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 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
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Ratner's property and mild mixing for smooth flows on surfaces. / Kanigowski, Adam; Kułaga-Przymus, Joanna.
In: Ergodic Theory and Dynamical Systems, Vol. 36, No. 8, 01.12.2016, p. 2512-2537.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.
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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 -