Periodic orbits in the thin part of strata

Abstract

Let S be a closed oriented surface of genus g ≥ 0 {g\geq 0} with n ≥ 0 {n\geq 0} punctures and 3 ⁢ g - 3 + n ≥ 5 {3g-3+n\geq 5} . Let 𝒬 {{\mathcal{Q}}} be a connected component of a stratum in the moduli space 𝒬 ⁢ ( S ) {{\mathcal{Q}}(S)} of area one meromorphic quadratic differentials on S with n simple poles at the punctures or in the moduli space ℋ ⁢ ( S ) {{\mathcal{H}}(S)} of abelian differentials on S if n = 0 {n=0} . For a compact subset K of 𝒬 ⁢ ( S ) {{\mathcal{Q}}(S)} or of ℋ ⁢ ( S ) {{\mathcal{H}}(S)} , we show that the asymptotic growth rate of the number of periodic orbits for the Teichmüller flow Φ t {\Phi^{t}} on 𝒬 {{\mathcal{Q}}} which are entirely contained in 𝒬 - K