Polygonal Billiards with One Sided Scattering

Skripchenko, Alexandra; Troubetzkoy, Serge

doi:10.5802/aif.2975

Cited by 7 publications

(6 citation statements)

References 7 publications

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“…Finally, in [ST15], the authors focused on a different part of the program suggested by Boshernitzan and Kornfeld. Namely, they studied billiards in rational polygons with so called spy mirrors and worked on the questions about complexity of the trajectories of these billiards.…”

Section: Results For Double Rotationsmentioning

confidence: 99%

“…Connection with billiards. It was shown in [ST15] (using ideas from [BK95]) that double rotations can be considered as the first return map of the billiard flow for the so-called billiards with spy mirrors. This implies the following natural question, partly related to the problem of geometrical interpretation described above: is there a way to estimate the diffusion rate of the billiard trajectory in terms of the Lyapunov spectrum of the induction cocycle?…”

Section: Open Questionsmentioning

confidence: 99%

“…Questions of complexity and combinatorial properties. Some study of the complexity in connection with double rotations was performed in [ST15] and [CN10]. More in detail, in [CN10] it was shown that the subword complexity of the subshift associated with a special subclass of double rotations that corresponds to a subspace of codimension 1 is linear: n + 1 ≤ p(n) ≤ 3n.…”

Section: Open Questionsmentioning

confidence: 99%

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A note on double rotations of infinite type

Artigiani¹,

Fougeron²,

Hubert³

et al. 2021

Preprint

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We introduce a new renormalization procedure on double rotations, which is reminiscent of the classical Rauzy induction. Using this renormalization we prove that the set of parameters which induce infinite type double rotations has Hausdorff dimension strictly smaller than 3. Moreover, we construct a natural invariant measure supported on these parameters and show that, with respect to this measure, almost all double rotations are uniquely ergodic.MSC Classification: 37E05. Bibliography: 17 items.

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Section: Results For Double Rotationsmentioning

confidence: 99%

Section: Open Questionsmentioning

confidence: 99%

Section: Open Questionsmentioning

confidence: 99%

See 1 more Smart Citation

A note on double rotations of infinite type

Artigiani¹,

Fougeron²,

Hubert³

et al. 2021

Preprint

Self Cite

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“…Buzzi & P. Hubert (2004) [5], H. Bruin (2007Bruin ( ,2012 [3], S. Marmi, P. Moussa, J-C. Yoccoz (2012) [14], D. Volk (2014) [22]. A further generalization when a general cover is used instead of a special partition (which inevitably leads to a multi-valued map) was considered in A. Skripchenko & S. Troubetzkoy (2015) [18]. In any case the problem with the boundary points already mentioned in the case of IET is becoming a serious obstacle here.…”

Section: Historical Remarksmentioning

confidence: 99%

Invariant measures of torus piecewise isometries

Blank¹

2021

Preprint

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By now, we have learned quite well how to study hyperbolic (locally expanding/contracting or both) chaotic dynamical systems, thanks to a large extent to the development of the so called operator approach. Contrary to this almost nothing is known about piecewise isometries, except for a special case of one-dimensional interval exchange mappings. The last case is fundamentally different from the general situation in the presence of an invariant measure (Lebesgue measure), which helps a lot in the analysis. We start by showing that already the restriction of the rotation of the plane to a torus demonstrates a number of rather unexpected properties. Our main results describe sufficient conditions for the existence/absence of invariant measures of torus piecewise isometries. Technically these results are based on the approximation of the maps under study by weakly periodic ones.

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“…In the last years, there has been significant research on billiards in polygons [3,6,15,16,18,24,25]. Dynamics of these models is extremely difficult to rigorously analyze which often happens with systems with intermediate, neither regular (integrable) nor chaotic, behavior.…”

Section: Introductionmentioning

confidence: 99%

Bridge to Hyperbolic Polygonal Billiards

Attarchi,

Bunimovich

2020

Preprint

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It is well-known that billiards in polygons cannot be chaotic (hyperbolic). Particularly Kolmogorov-Sinai entropy of any polygonal billiard is zero. We consider physical polygonal billiards where a moving particle is a hard disc rather than a point (mathematical) particle and show that typical physical polygonal billiard is hyperbolic at least on a subset of positive measure and therefore has a positive Kolmogorov-Sinai entropy for any positive radius of the moving particle (provided that the particle is not so big that it cannot move within a polygon). This happens because a typical physical polygonal billiard is equivalent to a mathematical (point particle) semi-dispersing billiard. We also conjecture that in fact typical physical billiard in polygon is ergodic under the same conditions.

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Polygonal Billiards with One Sided Scattering

Abstract: We study the billiard on a square billiard table with a one-sided vertical mirror. We associate trajectories of these billiards with double rotations and study orbit behavior and questions of complexity.

Cited by 7 publications

References 7 publications

A note on double rotations of infinite type

A note on double rotations of infinite type

Invariant measures of torus piecewise isometries

Bridge to Hyperbolic Polygonal Billiards

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