Ratner's property for special flows over irrational rotations under functions of bounded variation. II

Abstract

We consider special flows over the rotation on the circle by an irrational $\alpha$ under roof functions of bounded variation. The roof functions, in the Lebesgue decomposition, are assumed to have a continuous singular part coming from a quasi-similar Cantor set (including the Devil's staircase case). Moreover, a finite number of discontinuities is allowed. Assuming that $\alpha$ has bounded partial quotients, we prove that all such flows are weakly mixing and enjoy weak Ratner's property. Moreover, we provide a sufficient condition for the roof function to obtain a stability of the cocycle Ratner's property for the resulting special flow.

arXiv comment15 pages, this paper is a continuation of arXiv:1302.3429 and arXiv:1002.2734. arXiv admin note: text overlap with arXiv:1002.2734 by other authors