Rigidity and flexibility of entropies of boundary maps associated to Fuchsian groups

Given a closed, orientable surface of constant negative curvature and genus $g \ge 2$, we study the topological entropy and measure-theoretic entropy (with respect to a smooth invariant measure) of generalized Bowen--Series boundary maps. Each such map is defined for a particular fundamental polygon for the surface and a particular multi-parameter. We present and sketch the proofs of two strikingly different results: topological entropy is constant in this entire family ("rigidity"), while measure-theoretic entropy varies within Teichm\"uller space, taking all values ("flexibility") between zero and a maximum, which is achieved on the surface that admits a regular fundamental $(8g-4)$-gon. We obtain explicit formulas for both entropies. The rigidity proof uses conjugation to maps of constant slope, while the flexibility proof -- valid only for certain multi-parameters -- uses the realization of geodesic flow as a special flow over the natural extension of the boundary map.


Introduction
Any closed, orientable, compact surface S of constant negative curvature can be modeled as S = Γ\D, where D = { z ∈ C : |z| < 1 } is the unit disk endowed with hyperbolic metric 2 |dz| /(1 − |z| 2 ) and Γ is a finitely generated Fuchsian group of the first kind acting freely on D and isomorphic to π 1 (S). Recall that geodesics in this model are half-circles or diameters orthogonal to S = ∂D, the circle at infinity. The geodesic flow ϕ t on D is defined as an R-action on the unit tangent bundle T 1 D that moves a tangent vector along the geodesic defined by this vector with unit speed. The geodesic flow ϕ t on D descends to the geodesic flow ϕ t on the factor S = Γ\D via the canonical projection of the unit tangent bundles. The orbits of the geodesic flow ϕ t are oriented geodesics on S.
A surface S of genus g ≥ 2 admits an (8g − 4)-sided fundamental polygon with a particular pairing of sides (1) obtained by cutting it with 2g closed geodesics that intersect in pairs (g of them go around the "holes" of S and another g go around the "waists"; see Figure 1). The existence of such a fundamental polygon F is an old result attributed to Dehn, Fenchel, Nielsen, and Koebe [34,21,9]. Adler and Figure 1. Necklace of 2g geodesics on S forming the sides of F for g = 2.
Flatto [5, Appendix A] give a careful proof of existence and special properties of F that we list below. We label the sides of F, which are geodesic segments, in a counterclockwise order by numbers 1 ≤ k ≤ 8g − 4 and label the vertices of F by V k so that side k connects V k to V k+1 (mod 8g − 4) (this gives us a marking of the polygon).
We denote by P k and Q k+1 the endpoints of the oriented infinite geodesic that extends side k to the circle at infinity S. The counter-clockwise order of endpoints on S is P 1 , Q 1 , P 2 , Q 2 , ..., P 8g−4 , Q 8g−4 . The identification of the sides of F is given by the side pairing rule We denote by T k the Möbius transformation pairing side k with side σ(k). The group Γ is generated by T 1 , ..., T 8g−4 . Note that F is not regular in general but satisfies the following properties: (1) the sides k and σ(k) have equal length; (2) the angles at vertices k and σ(k) + 1 add up to π. We call fundamental (8g − 4)-gons F satisfying properties (1) and (2) canonical. Property (2) implies the "extension condition," which is crucial for our analysis: the extensions of the sides of F do not intersect the interior of the tessellation γF, γ ∈ Γ (see Figure 2).
The polygon F is related to a regular (8g − 4)-gon F reg centered at the origin with all interior angles equal to π 2 (see [5, Figure 1]) by the following construction. Let S = Γ\D be any compact surface of genus g ≥ 2 and S reg = Γ reg \D be the surface of the same genus admitting the regular fundamental polygon F reg . By the Fenchel-Nielsen Theorem [31], there exists an orientation-preserving homeomorphism h from D onto D such that Γ = h•Γ reg •h −1 . The map h S is a homeomorphism of S preserving the order of the points P ∪ Q, where P = {P 1 , ..., P 8g−4 } and Q = {Q 1 , ..., Q 8g−4 }. 1 The sides of F are constructed by connecting h(P k ) and h(Q k+1 ) by geodesics, where P k and Q k+1 correspond to F reg . Conversely, for every marked canonical (8g − 4)-gon and the associated Fuchsian group Γ, by Euler's formula the genus of Γ\D is g.
