Non-ergodic $\mathbb{Z}$ -periodic billiards and infinite translation surfaces

Abstract

We give a criterion which allows to prove non-ergodicity for certain infinite periodic billiards and directional flows on Z-periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with periodically spaced vertical barriers and to the Ehrenfest wind-tree model, which is a planar billiard with a Z 2 -periodic array of rectangular obstacles. We prove that, in these two examples, both for a full measure set of parameters of the billiard tables and for tables with rational parameters, for almost every direction the corresponding billiard flow is not ergodic and has uncountably many ergodic components. As another application, we show that for any recurrent Z-cover of a square tiled surface of genus two the directional flow is not ergodic and has no invariant sets of finite measure for a full measure set of directions. In the language of essential values, we prove that the skewproducts which arise as Poincaré maps of the above systems are associated to non-regular Z-valued cocycles for interval exchange transformations.

Recurrence and non‐ergodicity in generalized wind‐tree models

“…As an application of Theorem (included in Corollary where more general models are considered) we have that the directional billiard flow on is not ergodic for almost every direction. This extends the main result of in which the case and almost all parameters is taken up. We deal also with the problem of recurrence on which was recently solved for in .…”

“…The following two results are closely related to Theorem 2 in and Theorem and Lemma 6.3 in . For the completeness of exposition we include their proofs in Appendix .…”

“…Theorem is a corollary of the more technical Theorem which gives a general criterion for non‐ergodicity. This result, inspired by , relates the number of positive Lyapunov exponents of a compact translation surface to the non‐ergodicity of its ‐covers. Theorem applies to generalized wind‐tree models (see Theorem and Corollary ).…”

“…In geometric terms, the billiard flow induces a linear flow on a doubly periodic surface which is a ‐cover of a compact translation surface X . It is known that, for every parameters and almost every direction, the linear flow is recurrent and non‐ergodic on . Symmetries of the obstacles imply that X is a ‐cover of a genus 2 surface.…”

Diffusion in the Lorentz Gas

“…For small spikes in one of the walls, this is a retro-reflector, reversing almost all incoming trajectories [169]; it is also one of the models shown to be non-ergodic in almost all directions in Ref. [161].…”