On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces

TL;DR

This paper investigates recurrence and ergodicity of billiard flows on noncompact, periodic polygonal surfaces, providing criteria and results for specific models like the wind-tree, and analyzing cocycles over irrational rotations.

Contribution

It establishes recurrence criteria for Z-periodic surfaces, extends results to Z^2-periodic cases, and analyzes ergodic decomposition for the wind-tree model under certain geometric restrictions.

Findings

Criteria for recurrence in Z-periodic billiards

Ergodic decomposition for wind-tree model in specific directions

Results on ergodicity of Z-valued cocycles over irrational rotations

Abstract

We study the recurrence and ergodicity for the billiard on noncompact polygonal surfaces with a free, cocompact action of $\Z$ or $\Z^2$. In the $\Z$-periodic case, we establish criteria for recurrence. In the more difficult $\Z^2$-periodic case, we establish some general results. For a particular family of $\Z^2$-periodic polygonal surfaces, known in the physics literature as the wind-tree model, assuming certain restrictions of geometric nature, we obtain the ergodic decomposition of directional billiard dynamics for a dense, countable set of directions. This is a consequence of our results on the ergodicity of $\ZZ$-valued cocycles over irrational rotations.