We consider special flows over the rotation on the circle by an irrational α 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 α 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm136-1-11,
author = {Adam Kanigowski},
title = {Ratner's property for special flows over irrational rotations under functions of bounded variation. II},
journal = {Colloquium Mathematicae},
volume = {135},
year = {2014},
pages = {125-147},
zbl = {06330477},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm136-1-11}
}
Adam Kanigowski. Ratner's property for special flows over irrational rotations under functions of bounded variation. II. Colloquium Mathematicae, Tome 135 (2014) pp. 125-147. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm136-1-11/