On the ergodic theory of the real Rel foliation

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 fields, for more general spaces $({\mathcal L}, m_{{\mathcal L}})$, where ${\mathcal L} \subset {{\mathcal {H}}}$ is an orbit-closure for the action of $G = \operatorname {SL}_2({\mathbb {R}})$ (i.e., an affine invariant subvariety) and $m_{{\mathcal L}}$ is the natural measure. These results are conditional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz. We also prove that the entropy of $Z_0$ with respect to any of the measures $m_{{{\mathcal L}}}$ is zero.