Extremal parameters and their duals for boundary maps associated to Fuchsian groups

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

“…Our two main results are rigidity of the topological entropy and flexibility of the measure-theoretic entropy. 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 marked canonical 4g-gons, see [29], following the earlier work [35,32].…”

“…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].…”

“…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.…”

“…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…”

“…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.…”