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arXiv · 1002.2734

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

K. Fraczek·M. Lemanczyk
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Abstract

We consider special flows over two-dimensional rotations by $(α,β)$ 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 $(α,β)$ with both $α$ and $β$ 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 $α$ and $β$ 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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K. Fraczek, M. Lemanczyk. 2010-12-15. Ratner's property and mixing for special flows over two-dimensional rotations. https://arxiv.org/abs/1002.2734

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