Effective Unique Ergodicity and Weak Mixing of Translation Flows

Abstract

This text is an introduction to the author's cohomological approach, based on Hodge theory, to (effective) unique ergodicity and weak mixing of translation flows. Compared to earlier expositions, it emphasizes the analogy between the two problems by introducing a point of view on weak mixing based on the appropriate twisted cohomology. In particular, a new cohomological proof, based on Hodge theory for the twisted cohomology, of Veech's classical criterion for weak mixing is presented here for the first time. The exposition also aims to give a general introduction to the ergodic theory of translation flows and related systems and includes references to background material and related developments in the theory, as well as several exercises, open problems and conjectures.