsian Markov maps simply by furnishing suitable fixed points of the maps.In general, we ask the following question.Question 4.16.Is there a general recipe to test whether a piecewise Fuchsian Markov map is orbit equivalent to the Fuchsian group generated by its pieces?

Invariant laminations and Bers boundary groups
The existence of mateable maps orbit equivalent to Fuchsian punctured sphere groups leads one to the hunt for groups on boundaries of Teichmüller spaces (of punctured spheres) that can be conformally mated with complex polynomials.Since Fuchsian realizations of Teichmüller spaces are non-compact, the aforementioned pursuit ought to be carried out on boundaries of Bers slices of Fuchsian punctured sphere groups (note that such a realization of the Teichmüller space is precompact in a suitable topology).As in the Fuchsian case, the first challenge one encounters in this program is to come up with the correct notion of 'mateable maps' for Bers boundary groups.
Let us fix a Fuchsian punctured sphere group Γ 0 equipped with a (higher) Bowen-Series map A Γ0 .We denote the Bers slice of Γ 0 by B(Γ 0 ).The map A Γ0 defines, for each (marked) group Γ in B(Γ 0 ), a piecewise Möbius Markov covering map A Γ (via quasiconformal conjugation) of the limit set Λ(Γ ) such that A Γ is orbit equivalent to Γ on Λ(Γ ).The map A Γ is the (higher) Bowen-Series map associated with the map Γ .Now let Γ ∈ ∂B(Γ 0 ).Guided by the Fuchsian situation, we call a continuous self-map A Γ : Λ(Γ) → Λ(Γ) the (higher) Bowen-Series map of Γ ∈ ∂B(Γ 0 ) if (1) A Γ is orbit equivalent to Γ, and (2) A Γ is the uniform limit of the (higher) Bowen-Series maps A Γ , as Γ ∈ B(Γ 0 ) converges to Γ in the strong topology.
The Bers density conjecture, now a theorem due to Brock-Canary-Minsky [Min10, BCM12] (see also [BB04]) states that the Bers slice is dense in the space of all Kleinian surface groups with one end carrying a fixed conformal structure.Thus, the closure of the Bers slice gives all such Kleinian surface groups.For any such group Γ on the Bers boundary, there is an end-invariant called the ending lamination-a geodesic lamination supporting a transverse measure.Further, there exists a topological semiconjugacy [Mj14a,Mj17], called a Cannon-Thurston map from the circle onto the limit set Λ(Γ) of Γ.It was shown in [Mj14b,Mj17] that the Cannon-Thurston map identifies precisely the end-points of the ending lamination.
It follows from [Min10,BCM12] (see also [BB04]) that any Bers boundary group is a strong limit of groups in the Bers slice.
n be a sequence of groups in B(Γ 0 ) (where φ n is a quasiconformal homeomorphism inducing the representation Γ 0 → Γ n ) converging strongly to Γ ∈ ∂B(Γ 0 ).By [MS13,MS17], [Mj22, Section 4.2], Cannon-Thurston maps of Γ n converge uniformly to the Cannon-Thurston map of Γ.Since the (higher) Bowen-Series map of Γ n is equal to n , by definition, the (higher) Bowen-Series map A Γ , if it exists, must be given by φ is the Cannon-Thurston map of Γ.Thus, the (higher) Bowen-Series map A Γ : Λ(Γ) → Λ(Γ), if it exists, must be semi-conjugate to the base (higher) Bowen-Series map A Γ0 : S 1 → S 1 via the Cannon-Thurston map of Γ (see [MM21, §7.1] for details):

AΓ
Put differently, in the light of [Mj14b], the existence of a (higher) Bowen-Series map A Γ requires the ending lamination L of Γ (where S 1 /L ∼ = Λ(Γ)) to be invariant under the action of A Γ0 ([MM21, Lemma 7.3]).On the other hand, such laminations are necessarily invariant under the action of Γ 0 .Simultaneous invariance of L under the group Γ 0 and the covering map A Γ0 can be thought of as a compatibility condition between Kleinian group dynamics and polynomial dynamics, which turns out to be very restrictive.
Theorem 5.1 (Bers boundary (higher) Bowen-Series maps are sparse).[MM21, Propositions 7.6, 7.8] Let Γ 0 be a punctured sphere Fuchsian group.Then, there are only finitely many quasiconformal conjugacy classes of groups Γ ∈ ∂B(Γ 0 ) for which the Cannon-Thurston map of Γ semi-conjugates the (higher) Bowen-Series map of Γ 0 to a self-map A Γ of Λ(Γ) that is orbit equivalent to Γ.These Kleinian groups arise out of pinching finitely many disjoint, simple, closed curves (on the surface D/Γ 0 ) out of an explicit finite list.In particular, all such groups Γ are geometrically finite.