This construction gives a less common representation of the Teichmüller space T (g) for g ≥ 2: it is the space of marked canonical (8g − 4)-gons in the unit disk D up to an isometry of D. 2 The topology on the space of polygons is that P k → P if and only if the lengths of all sides converge and the measures of all angles converge. It is well-known [16,17] that dim(T (g)) = 6g − 6; the following is a heuristic argument for this fact using (8g − 4)-gons. The lengths of the identified pairs of sides are given by 4g − 2 real parameters, and 2g − 1 parameters represent the angles since four angles at each vertex are determined by one parameter. The dimension of the space SU (1, 1) of isometries of D is 3, so we have altogether (4g − 2) + (2g − 1) − 3 = 6g − 6 parameters.
The boundary map. The main object of this paper is the multi-parameter family of generalized Bowen-Series boundary maps studied in [19,2,1,3,4]. For each F ∈ T (g), recall that side k of F is contained in the geodesic from P k to Q k+1 and that T k maps side k to side σ(k); we define the boundary map f A : S → S by where When all A k = P k we denote the map by f P (see left of Figure 4 for its graph). The map f P was extensively studied by Adler and Flatto in [5]; they call it "the Bowen-Series boundary map," although Bowen and Series' construction [13] used 4g-gons. 1 In some works h S is also called a "boundary map;" this is different from our usage. 2 For a related representation of T (g) as the space of marked canonical 4g-gons, see [29], following the earlier work [35,32].
We study two dynamical invariants of f A : the topological entropy and the measuretheoretic entropy with respect to a smooth (that is, Lebesgue-equivalent) measure. Our two main results are rigidity of the topological entropy (Theorem 1) and flexibility of the measure-theoretic entropy (Theorem 2).
Theorem 1 (Rigidity of topological entropy [4]). Let S = Γ\D be a surface of genus g ≥ 2 with fundamental polygon F ∈ T (g). For any Remark. The maps f A are not necessarily topologically conjugate since, according to [19], the combinatorial structure of the orbits associated to the discontinuity points A k can differ.
All f A are piecewise continuous and piecewise monotone. Some maps in this family (such as those considered by Bowen and Series [13] and further studied by Adler and Flatto [5]) are Markov, and so the topological entropy can be calculated as the logarithm of the maximal eigenvalue of a transition matrix [33,Theorem 7.13] in these cases. Not all maps f A admit a Markov partition, but (3) holds for all A regardless. The proof of Theorem 1 uses conjugation to maps of constant slope.
The flexibility result requires a restriction on the set of parameters, as defined below. Such restrictions are common in previous work on boundary maps: [5] uses only A = P and A = Q, [1] focuses on extremal parameters and their so-called duals, and [2] requires short cycles.

Definition.
A multi-parameter A = {A 1 , ..., A 8g−4 } is extremal if for each k either A k = P k or A k = Q k . A multi-parameter A satisfies the cycle property if each A k ∈ (P k , Q k ) and there exist positive integers m k , n k such that f m k A (T k A k ) = f n k A (T k−1 A k ). If m k = n k = 1 then we say that A satisfies the short cycle property.
Notice that the above properties are preserved by the Fenchel-Nielsen homeomorphism h S (see [20]), so it makes sense to talk about a multi-parameter A satisfying these properties in the Teichmüller space T (g).
If A is extremal or has short cycles, then f A admits a unique smooth invariant ergodic measure µ A related to the Liouville measure for geodesic flow.
