Asymptotics of Weil-Petersson geodesics II: bounded geometry and unbounded entropy

TL;DR

This paper explores the relationship between bounded geometry and bounded combinatorics for Weil-Petersson geodesics, and demonstrates that the geodesic flow can have invariant subsets with arbitrarily large topological entropy.

Contribution

It establishes the equivalence between bounded geometry and bounded combinatorics for Weil-Petersson geodesics and introduces a broader notion of non-annular bounded combinatorics.

Findings

Bounded geometry is equivalent to bounded combinatorics for Weil-Petersson geodesics.

A generalized non-annular bounded combinatorics condition is introduced.

The Weil-Petersson geodesic flow can have invariant subsets with arbitrarily large topological entropy.

Abstract

We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil-Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equivalent condition for Weil-Petersson geodesics. As an application, we show the Weil-Petersson geodesic flow has compact invariant subsets with arbitrarily large topological entropy.