2011
DOI: 10.1515/crelle.2011.052
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The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion

Abstract: Abstract. We study periodic wind-tree models, unbounded planar billiards with periodically located rectangular obstacles. For a class of rational parameters we show the existence of completely periodic directions, and recurrence; for another class of rational parameters, there are directions in which all trajectories escape, and we prove a rate of escape for almost all directions. These results extend to a dense G δ of parameters.

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Cited by 34 publications
(31 citation statements)
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“…The cocycles that arise in this setting are piecewise constant functions with values in Z d . First results in these geometric settings were only recently proved in [6,14,15,17,18].…”
Section: T ϕ (X Y) = (T X Y + ϕ(X))mentioning
confidence: 99%
“…The cocycles that arise in this setting are piecewise constant functions with values in Z d . First results in these geometric settings were only recently proved in [6,14,15,17,18].…”
Section: T ϕ (X Y) = (T X Y + ϕ(X))mentioning
confidence: 99%
“…Remark 7.3. Historically, this Corollary was proven in [24] for the wind-tree model. It was a motivation to get a more general criterion like Corollary 6.3.…”
Section: F I G U R E 2 Billiard Table (λ )mentioning
confidence: 94%
“…If P=[0,a]×[0,b] then the corresponding billiard table we denote by E(Λ,a,b). The recurrence, ergodicity and diffusion times of standard Ehrenfest wind‐tree model, with Λ=Z2, were discussed recently in . In particular, it was recently shown that for every pair of parameters (a,b) and almost every direction θ the billiard flow on E(Z2,a,b) is recurrent and non‐ergodic and its rate of diffusion is t2/3.…”
Section: General Wind‐tree Modelmentioning
confidence: 99%
“…[160]. Here it was shown that for rectangles with rational lengths that (in lowest form) have odd numerator and even denominator, there is a dense set of rational directions for which the dynamics is periodic, and that for almost all directions the dynamics is recurrent.…”
Section: Polygonal Scatterersmentioning
confidence: 96%