Dynamical Systems Seminar

The Rokhlin problem asks whether measure theoretic 2-mixing implies k-mixing, for any k greater then 2. In the talk we consider certain classes of parabolic (that is with orbits diverging polynomially) flows on surfaces and show that the answer to the Rokhlin problem is affirmative for those classes. The methods of proofs rely heavily on recent results of Kanigowski and Ravotti from 2024 and an adaptation of methods introduced there. We will explain the main mechanism of mixing in parabolic flows (stretching of Birkhoff sums), outline some of the results in the field, and sketch proofs. Specifically, we extend the recent results of Kanigowski and Ravotti from 1-torus to higher dimensions, and apply the results to analytic mixing time changes of translation flows on m-torus, m greater then 3, and mixing time changes of Heisenberg nilflows to show that the flows are mixing of any order. This generalizes the corresponding results of Fayad, and Avila, Forni and Ulcigrai from mixing to mixing of any order. The talk is based on work in progress Multiple mixing for smooth time-changes of irrational flows on m-torus, Heisenberg flows and general nilflows.