Remark 5.2.Consider the Bowen-Series map A G d associated with the Fuchsian group G d equipped with the preferred fundamental domain R given by the ideal polygon with vertices at the 2d-th roots of unity (see Subsection 3.2).In this case, the explicit finite list of Theorem 5.1 is (see [MM21, Proposition 7.6]), and hence every A G d −invariant geodesic lamination on D/G d ∼ = S 0,d+1 is a subset of S d .For d = 3, this gives exactly two invariant laminations: {{g 2 }, {g −1 3 • g 1 }}.The curve corresponding to g 2 (respectively, g −1 3 •g 1 ) on the four times punctured sphere depicted in Figure 2 is the 'vertical' (respectively, 'horizontal') curve which separates the punctures The (higher) Bowen-Series map of a Bers boundary group (when it exists) is piecewise Möbius and hence admits a canonical extension A Γ to a subset of the filled limit set K(Γ) of the group (i.e., the complement of the completely invariant component of its domain of discontinuity).Now let P be a complex polynomial in the principal hyperbolic component H k , where k = deg{A Γ : Λ(Γ) −→ Λ(Γ)}.Then, the action of P on its Julia set J (P ) is topologically conjugate to z k | S 1 .On the other hand, A Γ | Λ(Γ) is a factor of A Γ0 | S 1 , which is in turn topologically conjugate to z k | S 1 .One can now glue the filled Julia set K(P ) (which is a closed Jordan disk) outside the filled limit set K(Γ) using a semi-conjugacy between P | J (P ) and A Γ | Λ(Γ) , and this produces a topological 2sphere.Moreover, the existence of this semi-conjugacy implies that the action of P on K(P ) and the action of A Γ on a subset of K(Γ) paste together to yield a continuous map on the copy of S 2 just defined.This map is called the topological mating of A Γ and P .We say that the canonical extension A Γ of the (higher) Bowen-Series map of a Bers boundary group is conformally mateable with a polynomial P in the principal hyperbolic component H k if the above topological 2-sphere admits a complex structure that turns the topological mating into a holomorphic map (cf.[MM21, §7.5]).
A sophisticated surgery procedure involving David homeomorphisms yields the following conformal mateability theorem.
For a group Γ ∈ ∂B(Γ 0 ) admitting a (higher) Bowen-Series map A Γ , the corresponding geodesic lamination L is invariant under the base (higher) Bowen-Series map A Γ0 .The associated equivalence relation L on S 1 satisfies the following properties. ( (3) L−equivalence classes are pairwise unlinked ; i.e., if X and Y are two distinct equivalence classes of L, then there exist disjoint intervals I X , I X ⊂ R/Z such that X ⊂ I X and Y ⊂ I Y .(4) If X is an L−equivalence class, then A Γ0 (X) is also an L-equivalence class.
(5) If X is an L−equivalence class, then X → A Γ0 (X) is a cyclic order preserving bijection.
On the other hand, the lamination associated with a complex polynomial P with connected Julia set also enjoys analogues of the properties listed above (where the role of A Γ0 is played by the base polynomial z d ).Roughly speaking, the lamination associated with P is a z d −invariant closed equivalence relation on S 1 such that the quotient of S 1 by the equivalence relation yields a topological model of the Julia set of P (cf.[Kiw04]).Remarkably, the topological conjugacy between A Γ0 | S 1 and z d | S 1 (for some d ≥ 2) provides us with a tool to pass from laminations in the group world to those in the polynomial world.This combinatorial link allows one to invoke standard realization results from polynomial dynamics and conclude that the limit set Λ(Γ) is indeed homeomorphic to the Julia set of a complex polynomial in a 'dynamically natural' way.
Theorem 5.4 (Equivariant homeomorphism between limit and Julia set).
[MM21, Theorem 7.16] Let Γ ∈ ∂B(Γ 0 ) be a group that admits a (higher) Bowen-Series map A Γ .Then there exists a complex polynomial P Γ (of degree equal to that of A Γ : Λ(Γ) −→ Λ(Γ)) such that the action of A Γ on the limit set Λ(Γ) is topologically conjugate to the action of P Γ on its Julia set.