Theorem 2 (Flexibility of measure-theoretic entropy [3]). Let S = Γ\D be a surface of genus g ≥ 2, and let A be extremal or satisfy the short cycle property. 3 (i) For each F ∈ T (g), the measure-theoretic entropy of f A with respect to its smooth invariant probability measure µ A is (ii) Among all F ∈ T (g), the maximum value of h µ A (f A ) is achieved on the surface for which F is regular; this value is Remark. The formula (4) shows that the measure-theoretic entropy remains constant under the change of the multi-parameter A within the considered class, so there is an aspect of rigidity to Theorem 2 as well. We emphasize that h µ A (f A ) is flexible in the Teichmüller space.
The paper is organized as follows. Sections 2 and 3 summarize the proofs of Theorems 1 and 2, respectively. Section 4.1 compares the two entropies (as was done in [3]), and Sections 4.

Rigidity of topological entropy
The notion of topological entropy was introduced by Adler, Konheim, and McAndrew in [6]. Their definition used covers and applied to compact Hausdorff spaces; Dinaburg [14] and Bowen [11] gave definitions involving distance functions and separated sets, which are often more suitable for calculation. While these formulations of topological entropy were originally intended for continuous maps acting on compact spaces, Bowen's definition can actually be applied to piecewise continuous, piecewise monotone maps on an interval, as explained in [25]. The theory naturally extends to maps of the circle, where piecewise monotonicity is understood to mean local monotonicity or, equivalently, having a piecewise monotone lift to R.
In [26], Parry introduced two probability measures for topological Markov chains: the first is the measure of maximal entropy and is commonly known as the "Parry measure"; the second measure is not shift-invariant but is uniformly expanding on cylinders. In [27], Parry used this second measure to conjugate a piecewise monotone, strongly transitive interval map (not necessarily Markov) with positive topological entropy to a map with constant slope. Alsedà, Llibre, and Misiurewicz [7,8] generalized this construction to piecewise continuous, piecewise monotone interval maps, obtaining semi-conjugacy for non-transitive maps. We apply the results of [27,8] to the family of maps f A , defining the second of these measures, ρ A , in (6).
Our original proof of Theorem 1 in [4] uses several symmetry properties of the regular fundamental polygon F reg and the conjugacy ψ P constructed for the boundary map f P that is associated to the regular polygon. The main fact required for Theorem 1 is [4, Lemma 15(a)], and in hindsight this can be proved without appealing to any symmetry. Thus we can prove the rigidity result for any polygon F without using the Fenchel-Nielsen homeomorphism to relate the setup to the regular polygon. (Figures 4 and 5 were made using the regular polygon with genus g = 2.) Step 1: topological entropy for extremal parameters. All f A (P k ) and f A (Q k ) belong to the set P ∪ Q (see [19,Proposition 2.2], originally [5,Theorem 3.4]), so the partition of S into intervals I 1 , ..., I 16g−8 given by is a Markov partition for f A for every extremal A. Although they share a Markov partition, each extremal A has its own transition matrix M A = (m i,j ) given by The transition matrices M P and M Q for genus 2 are shown in Figure 3.   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1  We compute the maximal eigenvalue of the transition matrices for all extremal parameters and hence the topological entropy for these Markov cases [4, Proposition 7 and Corollary 8]: • For any extremal multi-parameter A, the number of A-admissible words of length n grows as approximately λ n , where and therefore the maximal eigenvalue of M A is exactly λ.
Step 2: conjugacy to a constant slope map. The following theorem combines several results of [27,8], stated here for circle maps (as in [24]) instead of interval maps: Theorem 3. Given a piecewise monotone, piecewise continuous, topologically transitive map f : S → S of positive topological entropy h > 0, there exists a unique (up to rotation of S) increasing homeomorphism ψ : S → S conjugating f to a piecewise continuous map with constant slope e h .
Although for each parameter A (not necessarily extremal) the map f A has, a priori, its own conjugacy and its own corresponding map of constant slope, we are especially interested in the cases A = P and A = Q.
