On the Space of Ergodic Measures for the Horocycle Flow on Strata of Abelian Differentials

TL;DR

This paper investigates the behavior of the horocycle flow on the stratum of translation surfaces, revealing the existence of non-ergodic limit measures and non-equidistributing orbits, which deepen understanding of dynamical properties in this setting.

Contribution

It constructs explicit sequences of ergodic measures supported on periodic orbits that converge to non-ergodic invariant measures, highlighting complex orbit structures.

Findings

Existence of non-ergodic invariant measures as limits of ergodic measures.

Some points in the stratum have horocycle orbits that do not equidistribute.

Demonstrates intricate measure and orbit behavior in the moduli space.

Abstract

We study the horocycle flow on the stratum of translation surfaces $\mathcal{H}(2)$. We show that there is a sequence of horocycle ergodic measures, each supported on a periodic horocycle orbit, which weakly converges to an invariant, but non-ergodic, measure by $\mathrm{SL}_2(\mathbb{R})$. As a consequence, we show that there are points in $\mathcal{H}(2)$ whose horocycle flow orbits do not equidistribute towards any invariant measure.