Measures of maximal entropy and Patterson-Sullivan measures
In this section, we study the measure-theoretic dynamics of Bowen-Series and higher Bowen-Series maps associated with Fuchsian punctured sphere groups, thus linking this theme to another seminal piece of work by Sullivan-the Patterson-Sullivan measure.Specifically, we show that measures of maximal entropy of (higher) Bowen-Series maps acting on the circle are push-forwards of appropriate Patterson-Sullivan measures supported on Gromov boundaries of free groups.
Informally speaking, the Sullivan-Patterson measure on the Gromov boundary of a group is the weak limit of a sequence of atomic measures supported on the words of length n, appropriately weighted by the distances of the group elements from a fixed base point (for the free group with the standard generating set, the sequence reduces to Formula (1)).Although we will not use the general theory of Patterson-Sullivan measures, we encourage the reader to consult [Pat76,Sul79,Coo93] for the construction and basic properties of these measures in the context of Fuchsian groups, Kleinian groups, and hyperbolic groups, respectively.For background on symbolic dynamics and topological/measure-theoretic entropy, we refer the reader to [Wal82,BS02].6.1.Maximal entropy measure for Bowen-Series maps.For definiteness, let us fix the Fuchsian (d+1)−times punctured sphere group Γ 0 = G d of Section 3.2 and the fundamental domain R given by the ideal polygon with vertices at the 2d−th roots of unity (d ≥ 2).Further let A ≡ A Γ0,BS : S 1 → S 1 be the Bowen-Series map of Γ 0 associated with the fundamental domain R.
The topological entropy of a dynamical system is a numerical topological conjugacy invariant that measures the complexity of the system.Roughly, it represents the exponential growth rate of the number of essentially different orbit segments of length n.Since A is topologically conjugate to z 2d−1 , the topological entropy of the A−action on S 1 is equal to ln(2d − 1).We are interested in studying the measure of maximal entropy (MME for short) for A; i.e., the unique A−invariant measure on S 1 whose measure-theoretic entropy is equal to the topological entropy ln(2d − 1) (see [AKU21a,AKU21b] for computation of topological entropy of Bowen-Series maps associated with cocompact Fuchsian groups and results regarding their measures of maximal entropy).6.1.1.MME of A in terms of topological dynamics.By Proposition 3.3, there exists a homeomorphism φ : S 1 → S 1 that conjugates p : z → z 2d−1 to A (this homeomorphism can be thought of as a generalization of the Minkowski question-mark function ?; see [LLMM22, §4.4.2] for the analogy in the anti-holomorphic context).We denote the Haar (normalized Lebesgue) measure on S 1 by m.Note that m is the unique measure of maximal entropy for the action of p on S 1 (a straightforward computation shows that the measure-theoretic entropy of p| S 1 with respect to m is equal to the topological entropy ln(2d − 1), and the uniqueness of this measure follows for instance from [Lyu83, Theorem 9]).Since the homeomorphism φ is a conjugacy, we have the following.Proposition 6.1.ν = φ * m, where ν is the unique measure of maximal entropy for the A−action on S 1 .6.1.2.MME of A in terms of symbolic dynamics.The partition of S 1 determined by the 2d−th roots of unity form a Markov partition for A. We denote this partition by {I 1 , I −1 , • • • , I d , I −d }, where I j is the counter-clockwise arc of S 1 connecting e 2πi (j−1) 2d and e 2πi j 2d , and I −j is the complex conjugate of I j , for j ∈ {1, • • • , d}.The transition matrix for this Markov partition is The above transition matrix gives rise to a one-sided subshift of finite type M with the usual ultra-metric (in base e).
Since A is expansive, one obtains a continuous surjection ψ : Σ + M → S 1 that semi-conjugates σ to A. We may and will assume that ψ carries the cylinder set [±j] ⊂ Σ + M to the Markov partition piece of A connecting e ±πi(j−1)/d to e ±πij/d .Remark 6.2.See [Sta04] for Markov partitions of Bowen-Series maps associated with more general Fuchsian punctured surface groups.These maps, however, are not continuous if the genus of the surface is greater than zero.
The unique measure of maximal entropy for the σ−action on Σ + M (which is called the Parry measure in symbolic dynamics) is given by the 'uniform' Markov measure µ that assigns mass 1 2d•(2d−1) n to each cylinder set of rank n + 1 (n ≥ 0).The corresponding topological entropy is also ln(2d − 1) (note that 2d − 1 is the largest eigenvalue of M ).The existence of the semi-conjugacy ψ now implies that Proposition 6.3.The measure of maximal entropy of A, which we denote by ν, is the push-forward of the Parry measure µ under ψ; i.e., ν = ψ * µ.
