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arXiv · 1008.0136

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

Jean-Pierre Conze·Eugene Gutkin
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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.

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Jean-Pierre Conze, Eugene Gutkin. 2010-08-01. On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces. https://doi.org/10.1007/s00021-011-0071-0

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