Geodesic
Ratner's property for special flows over irrational rotations under functions of bounded variation. II
Adam Kanigowski1
1 Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland
Colloquium Mathematicum, Tome 136 (2014) no. 1, pp. 125-147

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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.
DOI : 10.4064/cm136-1-11
Keywords: consider special flows rotation circle irrational alpha under roof functions bounded variation roof functions lebesgue decomposition assumed have continuous singular part coming quasi similar cantor set including devils staircase moreover finite number discontinuities allowed assuming alpha has bounded partial quotients prove flows weakly mixing enjoy weak ratner property moreover provide sufficient condition roof function stability ratners cocycle property resulting special flow

Adam Kanigowski 1

1 Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland 
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Adam Kanigowski. Ratner's property for special flows over
 irrational rotations under functions of
 bounded variation. II. Colloquium Mathematicum, Tome 136 (2014) no. 1, pp. 125-147. doi: 10.4064/cm136-1-11

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