• The map f P : S → S is piecewise monotone, piecewise continuous, topologically transitive, and with positive topological entropy (see [13, Lemma 2.5] and Step 1), so by Theorem 3 there exists an increasing homeomorphism ψ P : S → S conjugating it to a map with constant slope, see Figure 4. The map ψ P is unique up to rotation of S, and the slope of P is exactly λ = e htop(f P ) .
• The map f Q : S → S also satisfies the conditions of Theorem 3, so there exists an increasing homeomorphism ψ Q : S → S, unique up to rotation of S, conjugating it to a map Q of constant slope. By Step 1, P and Q have the same slope. Figure 4. Plots of f P (x) (left) and P (x) (right) for g = 2.
When f A is Markov-in particular when A is extremal-the construction of the conjugacy ψ A follows the classical work of Parry [27], also used in the proof of [8, Lemma 5.1]: let λ, v be the maximal eigenpair for the transition matrix M A and define the measure ρ A on the shift space X A for any non-empty cylinder C ω A (see [4,Section 4] One defines the push-forward measure ρ A on S = [−π, π] as The conjugacy ψ A : S → S in Markov cases is then given by It turns out that the maps ψ P and ψ Q thus constructed coincide: This is the most technically difficult part of the argument. The proof is presented in [4, Appendix A]. We also prove many symmetric properties of ψ P ([4, Propositions 14 and 15]) that are visually evident in [4, Figure 4].
Step 3: completion of the proof. Denote P k = ψ P (P k ) and Q k = ψ P (Q k ). By construction, the map ψ P is the interval where f P acts by T k . The crucial fact that is necessary for our proof of rigidity is that ψ P To see this, note that ψ Q • T k • ψ −1 Q is linear on [Q k , Q k+1 ) by construction and that, by Theorem 4, ψ P = ψ Q . Thus P is linear with slope λ, and so, by [24,Theorem 3 ], the topological entropy of f A is log λ.

Flexibility of measure-theoretic entropy
A few years ago, Anatole Katok suggested a new area of research-or, at the very least, a new viewpoint-called the "flexibility program," which can be broadly formulated as follows: under properly understood general restrictions, within a fixed class of smooth dynamical systems, some dynamical invariants take arbitrary values. Taking this point of view, it is natural to ask how the measure-theoretic entropy h µ P (f P ) changes in T (g). Theorem 2 addresses this question, and we sketch its proof in this section.  Figure 5. Left: part of the graphs of f P (blue) and f Q (dashed orange). Right: P (blue) and Q (dashed orange).
Step 1: the smooth invariant measure. If A is extremal (e.g., A = P ) or has short cycles, then the smooth invariant measure µ A for f A can be described as twostep projection of the Liouville measure for the geodesic flow [2,1].
• On the unit tangent bundle T 1 D, parameterized by (x, y, θ) with x + iy ∈ D the base-point of tangent vector and θ its angle with the real axis, the Liouville volume dω = 4 dx dy dθ (1 − x 2 − y 2 ) 2 comes from the hyperbolic measure on D and is invariant under geodesic flow.
• The measure dν = |du| |dw| |u − w| 2 is a smooth measure on the space of oriented geodesics on D (modeled as {(u, w) ∈ S × S : u = w}) and is preserved by Möbius transformations. 4 Sometimes called "geodesic current," this measure was most probably first considered by E. Hopf [18] and was later used by Sullivan [30], Bonahon [10], Adler-Flatto [5], and the current authors [19,2]. • Using coordinates (u, w, s) on T 1 D, where s is arclength along a geodesic, instead of coordinates (x, y, θ), the measure dm = dν ds is also preserved by geodesic flow and is a multiple of the Liouville volume dω. Specifically, dm = 1 2 dω (see [10, Appendix A2]). 5 Generalizing Adler and Flatto's "rectilinear map" from [5], we define the map In [19], the authors showed that F A admits a global attractor Ω A with finite rectangular structure if A = P or if A satisfies the short cycle property. Adler and Flatto had previously shown that F P has as an invariant domain with finite rectangular structure (which we call Ω P , see Figure 6), and this was extended to F A for all extremal parameters in [1]. The restriction of F A to Ω A is the natural extension map of f A ; we will often denote F A Ω A by simply F A .
