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

TL;DR

This paper studies special flows over two-dimensional rotations with specific roof functions, proving weak mixing generally, strong mixing for uncountably many parameters, and mild mixing under certain bounded partial quotient conditions.

Contribution

It establishes weak and strong mixing properties and introduces weak Ratner's property for special flows over 2D rotations with piecewise smooth roof functions.

Findings

Flows are always weakly mixing.

Strong mixing holds for uncountably many rotation parameters.

Certain flows are mildly mixing due to weak Ratner's property.

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 $\int_{\T^2}f_x(x,y)\,dx\,dy\neq 0\neq \int_{\T^2}f_y(x,y)\,dx\,dy.$ 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.