On the ergodic theory of the real Rel foliation

TL;DR

This paper proves that certain flows on translation surface moduli spaces are mixing of all orders and ergodic, with results conditional on a future measure classification theorem, and shows these flows have zero entropy.

Contribution

It establishes the ergodic and mixing properties of Rel flow-induced actions on moduli spaces of translation surfaces, extending understanding of their dynamical behavior.

Findings

Rel flows are mixing of all orders and ergodic.

Flows have zero entropy.

Results depend on a forthcoming measure classification theorem.

Abstract

Let $\mathcal{H}$ be a stratum of translation surfaces with at least two singularities, let $m_{\mathcal{H}}$ denote the Masur-Veech measure on $\mathcal{H}$, and let $Z_0$ be a flow on $(\mathcal{H}, m_{\mathcal{H}})$ obtained by integrating a Rel vector field. We prove that $Z_0$ is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector field, for more general spaces $(\mathcal{L}, m_{\mathcal{L}})$, where $\mathcal{L} \subset \mathcal{H}$ is an orbit-closure for the action of $G = \mathrm{SL}_2(\mathbb{R})$ (i.e., an affine invariant subvariety) and $m_{\mathcal{L}}$ is the natural measure. Our results are conditional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz.We also prove that the entropy of the action of $Z_0$ on $(\mathcal{L}, m_{\mathcal{L})$ has zero entropy.