• The normalized measure is by construction a smooth invariant probability measure for F A . Because f A is a factor of F A (projecting on the second coordinate), its smooth invariant probability measure µ A is obtained as a projection of ν A .
Step 2: formula for the entropy. The geodesic flow on S can be realized as a special flow over a cross-section that is parametrized by Ω A , and the first return map to this cross-section acts exactly as F A : Ω A → Ω A (see [ Since F A is the natural extension of f A , the entropies h ν A (F A ) and h µ A (f A ) are equal, and since Area(F) = 2π(2g − 2) by the Gauss-Bonnet formula, we have The integral Ω A dν can be explicitly computed using the finite rectangular structure of Ω A , but this does not easily relate to the (hyperbolic) perimeter of F. Adler-Flatto [5] introduced another map, called the "curvilinear map" (or "geometric map" in [2]): denoting by uw the geodesic from u to w, the map is defined on the set Ω geo := { (u, w) : uw intersects F } ⊂ S × S \ ∆ Figure 6. Arithmetic set Ω P in red-and-purple, geometric set Ω geo in blue-and-purple (for an irregular polygon with g = 2).
(shown in Figure 6) and is given by There is a bijection Φ A : Ω geo → Ω A that acts piecewise by Möbius transformations (see [2,Proposition 3.4] for short cycles and [1, Proposition 12] for extremal) and since Möbius transformations preserve the measure ν, it follows that Given (10), we now want to show that Ωgeo dν is equal to the (hyperbolic) perimeter of F. We use the following fact proved by F. Bonahon [10, Appendix A3]: • For any oriented geodesic segment s on D, where Ψ + (s) is the set of oriented geodesics intersecting s with the oriented angle at the intersection between 0 and π. The domain Ω geo of the geometric map F geo can be decomposed as Ω geo = 8g−4 k=1 G k , where G k = { (u, w) : uw exits F through side k } = Ψ + (side k) (these "strips" are shown in [2, Figure 3]). Thus from Bohanon's result we immediately get length(side k) = Perimeter(F).
Combining this with (10), one can replace Ω A dν by the perimeter of F in the denominator of (9), and we obtain the formula (4), that is, Step 3: maximum of the entropy. To prove that Theorem 2(ii) follows from (4) we only need to show that for each genus g the perimeter of F in T (g) is minimized on the regular polygon.
• Isoareal Inequality: among all hyperbolic polygons with a given area and number of sides, the regular polygon has the smallest perimeter. More precisely, Ku-Ku-Zhang prove [22, Theorem 1.2(a)] that for a hyperbolic n-gon P n , with equality achieved on a regular polygon. The Isoareal Inequality follows immediately: Area(P n ) and n are constant, so the right-hand side 4d n Area(P n ) is constant and thus the perimeter is minimized when P n is regular. • In our setting, F = P n with n = 8g − 4, and Area(F) = 2π(2g − 2) is constant in T (g), so the Isoareal Inequality implies that the perimeter of F is minimized when F is regular. The expression for the maximum value H(g) in Theorem 2 comes directly from (4), with arccosh 1 + 2 cos π 4g−2 being the length of a single side of the regular (8g − 4)-gon. This completes the proof of Theorem 2(ii).
