Dynamics Seminar

Event Information Weak mixing of interval exchange transformations and translation flows
15:00 on Wednesday September 13, 2017
16:00 on Wednesday September 13, 2017
HU1018, 215 Huron St.
Martin Leguil

University of Toronto

An interval exchange transformation $f\colon I \to I$ can be parametrized by two data: a length vector $\lambda \in \mathbb{R}_+^d$ and a permutation $\pi\in \mathfrak{S}_d$. We denote $f=f(\lambda,\pi)$. Such a map can be seen as a discrete version of the geodesic flow on some translation surface. A criterion due to Veech relates the weak mixing property to the dynamics of a cocycle in the parameter space of interval exchange transformations. Based on this fact, Avila and Forni developed a probabilistic argument to show that when $\pi$ is irreducible and is not a rotation, $f(\lambda,\pi)$ is weak mixing for almost every $\lambda$. In a joint work with A. Avila, we prove that in fact, the set of $\lambda$ such that $f(\lambda,\pi)$ is not weak mixing does not have full Hausdorff dimension. Important ingredients of the proof are the ``fast-decay'' property of the dynamics in parameter space, and a large deviation result which improves the estimates of Avila-Forni.