6.1.3.MME of A in terms of Patterson-Sullivan measure.Since the Bowen-Series map A is cooked up from the Fuchsian group Γ 0 , it is natural to ask whether the measure of maximal entropy ν of A is related to the Patterson-Sullivan measure class of Γ 0 .The following proposition gives a negative answer to this question (recall that a Patterson-Sullivan measure of Γ 0 lies in the class of the Haar measure m).
Proposition 6.4.The measure ν is not mutually absolutely continuous with respect to the Haar measure m; i.e., ν and m do not lie in the same measure class.
Proof.We learned this from Caroline Series.The proposition follows from the facts that (1) the action of the Fuchsian group Γ 0 on S 1 = ∂D is of type III 1 .
(2) the action of the polynomial z k on S 1 = ∂D is of type III ln(k) .
Fortunately, the free group on d generators F d ∼ = Γ 0 provides us with a Patterson-Sullivan measure (supported on the Gromov boundary of F d ) that is intimately related to ν.
We denote the Cayley tree of F d by X, and equip it with the word metric.The group acts on the tree by isometries.The Gromov boundary of F d is denoted by ∂X.Note that we can naturally identify ∂X with the shift space Σ + M .Visualizing the Cayley tree X as dual to the Γ 0 −tessellation of D (associated with the fundamental domain R), one sees in light of the identification ∂X ∼ = Σ + M that the map ψ is the (Floyd-)Cannon-Thurston map from ∂X to S 1 (cf.[Flo80]).Definition 6.5.Let X denote a Cayley graph of a group Γ.Let g ∈ Γ (thought of as a vertex of X).The cone of g consists of the vertices h ∈ X such that any geodesic [1, g] followed by any geodesic The next result enables us to connect ν to a suitable Patterson-Sullivan measure on ∂X.Lemma 6.6.The Patterson-Sullivan measure on ∂X (with respect to the base point 1 and the standard generating set) is given by the Parry measure µ.
Proof.Note that the number of words in F d of length r is 2d • (2d − 1) r−1 , for r ≥ 1.Hence, the Patterson-Sullivan measure on ∂X (with respect to the base point 1 and the standard generating set) is a weak limit of the measures (1) A straightforward computation now shows that the µ n+r −mass of the cone at a group element of length r is: 1 Proof.First note that the (Floyd-)Cannon-Thurston map ψ semi-conjugates the F d −action on ∂X to the Γ 0 −action on S 1 [Flo80].We will denote the element of F d corresponding to γ ∈ Γ 0 by γ.
By Proposition 6.7 and the previous paragraph, the measure γ * ν on S 1 is the push-forward of the measure γ * µ on ∂X under ψ.Moreover, as µ is a Patterson-Sullivan measure on ∂X, it follows that γ * µ and µ are mutually absolutely continuous (see [Coo93, Theorem 5.4, Theorem 8.2]).It is now easy to see using the definition of push-forward of a measure that the measures ν = ψ * (µ) and γ * ν = ψ * ( γ * µ) are mutually absolutely continuous.Remark 6.9.For γ ∈ Γ 0 , the Radon-Nikodym derivative d(γ * ν)/dν can be written in terms of ψ and the Radon-Nikodym derivative d( γ * µ)/dµ, which in turn can be computed from measures of cylinder sets (see [Coo93,§8] for a general method of describing such Radon-Nikodym derivatives in terms of Busemann functions).Now observe that the Bowen-Series map A does not depend only on the group Γ 0 , but also on the choice of the fundamental domain R. The translation of R by an element γ ∈ Γ 0 is a different fundamental domain γ • R for Γ 0 .We denote the Bowen-Series map of Γ 0 associated with the fundamental domain γ • R by A γ .Clearly, A γ = γ • A • γ −1 .Moreover, the unique measure of maximal entropy for the A γ −action on S 1 is given by γ * ν.Proposition 6.8 now implies the following.
