Dynamics of the Teichmueller flow on compact invariant sets

TL;DR

This paper investigates the growth rate and entropy of periodic orbits of the Teichmueller flow on the moduli space of quadratic differentials, establishing that the maximum growth rate and entropy are both equal to 6g-6+2m.

Contribution

It proves that the supremum of the asymptotic growth rate of periodic orbits and the topological entropy on compact invariant sets equals 6g-6+2m.

Findings

Maximum growth rate and entropy are both 6g-6+2m.

The supremum over all compact subsets of the flow's periodic orbits matches the topological entropy.

The results connect the geometric complexity of the flow with the topology of the surface.

Abstract

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).