Step 4: flexibility of the entropy. The space T (g) is homeomorphic to R 6g−6 , and a standard way to parametrize T (g) is through Fenchel-Nielsen coordinates (see the classical manuscript recently published in [15]). The surface S can be decomposed into 2g − 2 pairs of pants by 3g − 3 non-intersecting closed geodesics; this decomposition is shown for g = 2 in the bottom of Figure 7 (for genus 3, see [3, Figure 4]). The lengths of these geodesics can be manipulated independently (they form 3g − 3 of the 6g − 6 coordinates) and can take arbitrarily large values. We take one of these geodesics to also be a geodesic from the necklace described in Section 1 that corresponds to one entire side of F (this shared geodesic is on the far right in both parts of Figure 7). Since the length of this side-one of the Fenchel-Nielsen coordinates-can be made arbitrarily large, the perimeter of F can also be made arbitrarily large, which by (4) means that h µ A (f A ) can be made arbitrarily small. Using the topology on the Teichmüller space T (g) as the space of marked canonical (8g − 4)-gons, we see that the perimeter of F varies continuously within T (g). From (4) we conclude the continuity of the entropy h µ A (f A ) within T (g). By the Intermediate Value Theorem, h µ A (f A ) must take on all values between 0 and its maximum; this is precisely the claim of Theorem 2(iii).

4.1.
Comparison of entropies. By the Variational Principle, the measure-theoretic entropy of a map can never exceed its topological entropy. For some classical systems, the Lebesgue measure is both a smooth invariant measure and also the measure of maximal entropy, but the boundary maps f A : S → S provide examples where the ergodic smooth invariant probability measure is not the measure of maximal entropy.

Proposition 5 ([3, Corollary 7]). If
A is extremal or has short cycles, the measuretheoretic entropy of f A with respect to its smooth invariant measure µ A is strictly less than the topological entropy of f A . From the expression for H(g) in Theorem 2(ii), the maximum possible value of the measure-theoretic entropy with respect to the smooth invariant measure is , and we show in [3,Section 4] that this is strictly larger than the topological entropy for all g ≥ 2. See Figure 8 for a graph of these values.

Examples of other Markov partitions.
Only countably many multi-parameters A will admit a Markov partition for f A . Specifically, Markov cases occur when the right and left orbits of A k , that is, are eventually periodic for each A k (these are called "upper" and "lower" orbits in [19]). For extremal A, this is immediate. If A has the cycle property, it is sufficient that the "cycle ends" are discontinuity points, as stated for short cycles in [2, Proposition 7.2]. When f A is Markov, the maximal eigenvalue of the associated transition matrix must be λ = 4g − 3 + (4g − 3) 2 − 1 because we know that h top (f A ) is the log of this value for all parameters A. Given a Markov partition {I 1 , ..., I n } with n elements, the entries of the associated n × n transition matrix are described in (5). In this section, we give four examples of maps and transition matrices with, respectively, (a) n = 2(8g − 4) (b) n = 8g − 4 (c) n = 3(8g − 4) (d) n = 5 2 (8g − 4) partition elements. Figure 9 shows these matrices for genus 2 (each white cell is a 0 in the matrix, and each black cell is a 1); each matrix is for a different map f A and uses a different partition {I 1 , ..., I n }, but all have the same maximal eigenvalue.
(a) Let A k = P k for all k. As described in [5,3], the partition {I 1 , ..., I 16g−8 } with is a Markov partition for f P (and, in fact, for every extremal A). For g = 2, the matrices M P and M Q for this partition are shown in Figure 3, and M P is shown again in Figure 9(a) as a colored grid. (b) For A = P Q = {P 1 , Q 2 , P 3 , Q 4 , ..., } we could use the partition {I 1 , ..., I 16g−8 } from (a) and [4], but the coarser partition {I 1 , ..., I 8g−4 } with I k = [P k , Q k+1 ] for odd k, I k = [Q k , P k+1 ] for even k is also Markov for f P Q . Using this partition we get a much smaller transition matrix (it is (8g − 4) × (8g − 4) rather than (16g − 8) × (16g − 8)) with exactly the same maximal eigenvalue. For g = 2, this 12 × 12 matrix is shown in Figure 9(b).
(c) Let A k be the midpoint of [P k , Q k ] for each k. As described in [2,, this f A has a Markov partition {I 1 , ..., I 24g−12 } with where B k := T σ(k−1) A σ(k−1) and C k := T σ(k+1) A σ(k+1)+1 are the first iterates in the right and left orbits. For g = 2, the corresponding 36 × 36 transition matrix is shown in Figure 9(c).
