Weak mixing in polygonal billiards

Abstract

We survey known and conjectural ergodic properties of billiards in rational and non-rational polygons. The directional flows of billiards in rational polygons were proved to be ergodic in almost all directions by Kerckhoff, Masur and Smillie in 1986. In the same paper, the authors derived the existence of a $G_{\delta }$ dense set of ergodic (non-rational) polygons. In recent joint work, we have proved analogous results for the weak mixing property. In joint work with Jon Chaika, we have proved the existence of a $G_{\delta }$ dense set of weakly mixing (non-rational) polygons and recently with Francisco Arana-Herrera and Jon Chaika, we have characterized the property of weak mixing in almost all directions of translation surfaces with applications to billiards in rational polygons. All these results are based on techniques of Teichmüller dynamics and their application to the weak mixing problem following the joint work with A. Avila from 2007. Recent advances rest on measure rigidity results of Eskin, Mirzakhani and Mohammadi and related work of Chaika and Eskin on the multiplicative ergodic theorem in Teichmüller dynamics and results of Filip and Wright on the relevant (Kontsevich–Zorich) renormalization cocycle.