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

TL;DR

This paper studies special flows over irrational circle rotations with bounded variation roof functions, demonstrating weak mixing and Ratner's property under certain conditions, including stability of the cocycle Ratner's property.

Contribution

It establishes weak Ratner's property for special flows with complex roof functions, including those with singular continuous parts and discontinuities, extending previous results.

Findings

All such flows are weakly mixing.

They satisfy weak Ratner's property.

A sufficient condition for cocycle Ratner's property stability is provided.

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.