Recurrence and non-ergodicity in generalized wind-tree models

TL;DR

This paper investigates generalized wind-tree models and $ ext{Z}^d$-covers over compact translation surfaces, demonstrating that under certain conditions, the linear flow is recurrent in a generic direction but not ergodic with respect to Lebesgue measure.

Contribution

It proves recurrence and non-ergodicity properties of linear flows in generalized wind-tree models under specific hypotheses.

Findings

Recurrence of linear flow in generic directions

Non-ergodicity of Lebesgue measure in these models

Results apply to $ ext{Z}^d$-covers over compact translation surfaces

Abstract

In this paper, we consider generalized wind-tree models and $\Z^d$-covers over compact translation surfaces. Under suitable hypothesis, we prove recurrence of the linear flow in a generic direction and non-ergodicity of Lebesgue measure.