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Journal of Modern Dynamics (JMD)
 

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

Pages: 609 - 635, Issue 4, October 2010      doi:10.3934/jmd.2010.4.609

 
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Krzysztof Frączek - Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland (email)
Mariusz Lemańczyk - Faculty of Mathematics and Computer Science, N. Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland (email)

Abstract: 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 $\int_{\T^2}f_x(x,y)dxdy\ne 0$ or $\int_{\T^2}f_y(x,y)dxdy\ne 0 $. Such flows are shown to be always weakly mixing and never partially rigid. It is 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 the so-called weak Ratner property. As a consequence, such flows turn out to be mildly mixing.

Keywords:  Measure-preserving flows, special flows, mild mixing, Ratner's property.
Mathematics Subject Classification:  Primary: 37A10; Secondary: 37C40.

Received: February 2010;      Revised: December 2010;      Available Online: January 2011.

 References