(d) For odd k let A k be the midpoint of [P k , Q k ], and for even k let A k be the image under T σ(k)+1 •T k of the midpoint of [P k+1 , Q k+1 ]. In [2,Section 8,Example 3] we show that this f A has a partition with 3(8g − 4) intervals following the same construction as (c). However, for this particular A we have that C k+1 = B k+1 if k is odd, which means that the intervals I 5 = I 3(2)−1 = [C 2 , B 2 ], I 11 = I 3(4)−1 = [C 4 , B 4 ], etc., are trivial. These comprise 4g − 2 of the 24g − 12 intervals.
Thus we can make a Markov partition for this f A using only 20g − 10 intervals. Explicitly, we use {I 1 , ..., I 20g−10 } given by For g = 2, the corresponding 30 × 30 transition matrix is shown in Figure 9(d).
The correspondence between parameters (α, β, γ, σ, τ, ρ) and polygons F can be made explicit as follows. Let e σ+γ e σ+γ −e ρ e −ρ , E = A −1 C −1 , and F = D −1 B −1 (the eigenvalues of E and F are, respectively, e ± and e ±φ , but the matrices that diagonalize E and F do not seem to have simple expressions). Then the sides of the polygon F lie along axes of Möbius transformations that are products of these matrices; explicitly, if s k is the transformation for which P k is the repelling fixed point and Q k+1 is the attracting fixed point, then Similarly, the combined length of sides 3 and 9 is 2 arccosh | tr(F )/2| = 2φ, and the length of side 4 is 2 arccosh | tr(A)/2| = 2α. Sides 6, 7, 10, 11, 12 have the same length as, respectively, sides 8, 1, 4, 9, 2 because those respective side pairs are identified. Thus the total perimeter of F is 2(2δ + 2 + 2φ + 2α) = 4(α + δ + + φ).
Remark. For many surfaces (e.g., all those with σ = τ = ρ = 0) sides 2 and 8 will each have length , sides 3 and 9 will each have length φ, etc., but in general we require only that sides 2 and 8 together have total length 2 , and so on, with only sides 1, 4, 7, 10 having simple expressions for their individual lengths.

Open questions
The flexibility results of Section 3 are proved under the assumption that the multiparameter A is extremal or has short cycles. In order to say anything about the measure-theoretic entropy h µ A (f A ), we must have a well-defined measure µ A , and this is not guaranteed in all cases. This could be established directly-Conjecture 1-or through the connection to geodesic flow and natural extension maps-Conjectures 2 and 3. Additionally, we use the conjugacy Φ A : Ω geo → Ω A between F geo and F A Ω A to prove that h µ A (f A ) = π · Area(F)/ν(Ω A ) from Step 2 of Section 3, and this is currently known only for extremal and short cycle multi-parameters. Conjecture 1. For any F ∈ T (g) and any A with A k ∈ [P k , Q k ], there exists a smooth f A -invariant ergodic probability measure µ A . The existence of this measure for maps f A admitting a Markov partition can be established directly using Adler's "Folklore Theorem," similarly to the case A = P discussed in [5,13,12]. For regular fundamental polygons, existence can be established using results about expanding, non-Markov maps in [36].

Conjecture 2.
For any A with A k ∈ [P k , Q k ], there exists a set Ω A ⊂ S × S \ ∆ with finite rectangular structure that is a domain of bijectivity for F A and moreover the global attractor of F A .
Conjecture 2 is part of the "Reduction Theory" proposed by Don Zagier and described for Fuchsian groups in [19,Introduction]. Additionally, understanding the structure of Ω A may help in proving the following: Conjecture 3. For any A with A k ∈ [P k , Q k ], the map F A Ω A is conjugate to F geo by a map Φ A : Ω geo → Ω A that acts piecewise by Möbius transformations.
If Conjecture 3 is true then Theorem 2 in fact holds for all A with A k ∈ [P k , Q k ].