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

Adam Kanigowski

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

We consider special flows over the rotation by an irrational α under the roof functions of bounded variation without continuous, singular part in the Lebesgue decomposition and sum of jumps not equal to zero. We show that all such flows are weakly mixing. Under the additional assumption that α has bounded partial quotients, we study the weak Ratner property. We establish this property whenever an additional condition (stable under sufficiently small perturbations) on the set of jumps is satisfied. While it is a classical result that the flows under consideration are not mixing, one more condition on the set of jumps turns out to be sufficient to obtain the absence of partial rigidity, hence mild mixing of such flows.

Original languageEnglish (US)
Pages (from-to)915-934
Number of pages20
JournalErgodic Theory and Dynamical Systems
Volume35
Issue number3
DOIs
StatePublished - Aug 28 2015

Fingerprint

Functions of Bounded Variation
Jump
Partial
Rigidity
Roofs
Henri Léon Lebésgue
Small Perturbations
Decomposition
Quotient
Sufficient
Decompose
Zero

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

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Ratner's property for special flows over irrational rotations under functions of bounded variation. / Kanigowski, Adam.

In: Ergodic Theory and Dynamical Systems, Vol. 35, No. 3, 28.08.2015, p. 915-934.

Research output: Contribution to journalArticle

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AB - We consider special flows over the rotation by an irrational α under the roof functions of bounded variation without continuous, singular part in the Lebesgue decomposition and sum of jumps not equal to zero. We show that all such flows are weakly mixing. Under the additional assumption that α has bounded partial quotients, we study the weak Ratner property. We establish this property whenever an additional condition (stable under sufficiently small perturbations) on the set of jumps is satisfied. While it is a classical result that the flows under consideration are not mixing, one more condition on the set of jumps turns out to be sufficient to obtain the absence of partial rigidity, hence mild mixing of such flows.

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