  Let Q(S) be the moduli space of area one holomorphic quadratic differentials
for an oriented surface S of genus g with m punctures and 3g-3+m>1. We show
that the supremum over all compact subsets K of Q(S) of the asymptotic growth
rate of the number of periodic orbits of the Teichmueller flow which are
contained in K equals h=6g-6+2m. Moreover, h is also the supremum of the
topological entropies of the restriction of the Teichmueller flow to compact
invariant subsets of Q(S).
