An estimate from above of the number of periodic orbits for semi-dispersed billiards

Stojanov, Luchezar

doi:10.1007/bf01219195

Cited by 32 publications

(22 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…For the now so-called Sinai billiard system, he proved that it has positive measure-theoretic (Kolmogorov-Sinai) entropy and is hyperbolic almost everywhere. See also [11,18,32,34] and also [14,26,[29][30][31] for relevant and recent results and references therein. In contrast, rays converge after reflecting from convex boundaries.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

See 1 more Smart Citation

On topological entropy of billiard tables with small inner scatterers

Chen

2010

Advances in Mathematics

View full text Add to dashboard Cite

We present in this paper an approach to studying the topological entropy of a class of billiard systems. In this class, any billiard table consists of strictly convex domain in the plane and strictly convex inner scatterers. Combining the concept of anti-integrable limit with the theory of Lyusternik-Shnirel'man, we show that a billiard system in this class generically admits a set of non-degenerate anti-integrable orbits which corresponds bijectively to a topological Markov chain of arbitrarily large topological entropy. The anti-integrable limit is the singular limit when scatterers shrink to points. In order to get around the singular limit and so as to apply the implicit function theorem, on auxiliary circles encircling these scatterers we define a length functional whose critical points are well-defined at the anti-integrable limit and give rise to billiard orbits when the scatterers are not points. Consequently, we prove the topological entropy of the first return map to the scatterers can be made arbitrarily large provided the inner scatterers are sufficiently small.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 98%

“…Subtracting (36) from (37) and using Proposition 5.1, we arrive at (33) and (34) are satisfied for some non-zero (dθ 1 , . .…”

Section: Dynamics Away From Scatterersmentioning

confidence: 93%

On topological entropy of billiard tables with small inner scatterers

Chen

2010

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…is a billiard deformation, then G also depends on α and is C r,r ′ -smooth. [24], see also [20]) If K(α) is a billiard deformation, then for a fixed ξ the function G ξ has exactly one minimum at u(α) = (u 1 (α), . .…”

Section: Symbolic Modelmentioning

confidence: 99%

Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard

Wright¹

2016

DCDS-A

View full text Add to dashboard Cite

We consider open billiards in the plane satisfying the no-eclipse condition. We show that the points in the non-wandering set depend differentiably on deformations to the boundary of the billiard. We use Bowen's equation to estimate the Hausdorff dimension of the non-wandering set of the billiard. Finally we show that the Hausdorff dimension depends differentiably on sufficiently smooth deformations to the boundary of the billiard, and estimate the derivative with respect to such deformations.(H) For distinct 1 ≤ i, j, k ≤ m, the convex hull of K i ∪ K j is disjoint from K k .

show abstract

“…Rotation vectors and sets provide essential information for understanding the behavior of trajectories of a billiard system [1], [11].…”

Section: Introductionmentioning

confidence: 99%

Rotation Sets of Billiards with N Obstacles on a Torus

Alsheekhhussain

2015

Differ Equ Dyn Syst

View full text Add to dashboard Cite

For billiards with N obstacles on a torus, we study the behavior of specific kind of its trajectories, the so called admissible trajectories. Using the methods developed in [1], we prove that the admissible rotation set is convex, and the periodic trajectories of admissible type are dense in the admissible rotation set. In addition, we show that the admissible rotation set is a proper subset of the general rotation set.

show abstract

An estimate from above of the number of periodic orbits for semi-dispersed billiards

Abstract: For a large class of semi-dispersed billiards an exponential estimate from above is found for the number of periodic points of the billiard ball map.

Cited by 32 publications

References 12 publications

On topological entropy of billiard tables with small inner scatterers

On topological entropy of billiard tables with small inner scatterers

Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard

Rotation Sets of Billiards with N Obstacles on a Torus

Contact Info

Product

Resources

About