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

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

2006-01-18

arxiv.org/abs/math/0601430v2

Mathematics

Dynamical Systems

22 pages

Scientific paper

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 $(\phi_t)_{t\in\R}$ preserves a smooth measure, its set of ergodic components consists of a family of periodic orbits and one component of positive measure on which $(\phi_t)_{t\in\R}$ is mildly mixing and is spectrally disjoint from all mixing flows.

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