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

Abstract

It is proved that all special flows over a 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 that emulates the Ratner condition introduced in M. Ratner (Horocycle flows, joinings and rigidity of products. Ann. of Math.118 (1983), 277–313). As a consequence we construct a smooth vector-field on $\mathbb{T}^2$ with one singularity point for which 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.