Logarithmic asymptotics for the number of periodic orbits of the Teichmüller flow on Veech’s space of zippered rectangles

The aim of this note is to obtain a logarithmic asymptotics for number of periodic orbits on Veech’s space of zippered rectangles. The space of zippered rectangles is a finite cover of a set of full measure in the moduli space of abelian differentials with prescribed singularities. In particular, all periodic orbit in moduli space can be lifted to the space of zippered rectangles. Let R be a Rauzy class of permutations on m symbols, let Ω0(R) be Veech’s moduli space of zippered rectangles, and let P t be the Teichmüller flow on Ω0(R) (see the next section for precise definitions). Denote by Per(R, T ) the set of periodic orbits for P t of period not exceeding T . If A is a finite set then #A will stand for its cardinality.