Periodic orbits in the thin part of strata

Let S be a closed oriented surface of genus $g\ge 0$ with $n\ge 0$ punctures and $3g-3+n\ge 5$. Let $Q$ be a connected component of a stratum in the moduli space Q(S) of area one meromorphic quadratic differentials on S with n simple poles at the punctures or in the moduli space H(S) of abelian differentials on S if n=0. For a compact subset K of Q(S) or of H(S), we show that the asymptotic growth rate of the number of periodic orbits for the Teichmueller flow on Q which are entirely contained in Q-K is at least h(Q)-1 where h(Q)>0 is the complex dimension of Q.