### Abstract

Let T =(T^{f}_{t})_{t}ϵℝ be a special flow built over an IET T :T→T of bounded type, under a roof function f with symmetric logarithmic singularities at a subset of discontinuities of T . We show that T satisfies the so-called switchable Ratner's property which was introduced in Fayad and Kanigowski [On multiple mixing for a class of conservative surface flows. Invent. Math. to appear]. A consequence of this fact is that such flows are mildly mixing (before, they were only known to be weakly mixing [Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. J. Mod. Dynam. 3 (2009), 35-49] and not mixing [Ulcigrai. Absence of mixing in areapreserving flows on surfaces. Ann. of Math. (2) 173 (2011), 1743-1778]). Thus, on each compact, connected, orientable surface of genus greater than one there exist flows that are mildly mixing and not mixing.

Original language | English (US) |
---|---|

Pages (from-to) | 2512-2537 |

Number of pages | 26 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 36 |

Issue number | 8 |

DOIs | |

State | Published - Dec 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Ergodic Theory and Dynamical Systems*,

*36*(8), 2512-2537. https://doi.org/10.1017/etds.2015.35

}

*Ergodic Theory and Dynamical Systems*, vol. 36, no. 8, pp. 2512-2537. https://doi.org/10.1017/etds.2015.35

**Ratner's property and mild mixing for smooth flows on surfaces.** / Kanigowski, Adam; Kułaga-Przymus, Joanna.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Ratner's property and mild mixing for smooth flows on surfaces

AU - Kanigowski, Adam

AU - Kułaga-Przymus, Joanna

PY - 2016/12/1

Y1 - 2016/12/1

N2 - Let T =(Tft)tϵℝ be a special flow built over an IET T :T→T of bounded type, under a roof function f with symmetric logarithmic singularities at a subset of discontinuities of T . We show that T satisfies the so-called switchable Ratner's property which was introduced in Fayad and Kanigowski [On multiple mixing for a class of conservative surface flows. Invent. Math. to appear]. A consequence of this fact is that such flows are mildly mixing (before, they were only known to be weakly mixing [Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. J. Mod. Dynam. 3 (2009), 35-49] and not mixing [Ulcigrai. Absence of mixing in areapreserving flows on surfaces. Ann. of Math. (2) 173 (2011), 1743-1778]). Thus, on each compact, connected, orientable surface of genus greater than one there exist flows that are mildly mixing and not mixing.

AB - Let T =(Tft)tϵℝ be a special flow built over an IET T :T→T of bounded type, under a roof function f with symmetric logarithmic singularities at a subset of discontinuities of T . We show that T satisfies the so-called switchable Ratner's property which was introduced in Fayad and Kanigowski [On multiple mixing for a class of conservative surface flows. Invent. Math. to appear]. A consequence of this fact is that such flows are mildly mixing (before, they were only known to be weakly mixing [Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. J. Mod. Dynam. 3 (2009), 35-49] and not mixing [Ulcigrai. Absence of mixing in areapreserving flows on surfaces. Ann. of Math. (2) 173 (2011), 1743-1778]). Thus, on each compact, connected, orientable surface of genus greater than one there exist flows that are mildly mixing and not mixing.

UR - http://www.scopus.com/inward/record.url?scp=84940092859&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84940092859&partnerID=8YFLogxK

U2 - 10.1017/etds.2015.35

DO - 10.1017/etds.2015.35

M3 - Article

AN - SCOPUS:84940092859

VL - 36

SP - 2512

EP - 2537

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 8

ER -