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A convexity criterion for unique ergodicity of interval exchange transformations. (English) Zbl 1440.37050
For \(d \in \mathbb{Z}^+\), \(a(s)=(s,s^2,\dots,s^d)\) can be viewed as a curve in \(\mathbb{R}_+^d\) if \(s > 0\). Y. Minsky and B. Weiss [Ann. Sci. Éc. Norm. Supér. (4) 47, No. 2, 245–284 (2014; Zbl 1346.37039)] proved for Lebesgue almost all \(s > 0\) that the interval exchange transformation associated to \((\sigma, a(s))\) is uniquely ergodic for the permutation \(\sigma=(d,\dots,1)\).
The present author extends this result to arbitrary permutations through a convexity criterion leading to unique ergodicity for points of a curve in the space of interval exchange transformations.
MSC:
37E10 Dynamical systems involving maps of the circle
37A25 Ergodicity, mixing, rates of mixing
28D10 One-parameter continuous families of measure-preserving transformations
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References:
[1] 10.1090/pspum/069/1858548
[2] 10.2307/1971341 · Zbl 0497.28012
[3] ; Minsky, Ann. Sci. Éc. Norm. Supér. (4), 47, 245 (2014)
[4] 10.2307/1971391 · Zbl 0486.28014
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