Weak mixing properties of interval exchange transformations and translation flows

2016/05/10 by Artur Avila, Avila, Artur, Martin Leguil +1
Mathematics · #37A05 #37A25 #37E35 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1605.03048

openalex publication_date 2016/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let d >1. In this paper we show that for an irreducible permutation π which is not a rotation, the set of [λ]∈ ℙ₊^d-1 such that the interval exchange transformation f([λ],π) is not weakly mixing does not have full Hausdorff dimension. We also obtain an analogous statement for translation flows. In particular, it strengthens the result of almost sure weak mixing proved by G. Forni and the first author. We adapt here the probabilistic argument developed in their paper in order to get some large deviation results. We then show how the latter can be converted into estimates on the Hausdorff dimension of the set of "bad" parameters in the context of fast decaying cocycles, following a strategy developed by V. Delecroix and the first author.

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