Escape orbits and ergodicity in infinite step billiards

Esposti, Mirko Degli; Magno, Gianluigi Del; Lenci, Marco

doi:10.1088/0951-7715/13/4/316

Cited by 10 publications

(8 citation statements)

References 16 publications

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“…The procedure whereby one passes from the internal-wave billiard on Ω to the linear flow on Ω 4 is also called billiard unfolding. (This technique has been applied very fruitfully to polygonal (ordinary) billiards; see, e.g., the reference list of [4].) Our assumptions on the billiard table Ω, given at the beginning of this section, can be restated as assumptions on Ω 4 as follows: the upper boundary of Ω 4 is the graph of a function b : [−1/2, 1/2] −→ R + , which is even, piecewise C 1 , concave, and (not necessarily strictly) decreasing on [0, 1/2].…”

Section: Reduction To One-dimensional Dynamicsmentioning

confidence: 99%

Internal-wave billiards in trapezoids and similar tables

Lenci¹,

Bonanno²,

Cristadoro³

2021

Preprint

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We call internal-wave billiard the dynamical system of a point particle that moves freely inside a planar domain (the table) and is reflected by its boundary according to this rule: reflections are standard Fresnel reflections but with the pretense that the boundary at any collision point is either horizontal or vertical (relative to a predetermined direction representing gravity). These systems are point particle approximations for the motion of internal gravity waves in closed containers, hence the name. For a class of tables similar to rectangular trapezoids, but with the slanted leg replaced by a general curve with downward concavity, we prove that the dynamics has only three asymptotic regimes: (1) minimality (all trajectories are dense); (2) there exist a global attractor and a global repellor, which are periodic and might coincide; (3) there exists a beam of periodic trajectories, whose boundary (if any) comprises an attractor and a repellor for all the other trajectories. Furthermore, in the prominent case where the table is an actual trapezoid, we study the sets in parameter space relative to the three regimes. We prove in particular that the set for (1) is a positive-measure fractal; the set for (2) has positive measure (giving a rigorous proof of the existence of Arnol'd tongues for internal-wave billiards); the set for (3) has measure zero.

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Section: Reduction To One-dimensional Dynamicsmentioning

confidence: 99%

Internal-wave billiards in trapezoids and similar tables

Lenci¹,

Bonanno²,

Cristadoro³

2021

Preprint

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“…The main direction in these developments is to replace the usual polygons by noncompact or infinite polygons. The work in this direction started already at the end of the last century [33,34], and flourished after the turn of the century. The noncompact polygons in [33,34] are semi-infinite stairways.…”

Section: Ramifications and Extensions Of The Polygonal Billiardmentioning

confidence: 99%

“…In particular, P is a rational noncompact polygon, and we have the obvious family of directional billiard flows b t θ , 0 ≤ θ ≤ π/2, on P . In addition to studying the ergodicity of these flows, the papers [33,34] investigate escaping orbits in P , a new phenomenon caused by the noncompactness of P .…”

Section: Ramifications and Extensions Of The Polygonal Billiardmentioning

confidence: 99%

Billiard dynamics: An updated survey with the emphasis on open problems

Gutkin

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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This is an updated and expanded version of our earlier survey article [E. Gutkin, "Billiard dynamics: a survey with the emphasis on open problems," Regular Chaotic Dyn. 8, 1-13 (2003)]. Section I introduces the subject matter. Sections II s3 s4 expose the basic material following the paradigm of elliptic, hyperbolic, and parabolic billiard dynamics. In Sec. V, we report on the recent work pertaining to the problems and conjectures exposed in the survey [E. Gutkin, "Billiard dynamics: a survey with the emphasis on open problems," Regular Chaotic Dyn. 8, 1-13 (2003)]. Besides, in Sec. V we formulate a few additional problems and conjectures. The bibliography has been updated and considerably expanded.

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“…A further interesting class of examples are infinite pseudo-integrable billiards ( fig. 3) that are known to be ergodic 5 for almost all initial directions [12].…”

Section: Introductionmentioning

confidence: 99%