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

2010/02/13 by Krzysztof Frączek, Mariusz Lemańczyk, Fraczek, K. +1
Mathematics · #37A10 #37C40 #37E35 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1002.2734

openalex publication_date 2010/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider special flows over two-dimensional rotations by (α,β) on \T² and under piecewise C² roof functions f satisfying von Neumann's condition ∫_\T²f_x(x,y) dx dy≠ 0≠ ∫_\T²f_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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