Effective Unique Ergodicity and Weak Mixing of Translation Flows

TL;DR

This paper introduces a cohomological approach using Hodge theory to analyze effective unique ergodicity and weak mixing in translation flows, providing new proofs and insights into their dynamics.

Contribution

It presents a novel cohomological proof of Veech's criterion for weak mixing and emphasizes the analogy between ergodicity and weak mixing through twisted cohomology.

Findings

New cohomological proof of Veech's criterion for weak mixing

Enhanced understanding of translation flows via Hodge theory

Connections between ergodicity and weak mixing established

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.