二维旋转特殊流动的Ratner性质和混合
Ratner's property and mixing for special flows over two-dimensional rotations
摘要：
本文用$(α,β)$在$\T^2$上和在分段$C^2$屋顶函数$f$满足von Neumann条件$\int_{\T^2}f_x(x,y)\,dx\,dy\neq 0\neq \int_{\T^2}f_y(x,y)\,dx\,dy.$下考虑二维旋转的特殊流动。对于既有$α$又有$β$的不可数集$(α,β)$，证明了强混合性质成立.本文还证明了，当指定一个屋顶函数子类和遍历旋转时，当$α$和$β$是有界偏导数时，相应的特殊流具有所谓的弱Ratner性质.因此，这种流动被证明是温和的混合。
英文摘要：
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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