TY - JOUR

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

AU - Kanigowski, Adam

AU - Kułaga-Przymus, Joanna

N1 - Publisher Copyright:
© Cambridge University Press, 2015.

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.

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U2 - 10.1017/etds.2015.35

DO - 10.1017/etds.2015.35

M3 - Article

AN - SCOPUS:84940092859

SN - 0143-3857

VL - 36

SP - 2512

EP - 2537

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 8

ER -