Flexibility of measure-theoretic entropy of boundary maps associated to Fuchsian groups

Abstract

Given a closed, orientable, compact surface S of constant negative curvature and genus $g \geq 2$ , we study the measure-theoretic entropy of the Bowen–Series boundary map with respect to its smooth invariant measure. We obtain an explicit formula for the entropy that only depends on the perimeter of the $(8g-4)$ -sided fundamental polygon of the surface S and its genus. Using this, we analyze how the entropy changes in the Teichmüller space of S and prove the following flexibility result: the measure-theoretic entropy takes all values between 0 and a maximum that is achieved on the surface that admits a regular $(8g-4)$ -sided fundamental polygon. We also compare the measure-theoretic entropy to the topological entropy of these maps and show that the smooth invariant measure is not a measure of maximal entropy.

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

“…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.…”

“…• 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. 3 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].…”

“…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 8 (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.…”

“…Figure 10 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) Ā = P (a) Let A k = P k for all k. As described in [5,3], the partition {I 1 , ..., I 16g−8 } with…”

“…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…”