On the ergodic theory of the real Rel foliation

Let H be a stratum of translation surfaces with at least two singularities, let mH denote the Masur-Veech measure on H, and let Z(0) be a flow on (H,mH) 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 (L,m(L)), where L subset of H is an orbit-closure for the action of G=SL2(R) (i.e., an affine invariant subvariety) and mL 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 m(L) has zero entropy.