Corollary 6.10.The measures of maximal entropy for the Bowen-Series maps associated with the fundamental domains γ • R (for γ ∈ Γ 0 ) are mutually absolutely continuous.In particular, all these measures have the same Hausdorff dimension.6.1.5.MME for matings of Bowen-Series maps and polynomials.Recall that Theorem 3.5 provides us with a conformal mating of the canonical extension A of the Bowen-Series map A (associated with the fundamental domain R of Γ 0 ) and the polynomial map z 2d−1 .Also note that the restriction of this conformal mating on its Jordan curve limit set is topologically conjugate to A| S 1 .The following description of the measure of maximal entropy of the conformal mating now follows from Propositions 6.1 and 6.7.Proposition 6.11.The unique measure of maximal entropy of the conformal mating of A and z 2d−1 restricted to the limit set is equal to the push-forward of the normalized Lebesgue measure m (which is the unique measure of maximal entropy of z 2d−1 | S 1 ) as well as the push-forward of the Patterson-Sullivan measure µ on ∂X (with respect to the base point 1 and the standard generating set) under appropriate conjugacies.In particular, the corresponding topological entropy is ln(2d − 1).6.1.6.Topological entropy of A from a group-theoretic perspective.The topological entropy ln(2d − 1) of A can be related to the volume entropy of the group F d , which measures the exponential growth rate of the number of words of length n in a group (equivalently, the exponential growth rate of the number of group elements in a ball of radius n around identity).Lemma 6.12.The volume entropy of F d with respect to the standard (symmetric) set of generators and the critical exponent for the F d −action on X are both equal to ln(2d − 1).
Proof.Recall that the number of words in F d of length r is 2d from which it follows that the volume entropy is ln(2d − 1).Now consider the Poincaré series with exponent s: Clearly, the series converges if and only if 2d − 1 < e s ⇐⇒ s > ln(2d − 1).In particular, the critical exponent is ln(2d − 1).
Remark 6.13.A connection between the topological entropy of Bowen-Series maps associated with cocompact Fuchsian groups and the volume entropy of suitable hyperbolic groups was established in [Los14].
In [Sul84, Theorem 1], Sullivan proved equality of critical exponents and Hausdorff dimensions of limit sets for geometrically finite Kleinian groups.While the analogous result for hyperbolic groups follows from general consideration (cf.We first note that the µ-measure of a cylinder of rank n is equal to 1 2d(2d−1) n−1 .On the other hand, the diameter of a cylinder of rank n is e −n .Thus, µ(B(a, e −n )) = 1 2d(2d − 1) n−1 =⇒ µ(B(a, r)) ∼ r ln(2d−1) .
The result now follows from standard results on Hausdorff dimension (for instance, see [Fal03, Proposition 4.9]).In fact, we have shown that the ln(2d−1)−dimensional Hausdorff measure is positive and finite.The second statement is obvious from the above proof.6.2.Maximal entropy measure for higher Bowen-Series maps.We now carry out a similar analysis for the measure of maximal entropy of a higher Bowen-Series map of a Fuchsian punctured sphere group.For simplicity of exposition, we work with the thrice punctured sphere case.
Let us fix the Fuchsian thrice punctured sphere group Γ 0 of Section 4.1 and a (closed) fundamental domain W given by the quadrilateral with vertices at the fourth roots of unity.Further let A ≡ A Γ0,hBS : S 1 → S 1 be the associated higher Bowen-Series map of Γ 0 .As A| S 1 is topologically conjugate to z 4 | S 1 , the topological entropy of A is equal to ln(4).We denote the unique measure of maximal entropy for A| S 1 by ν.

Topological dynamics.
As A is an expansive circle covering of degree 4, there exists a homeomorphism φ : S 1 → S 1 that conjugates p : z → z 4 to A. Using the conjugacy φ, one can write the measure of maximal entropy ν for A| S 1 as the push-forward measure φ * m. 6.2.2.Symbolic dynamics.The pieces of A are given by g ±1 , h ±1 , g Their intervals of definition yield a Markov partition (counter-clockwise starting at 1) for A with transition matrix The above transition matrix gives rise to a one-sided subshift of finite type and σ is the left-shift map.As before, we metrize Σ + M with the usual ultra-metric (in base e).Since A is expansive, one obtains a continuous surjection ψ : Σ + M → S 1 that semi-conjugates σ to A, and sends the cylinders of rank 1 to the Markov partition pieces of A.
The Parry measure (i.e., the unique measure of maximal entropy) for the σ−action on Σ + M is given by the 'uniform' Markov measure µ, that assigns mass 1 6•4 n to each cylinder set of rank n + 1 (n ≥ 0).The corresponding topological entropy is also ln(4) (note that 4 is the largest eigenvalue of M ), and ν = ψ * µ.
6.2.3.Patterson-Sullivan measure.We now turn our attention to the the free group F 2 ∼ = g * h with the generating set {g ±1 , h ±1 , g•h −1 , h•g −1 } (which are precisely the pieces of A).
We denote the Cayley graph of F 2 with respect to the above (non-standard) generating set by X, and equip it with the word metric.Note that we can naturally identify the Gromov boundary ∂X with the shift space Σ + M .With this identification, the boundary at infinity of the cone at a generator is the corresponding cylinder set in Σ + M .
