On Nonconvex Caustics of Convex Billiards

Knill, Oliver

doi:10.1007/s000170050038

Cited by 25 publications

(22 citation statements)

References 24 publications

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“…It follows immediately from the definition that for any symmetric billiard configuration (K, T ), there is a natural bijection between the invariant circles of the twist map associated with the T -billiard dynamics in K, and the invariant circles of the twist map associated with the K-billiard dynamics in T . However, in general, not every invariant circle corresponds to a convex caustic (see e.g., [13] and [19]). Thus, the existence of a convex dual caustic in Theorem 1.4 above is not obvious.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Moreover, as C is a caustic, Γ is φ K -invariant. We emphasize that the converse is false in general, i.e., not every invariant circle corresponds to a convex caustic (see e.g., [13,19]). Note that given a φ K -invariant circle Γ ⊂ A K (∂K × ∂T ) + , its image under Ψ defines a φ T -invariant circle Ψ(Γ) ⊂ A T (∂K × ∂T ) − .…”

Section: Minkowski Caustics and Invariant Circlesmentioning

confidence: 99%

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Duality of caustics in Minkowski billiards

et al. 2018

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In this paper we study convex caustics in Minkowski billiards. We show that for the Euclidean billiard dynamics in a planar smooth, centrally symmetric, strictly convex body K, for every convex caustic which K possesses, the "dual" billiard dynamics in which the table is the Euclidean unit ball and the geometry that governs the motion is induced by the body K, possesses a dual convex caustic. Such a pair of caustics are dual in a strong sense, and in particular they have the same perimeter, Lazutkin parameter (both measured with respect to the corresponding geometries), and rotation number. We show moreover that for general Minkowski billiards this phenomenon fails, and one can construct a smooth caustic in a Minkowski billiard table which possesses no dual convex caustic.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Minkowski Caustics and Invariant Circlesmentioning

confidence: 99%

Duality of caustics in Minkowski billiards

et al. 2018

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“…Most of the literature deals with convex caustics, since they are easier to understand and related to ordered trajectories. Two exceptions are [8, §3] and [10].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Non-persistence of resonant caustics in perturbed elliptic billiards

Pinto-de-Carvalho

Ramírez-Ros

2012

Ergod. Th. Dynam. Sys.

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Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics -the ones whose tangent trajectories are closed polygons-are destroyed under generic perturbations of the billiard table. We prove that none of the resonant elliptical caustics persists under a large class of explicit perturbations of the original ellipse. This result follows from a standard Melnikov argument and the analysis of the complex singularities of certain elliptic functions.

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“…In [15], Knill proved that a billiard map preserves the circle α = π/2 if and only if the oval boundary Γ has constant width. The rotation number of this circle is 1/2 as each of its points belongs to a 2-periodic orbit.…”

Section: Classical Billiards On Ovalsmentioning

confidence: 99%