Ratner's property and mixing for special flows over two-dimensional rotations

Details Ratner's property and mixing for special flows over two-dimensional rotations Ratner's property and mixing for special flows over two-dimensional rotations

2010-02-13

arxiv.org/abs/1002.2734v2

Mathematics

Dynamical Systems

Scientific paper

We consider special flows over two-dimensional rotations by $(\alpha,\beta)$ on $\T^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition $\int_{\T^2}f_x(x,y)\,dx\,dy\neq 0\neq \int_{\T^2}f_y(x,y)\,dx\,dy.$ Such flows are shown to be always weakly mixing and never partially rigid. For an uncountable set of $(\alpha,\beta)$ with both $\alpha$ and $\beta$ of unbounded partial quotients the strong mixing property is proved to hold. It is also proved that while specifying to a subclass of roof functions and to ergodic rotations for which $\alpha$ and $\beta$ are of bounded partial quotients the corresponding special flows enjoy so called weak Ratner's property. As a consequence, such flows turn out to be mildly mixing.

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