Remark 6.15.The higher Bowen-Series map A gives rise to a Markov map A acting on the Gromov boundary ∂X (such that A is orbit equivalent to the F 2 −action on ∂X) in the following way: for α ∈ {g ±1 , h ±1 , g • h −1 , h • g −1 }, the map A acts on the boundary at infinity of Cone(α) as α −1 .Lemma 6.16.The Patterson-Sullivan measure on ∂X (with respect to the base point 1 and the generating set {g ±1 , h ±1 , g • h −1 , h • g −1 }) is given by the Parry measure µ.
Proof.It is easy to see from the generators and relations (equivalently, from the Cayley graph depicted in Figure 6) that the number of words of length r in F 2 (with respect to generating set {g ±1 , h ±1 , g • h −1 , h • g −1 }) is 6 • 4 r−1 (r ≥ 1).A computation similar to the one in the proof of Lemma 6.6 now readily shows that the Patterson-Sullivan measure in question (on ∂X) assigns mass 1 6•4 r−1 to each cylinder set (in ∂X) of rank r.Thus, the Patterson-Sullivan measure agrees with µ on each cylinder set.Proposition 6.17.The measure of maximal entropy ν of the higher Bowen-Series map A is the push-forward of the Patterson-Sullivan measure µ on ∂X (with respect to the base point 1 and the generating set {g ±1 , h ±1 , g • h −1 , h • g −1 }) under ψ.6.2.4.Γ 0 −invariance of the class of ν.Proposition 6.17, F d −invariance of the measure class of the Patterson-Sullivan measure µ (on ∂X) [Coo93, Theorem 5.4, Theorem 8.2], and the fact that the map ψ semi-conjugates the F d −action on ∂X to the Γ 0 −action on S 1 together imply the following.Proposition 6.18.For each γ ∈ Γ 0 , the measures ν and γ * ν are mutually absolutely continuous.6.2.5.MME for mating.According to Theorem 4.11, there exists a conformal mating of the canonical extension A of the higher Bowen-Series map A (associated with the closed fundamental domain W of Γ 0 ) and the polynomial map z 4 such that the restriction of this conformal mating on its Jordan curve limit set is topologically conjugate to A| S 1 .The interpretation of the measure ν in terms of m and µ implies the following.Proposition 6.19.The unique measure of maximal entropy of the conformal mating of A and z 4 restricted to the limit set is equal to the push-forward of the MME of z 4 on S 1 as well as the push-forward of the Patterson-Sullivan measure on ∂X (with respect to the base point 1 and the generating set {g ±1 , h ±1 , g • h −1 , h • g −1 }) under appropriate conjugacies.In particular, the corresponding topological entropy is ln(4).6.2.6.Topological entropy, volume entropy, and Hausdorff dimension.Since there are 6 • 4 r−1 words of length r in F 2 (with respect to generating set {g ±1 , h ±1 , g • h −1 , h • g −1 }), the arguments used in the proof of Lemmas 6.12, 6.14 apply mutatis mutandis to the current setting and prove equality of volume entropy, critical exponent, and Hausdorff dimension of the Gromov boundary.Moreover, this number coincides with the topological entropy of A| S 1 .Lemma 6.20.
(1) The volume entropy of F 2 with respect to the generating set {g ±1 , h ±1 , g • h −1 , h • g −1 } and the critical exponent for the F 2 −action on X are both equal to ln(4).(2) The Hausdorff dimension of ∂X equipped with the visual metric (in base e) with respect to the generating set {g ±1 , h ±1 , g • h −1 , h • g −1 } is equal to ln(4).Moreover, the ln(4)−dimensional Hausdorff measure H ln(4) and µ (on ∂X) are mutually absolutely continuous.

6.3.
Hausdorff dimension of measure of maximal entropy: thrice punctures sphere.Recall from Proposition 6.4 that the measure of maximal entropy of the Bowen-Series map of a Fuchsian punctured sphere group does not lie in the Lebesgue measure class.In this section, we will prove a sharper version of this fact in the thrice punctured sphere case.Specifically, we will show that the Hausdorff dimension of the MME ν of the (higher) Bowen-Series map of a Fuchsian thrice punctured sphere group is strictly less than 1.This statement should be compared with the analogous result that except for some very special cases, the Hausdorff dimension of the measure of maximal entropy of a rational map is strictly smaller than the Hausdorff dimension of the Julia set [Zdu90].
