Existence and uniqueness of the measure of maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials

Abstract

The Teichmüller geodesic flow {g t }, first studied by H. Masur [15] and W. Veech [21], acts on the moduli space of Riemann surfaces endowed with a holomorphic differential. More precisely, let S be a closed surface of genus g ≥ 2. One introduces on S a complex structure σ and a holomorphic differential ω. The pair (σ, ω) is considered to be equivalent to another pair of the same nature (σ 1 , ω 1 ) if there is a diffeomorphism of S sending (σ, ω) to (σ 1 , ω 1 ). The moduli space M(g) consists of the equivalence classes, and the flow {g t } on M(g) is induced by the action on the pairs (σ, ω) defined by the formula g t (σ, ω) = (σ ′ , ω ′ ), where ω ′ = e t ℜ(ω) + ie −t ℑ(ω), while the complex structure σ ′ is determined by the requirement that ω ′ be holomorphic. If (σ, ω) and (σ ′ , ω ′ ) are equivalent, then the differentials ω and ω ′ have the same orders of zeros and the same area. Therefore, these orders and area are well-defined on M(g). Moreover, they are preserved by the Teichmüller flow {g t }. Take an arbitrary non-ordered collection κ = (k 1 , . . . , k r ) with k i ∈ N, k 1 + · · · + k r = 2g − 2, and denote by M κ the subspace of M(g) corresponding to the differentials of area 1 (i.e., (i/2) ω ∧ω = 1) with orders of zeros k i , i = 1, . . . , r; M κ is said to be a stratum in M(g). Each stratum

On a measure with maximal entropy for a suspension flow over a countable alphabet Markov shift

“…In other words, taking into account that our evaluation of the topological entropy is valid not only for the specific suspension flow related to the Teihmüller flow, but for every suspension flow that has an invariant measure with the above properties, is it true that the entropy of this flow with respect to such a measure coincides with its topological entropy? It could seem that the whole information contained in the assumptions on μ was already used in [2]. In fact this is not true.…”

“…Remark 2.5 While the former condition can be essentially relaxed, it enables us to use results from [2], where this condition is present. In contrast, the latter condition, which is also present in [2], is significant.…”

“…Suspension constructions (flows and cascades) over countable alphabet Markov shifts turn out to be a powerful tool in the study of some classes of smooth dynamical systems on manifolds [1,2,6,10].…”

“…In fact this is not true. In the present paper we are based on the same assumptions as in [2], but exploit them in a different way (see the proof of Theorem 2.10 below).…”

“…The present paper is motivated by the study in [1,2] of a measure with maximal entropy for the Teihmüller flow on the moduli space of Abelian differentials. The following result was obtained there: a measure with maximal entropy for this flow exists, is unique, and moreover, it is the well-known invariant measure constructed by H. Masur and W. Veech in the 80th.…”