We study the recurrence and ergodicity for the billiard on noncompact
polygonal surfaces with a free, cocompact action of Z or Z2. In the
Z-periodic case, we establish criteria for recurrence. In the more difficult
Z2-periodic case, we establish some general results. For a particular
family of Z2-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.Comment: 48 pages, 12 figure