In what follows, we will cook up a self-map of the interval [0, 1] from the (higher) Bowen-Series map under consideration, and relate the Hausdorff dimension of the MME of the (higher) Bowen-Series map to that of the MME of the associated selfmap.This will allow us to obtain the desired upper bounds.We note that while this intermediate step is not essential for the Bowen-Series map (in this case, one can obtain the upper bound of Proposition 6.22 by working directly with the Bowen-Series map), this method yields additional information in the higher Bowen-Series case.Indeed, this reduction step connects the MME of the higher Bowen-Series map to a classical measure from number theory, which makes known results applicable to the current setting and gives a better estimate for the Hausdorff dimension of the MME.6.3.1.The Bowen-Series case.Since the Teichmüller space of a thrice punctured sphere is a singleton, we may, without loss of generality, work with the Bowen-Series map A G2 introduced in Subsection 3.2.Note that every non-identity element g in the free group G 2 admits a unique shortest representation with respect to the symmetric generating set {g ±1 1 , g ±1 2 } introduced in Subsection 3.2.The length of this shortest representation is called the length of g (the length of the identity element is defined to be zero).Recall that the ideal polygon in D with vertices at the fourth roots of unity is a fundamental domain for the G 2 −action on D, and hence its translates under elements of G 2 yield a tiling T G2 of D. We call this fundamental domain the rank 0 tile and its translate under an element g ∈ G 2 of length k a rank k tile for the above tessellation.
We will use a specific symmetric property of the tessellation T G2 which we now describe.Following Section 3, we denote the ideal polygon in D with vertices at the fourth roots of unity by R, and its edges by C ±1 , C ±2 .Let us further denote the anti-Möbius reflections in these edges by ρ ±1 , ρ ±2 , and the reflection group generated by ρ ±1 , ρ ±2 by G.Note that the polygon R is invariant under the actions of R and ι, where R is rotation by angle π/2 and ι is the reflection in the real axis.It follows that conjugation by R and ι act as permutations on the generating set {ρ ±1 , ρ ±2 } of G, and hence R and ι conjugate G to itself.Hence, the G−tessellation T G of D arising from the fundamental domain R (closure taken in D) is preserved by both R and ι.Note furthermore that the relations g i = ι • ρ i = ρ −i • ι, i ∈ {1, 2}, and ι−invariance of the tessellation T G imply that the tessellations T G and T G2 are the same.It follows that the tessellation T G2 of D is symmetric with respect to π/2−rotation R.
For the current purpose, it will be more convenient to work with the upper half-plane model.To this end, consider the Möbius transformation M (z) = i(1 − z)/(1 + z) which carries the unit disk onto the upper half-plane such that M (1) = 0, M (i) = 1, M (−1) = ∞, and M (−i) = −1.Hence, M sends the ideal polygon in D with vertices at the fourth roots of unity to the ideal polygon in H with vertices at −1, 0, 1, and ∞.The map M conjugates G 2 to a discrete subgroup G G G 2 of PSL 2 (R), and transports the G 2 −tessellation of D defined in the previous paragraph to a G G G 2 −tessellation of H.One defines tiles of this tessellation and their ranks as in the previous paragraph.Moreover, M conjugates the Bowen-Series map A G2 to the map By construction, τ maps [0, 1 3 ] to [0, 1], [ 1 3 , 1 2 ] to [1, +∞], and [ 1 2 , 1] to [−∞, −1] (see Figure 7).
Since Euclidean isometric rotation R : z → iz (about the origin) respects the G 2 − tessellation of D, it follows that the conformal rotation R H (w) = M (i • M −1 (w)) = 1+w 1−w ∈ PSL 2 (R) (about i) respects the corresponding G G G 2 −tessellation of H.
This allows one to construct a self-map of [0, 1) associated with τ : F : [0, 1) → [0, 1), (See Figure 8.) The symmetry of the G G G 2 -tessellation of H under the conformal rotation R H implies that F sends the ideal vertices of tiles of a given rank to the ideal vertices of tiles of the previous rank.second iterate of the piecewise reflection Markov map Hence, the circle endomorphisms A and R have the same measure of maximal entropy.
We will now relate the map R to a well-studied orientation-reversing double covering of S 1 .Note that as any pair of hyperbolic ideal triangles are Möbius equivalent, the triangle ∆136 of Figure 3 is Möbius equivalent to the regular ideal polygon Π ⊂ D with vertices at the third roots of unity.The Nielsen map ρ ρ ρ 2 : S 1 → S 1 of the regular ideal triangle reflection group is defined as anti-Möbius reflections in the three sides of Π on the three corresponding arcs of S 1 (see Figure 10    (mod 1) t ∈ 1 2 , 1 .

