On the ergodic theory of the real Rel foliation

On the ergodic theory of the real Rel foliation

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Let ${{mathcal {H}}}$ be a stratum of translation surfaces with at least two singularities, let $m_{{{mathcal {H}}}}$ Flight Socks 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 $({mathcal L}, m_{{mathcal L}})$ , Safety Key Board 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.