Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces

Abstract

We consider typical area-preserving flows on higher genus surfaces and prove that the flow restricted to mixing minimal components is mixing of all orders, thus answering affirmatively Rokhlin’s multiple mixing question in this context. The main tool is a variation of the Ratner property originally proved by Ratner for the horocycle flow, i.e. the switchable Ratner property introduced by Fayad and Kanigowski for special flows over rotations. This property, which is of independent interest, provides a quantitative description of the parabolic behavior of these flows and has implications for joining classification. The main result is formulated in the language of special flows over interval exchange transformations with asymmetric logarithmic singularities. We also prove a strengthening of one of Fayad and Kanigowski’s main results, by showing that Arnold’s flows are mixing of all orders for almost every location of the singularities.