On mild mixing of special flows over irrational rotations under piecewise smooth functions

Abstract

It is proved that all special flows over the rotation by an irrational $\alpha$ with bounded partial quotients and under $f$ which is piecewise absolutely continuous with a non-zero sum of jumps are mildly mixing. Such flows are also shown to enjoy a condition which emulates the Ratner condition introduced in \cite{Rat}. As a consequence we construct a smooth vector--field on $\T^2$ with one singularity point such that the corresponding flow $({\varphi }_t{)}_{t\in \mathbb{R}}$ preserves a smooth measure, its set of ergodic components consists of a family of periodic orbits and one component of positive measure on which $({\varphi }_t{)}_{t\in \mathbb{R}}$ is mildly mixing and is spectrally disjoint from all mixing flows.

22 pages