Ratner's property for special flows over irrational rotations under functions of bounded variation. II
Tom 136 / 2014
Colloquium Mathematicum 136 (2014), 125-147 MSC: Primary 37A25; Secondary 37E35, 37A10. DOI: 10.4064/cm136-1-11
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 the weak Ratner property. Moreover, we provide a sufficient condition on the roof function for stability of Ratner's cocycle property of the resulting special flow.