Weak mixing properties of interval exchange transformations and translation flows

TL;DR

This paper investigates the weak mixing properties of interval exchange transformations and translation flows, showing that the set of parameters lacking weak mixing has Hausdorff dimension less than full, thus strengthening previous almost sure results.

Contribution

It extends weak mixing results by demonstrating the Hausdorff dimension of non-weakly mixing parameters is not full, using probabilistic and large deviation techniques.

Findings

Set of non-weakly mixing parameters has less than full Hausdorff dimension.

Probabilistic methods adapted to estimate large deviations.

Results apply to both interval exchange transformations and translation flows.

Abstract

Let $d >1$. In this paper we show that for an irreducible permutation $\pi$ which is not a rotation, the set of $[\lambda]\in \mathbb{P}_+^{d-1}$ such that the interval exchange transformation $f([\lambda],\pi)$ 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.