Figure 1 .
Figure 1.Bowen-Series Maps for surfaces of higher genus

Figure 2 .
Figure 2. The preferred fundamental domain R of G 3 , which uniformizes a four times punctured sphere, is shown.The fundamental domain has all six vertices on S 1 , and they cut the circle into six arcs.The corresponding Bowen-Series map acts on these arcs by the generators g ±1 j displayed next to them.
the following hold.(1)W = D ∪ P , and for each j ∈ {1, • • • , k − 1}, D ∪ g j (P ) is a (closed) fundamental domain for Γ. (2) On the clockwise arc from j to j + 1, the higher Bowen-Series map A Γ,hBS equals the Bowen-Series map of Γ associated with the (closed) fundamental domain D ∪ g j (P ) (j ∈ {1, • • • , k − 1}), and on the counterclockwise arc from 1 to k, A Γ,hBS equals the Bowen-Series map of Γ associated with the fundamental domain W = D ∪ P .

Figure 5 .
Figure 5.The two piecewise Fuchsian Markov maps B (left) and C (right) are depicted.They are not orbit equivalent to the Fuchsian groups generated by their pieces.
−1) r−1 as n → +∞.It follows that the Patterson-Sullivan measure on ∂X assigns mass 1 2d(2d−1) r−1 to each cylinder set (in ∂X) of rank r.In view of the definition of µ, the proof is now complete.Since ν = ψ * µ, we conclude the following result.Proposition 6.7.The measure of maximal entropy ν of the Bowen-Series map A is the push-forward of the Patterson-Sullivan measure µ on ∂X (with respect to the base point 1 and the standard generating set) under the (Floyd-)Cannon-Thurston map ψ. 6.1.4.Γ 0 −invariance of the MME of A. We now exploit the connection between ν and Patterson-Sullivan measures to exhibit Γ 0 −invariance of the measure class of ν.Proposition 6.8.For each γ ∈ Γ 0 , the measures ν and γ * ν are mutually absolutely continuous.
[Coo93,  Theorem 8.3] [KB02, Theorem 15.8]), we can give a simple proof in the present setting.Lemma 6.14.The Hausdorff dimension of the Gromov boundary of F d equipped with the visual metric (in base e) is equal to ln(2d − 1).Moreover, the ln(2d − 1)−dimensional Hausdorff measure H ln(2d−1) (on ∂X) and µ are mutually absolutely continuous.Proof.The visual metric (in base e) on ∂X is bi-Lipschitz to the ultra-metric given by d(a, b) = e −|c| , where c is the bifurcation point for the geodesic rays [1, a) and [1, b).Hence, it suffices to compute the Hausdorff dimension of ∂X with respect to this ultra-metric.

Figure 6 .
Figure 6.The words of length one and two in the Cayley graph of F 2 with respect to the generating set {g±1 , h ±1 , g • h −1 , h • g −1 } are displayed.
for a pictorial illustration and [LLMM22, §2], [LLMM19, §4.1] for the precise definition and properties of this map).The Möbius equivalence of ∆136 and Π implies that the map R is Möbius conjugate to ρ ρ ρ 2 .Moreover, the fact that Möbius maps are bi-Lipschitz tells us that the Hausdorff dimension of the MME ν of the higher Bowen-Series map A is equal to the Hausdorff dimension of the MME of ρ ρ ρ 2 .

Figure 10 .
Figure 10.The action of the Nielsen map ρ ρ ρ 2 of the ideal triangle group is depicted.

Figure 10 :
Figure 10: The Nielsen map ρ ρ ρ 2 acts on the arcs > 1e 2πi 3 , > e 2πi 3 e 4πi 3 , and > e 4πi 3 1 of S 1 as reflections in the bi-infinite hyperbolic geodesics 1e 2πi 3 , e 2πi 3 e 4πi 3 , and e 4πi 3 1, respectively.It naturally extends as a piecewise anti-Möbius map to the complement of the ideal triangle Π (in grey) in D.Applying the construction of Subsection 6.3.1 to the upper half-plane model of ρ ρ ρ 2 (such that the ideal triangle in D with vertices at the third roots of unity corresponds to the ideal triangle in H with vertices at 0, 1, ∞) combined with the arguments of Lemma 6.21, one can show that the Hausdorff dimension of the MME of ρ ρ ρ 2 is equal to the Hausdorff dimension of the MME of the orientation-reversing degree two map