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

Adam Kanigowski, Joanna Kułaga-Przymus

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Abstract

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.

Original languageEnglish (US)
Pages (from-to)2512-2537
Number of pages26
JournalErgodic Theory and Dynamical Systems
Volume36
Issue number8
DOIs
Publication statusPublished - Dec 1 2016

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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