Hyperbolic Billiards on Surfaces of Constant Curvature

Gutkin, Boris; Smilansky, Uzy; Gutkin, Eugène

doi:10.1007/s002200050748

Cited by 29 publications

(43 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Previous works, such as that of Kimura (1999), brought to light the fact that the centroid of an isolated dipole on a surface of constant curvature follows a geodesic path, as does a billiard on such a surface (Gutkin et al 1999). However, when multiple dipoles are present, the farfield interactions play a central role in the governing dynamics and cause the trajectories to deviate from great circle paths.…”

Section: Discussionmentioning

confidence: 99%

Interacting dipole pairs on a rotating sphere

Newton

Shokraneh

2008

Proc. R. Soc. A.

View full text Add to dashboard Cite

The evolution, interaction and scattering of 2N point vortices grouped into equal and opposite pairs (N-dipoles) on a rotating unit sphere are studied. A new coordinate system made up of centres of vorticity and centroids associated with each dipole is introduced. With these coordinates, the nonlinear equations for an isolated dipole diagonalize and one directly obtains the equation for geodesic motion on the sphere for the dipole centroid. When two or more dipoles interact, the equations are viewed as an interacting billiard system on the sphere -charged billiards -with long-range interactions causing the centroid trajectories to deviate from their geodesic paths. Canonical interactions are studied both with and without rotation. For two dipoles, the four basic interactions are described as exchange-scattering, non-exchange-scattering, loop-scattering (head on) and loop-scattering (chasing) interactions. For three or more dipoles, one obtains a richer variety of interactions, although the interactions identified in the two-dipole case remain fundamental.

show abstract

Section: Discussionmentioning

confidence: 99%

Interacting dipole pairs on a rotating sphere

Newton

Shokraneh

2008

Proc. R. Soc. A.

View full text Add to dashboard Cite

show abstract

“…Wojtkowski's approach was further extended by Bunimovich, V. Donnay, and R. Markarian [26]. Using these ideas, B. Gutkin, U. Smilanski and the author constructed hyperbolic billiard tables on surfaces of arbitrary constant curvature [53]. Let Y be a semi-dispersive n-gon with only one neutral component.…”

Section: Hyperbolic Billiard Dynamicsmentioning

confidence: 99%

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

Gutkin

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Chaotic billiards in general Riemannian surfaces were studied, for example, in [Vet84], [KSS89], and [Zha17]. For billiards in surfaces of constant curvature, see also [BL97] and [GSG99]. In this paper, we focus on uniform hyperbolicity (see Definition 1.1) for billiards in general surfaces.…”

Section: Introduction and Notationsmentioning

confidence: 99%

Uniform hyperbolicity in nonflat billiards

Kourganoff

2018

Discrete &Amp; Continuous Dynamical Systems - A

View full text Add to dashboard Cite

Uniform hyperbolicity is a strong chaotic property which holds, in particular, for Sinai billiards. In this paper, we consider the case of a nonflat billiard, that is, a Riemannian surface with boundary. Each trajectory follows the geodesic flow in the interior of the billiard, and bounces when it meets the boundary. We give a sufficient condition for a nonflat billiard to be uniformly hyperbolic. As a particular case, we obtain a new criterion to show that a closed surface has an Anosov geodesic flow.

show abstract

Hyperbolic Billiards on Surfaces of Constant Curvature

Cited by 29 publications

References 7 publications

Interacting dipole pairs on a rotating sphere

Interacting dipole pairs on a rotating sphere

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

Uniform hyperbolicity in nonflat billiards

Contact Info

Product

Resources

About