Examples of recurrent or transient stationary walks in ℝ^{𝕕} over a rotation of 𝕋²

International audienceWe study billiard dynamics on non-compact polygonal surfaces with a free, cocompact action of Z or Z(2). In the Z-periodic case, we establish criteria for conservativity. In the Z(2)-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 Z(d)-valued cocycles over irrational rotations

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“…The recurrence of α n , β n does not necessarily imply that ϕ n is recurrent. See, e. g., [4] for an example.…”

Section: Ergodic Theory For Skew Productsmentioning

confidence: 99%

“…See also [20], [2]. The following lemma from [4] gives a simple sufficient condition for recurrence of R d -valued cocycles. Lemma 1.…”

Section: Ergodic Theory For Skew Productsmentioning

confidence: 99%

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

Conze

Gutkin

2012

Ergod. Th. Dynam. Sys.

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“…This result is already known for m = 1 (see [AH,Theorem 1]). Moreover, as explained to us by V. Delecroix, a criterion of recurrence due to N. Chevallier and J.-P. Conze [CC,Corollary 1.2] allows us to conclude that the billiard flow φ θ t is recurrent in Π for almost every direction θ ∈ [0, 2π) if the polynomial diffusion rate (see above) δ(Π) < 1/2. However, we only know that the polynomial diffusion rate is less than 1/2 for almost every Π ∈ WT (m) and only for m > 2.…”

Section: Recurrencementioning

confidence: 97%

Counting problem on wind-tree models

Pardo

2018

Geom. Topol.

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We study periodic wind-tree models, billiards in the plane endowed with Z 2 -periodically located identical connected symmetric right-angled obstacles. We show asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to Z 2 -translations) on the wind-tree billiard. We also compute explicitly the associated Siegel-Veech constant for generic wind-tree billiards depending on the number of corners on the obstacle.

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“…Next we describe the idea of the proof of Theorem 12. Note that (8) looks similar to (22). The main ingredient in the proof of Theorem 12 involves a result on the distribution of small divisors of multiplicative form Π|k i | (k, α) .…”

Section: 5mentioning

confidence: 99%

Limit theorems for toral translations

Dolgopyat

Fayad²

2015

Hyperbolic Dynamics, Fluctuations and Large Deviations

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Contents 1. Introduction 2. Ergodic sums of smooth functions with singularities 2.1. Smooth observables 2.2. Observables with singularities 3. Ergodic sums of characteristic functions. Discrepancies 3.1. The maximal discrepancy 3.2. Limit laws for the discrepancy as α is random 4. Poisson regime 5. Outlines of proofs 5.1. Poisson processes 5.2. Uniform distribution on the space of lattices 5.3. The Poisson regime 5.4. Application of the Poisson regime theorems to the ergodic sums of smooth functions with singularities 5.5. Limit laws for discrepancies. 6. Shrinking targets 6.1. Dynamical Borel-Cantelli lemmas for translations. 6.2. On the distribution of hits. 6.3. Proofs outlines. 7. Skew products. Random walks. 7.1. Basic properties. 7.2. Recurrence. 7.3. Ergodicity. 7.4. Rate of recurrence.

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Examples of recurrent or transient stationary walks in ℝ^{𝕕} over a rotation of 𝕋²

Cited by 10 publications

References 0 publications

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

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

Counting problem on wind-tree models

Limit theorems for toral translations

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