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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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