  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.

Dynamical Systems