Recurrence and non‐ergodicity in generalized wind‐tree models

Abstract

In this paper, we consider generalized wind‐tree models and double-struckZd‐covers over compact translation surfaces. Under suitable hypothesis, we prove the recurrence of the linear flow in a generic direction and the non‐ergodicity of Lebesgue measure.

Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions

“…If follows that |f (t n ) − f (l n )| < ε provided that n ≥ max{N 1 , N 2 }. Therefore (19) holds and the proof is complete.…”

“…Their result turns out to be a fundamental tool for proving dynamical properties of directional flows on non-compact periodic translation surfaces, cf. for example [3], [8], [21] and [19] for the Ehrenfest wind tree model.…”

“…In this section we relate the trapping phenomenon for light rays in almost all directions with Oseledets genericity along a curve of translation surfaces. We rely for this section on some basic steps of the proof of Theorem 4.1 which were developed in [20] and a technical result recently proved in [19].…”

“…Oseledets genericity plays a central role in verifying this type of assumptions in Proposition 4.3. The technical tool which allows us to check the assumptions in our specific context will be provided by the following general proposition recently proved by Frączek and Hubert in [19]. We should mention that the idea behind this result is not new (can be found also in [21] and [8]) and exploits the phenomenon of bounded deviation discovered by Zorich in [42,43].…”

On Ergodicity of Foliations on $${\mathbb{Z}^d}$$ Z d -Covers of Half-Translation Surfaces and Some Applications to Periodic Systems of Eaton Lenses

“…Then x = ϕ θ τ ( j) (x 0 ) (x 0 ) for some x 0 ∈ J and there is 0 ≤ s < τ(x) such that ϕ θ s x is a singular point. Therefore, ϕ θ τ ( j) (x 0 )+s x 0 is a singular point and τ ( j) (x 0 ) + s < ( j +1)|τ | ≤ h|τ | ≤ , contrary to the assumption.The following result follows directly from Lemmas A.3 and A.4 in[14].…”