On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces

Abstract

AbstractWe study billiard dynamics on non-compact polygonal surfaces with a free, cocompact action of ℤ or ℤ2. In the ℤ-periodic case, we establish criteria for conservativity. In the ℤ2-periodic case, we study a particular family of such surfaces, the rectangular Lorenz gas. Assuming that the obstacles are sufficiently small, we obtain the ergodic decomposition of directional billiards for a finite but asymptotically dense set of directions. This is based on our study of ergodicity for ℤd-valued cocycles over irrational rotations.