Note on the periodic points of the billiard

Stojanov, Luchezar

doi:10.4310/jdg/1214447542

Cited by 33 publications

(27 citation statements)

References 2 publications

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“…We also prove that, for every outer billiard table, the set of 3-periodic orbits has zero measure (see Theorem 4). For inner billiards, this is a wellknown result having a number of different proofs [11,12,15,16]. We also give different proofs, one of them follows ideas of [16] and another those of [1].…”

Section: T(x)mentioning

confidence: 97%

On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards

Genin

Tabachnikov

2007

Journal of Modern Dynamics

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Section: T(x)mentioning

confidence: 97%

On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards

Genin

Tabachnikov

2007

Journal of Modern Dynamics

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“…Then the billiard is not k-reflective. Theorem 1.8 is the complexification of the above-mentioned results by M.Rychlik et al on triangular orbits in real billiards, see [2,10,11,14,16]. Theorem 1.9 has immediate application to the real Ivrii's conjecture.…”

Section: Introductionmentioning

confidence: 80%

On Odd-periodic Orbits in Complex Planar Billiards

Glutsyuk

2014

J Dyn Control Syst

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The famous conjecture of V. Ya.Ivrii (1978) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study the complex version of Ivrii's conjecture for odd-periodic orbits in planar billiards, with reflections from complex analytic curves. We prove positive answer in the following cases: 1) triangular orbits; 2) odd-periodic orbits in the case, when the mirrors are algebraic curves avoiding two special points at infinity, the so-called isotropic points. We provide immediate applications to the real piecewise-algebraic Ivrii's conjecture and to its analogue in the invisibility theory.

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“…For the proof of Ivrii's conjecture it suffices to show that for every k ∈ N the set ot k-periodic orbits has measure zero. For k = 3 this was proved in [2,22,23,27,29]. For k = 4 in dimension two this was proved in [9,10].…”

mentioning

confidence: 95%

On 4-Reflective Complex Analytic Planar Billiards

Glutsyuk

2016

J Geom Anal

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The famous conjecture of V.Ya.Ivrii [14] says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study its complex analytic version for quadrilateral orbits in two dimensions, with reflections from holomorphic curves. We present the complete classification of 4-reflective complex analytic counterexamples: billiards formed by four holomorphic curves in the projective plane that have open set of quadrilateral orbits. This extends the previous author's result [7] classifying 4-reflective complex planar algebraic counterexamples. We provide applications to real planar billiards: classification of 4-reflective germs of real planar C 4 -smooth pseudo-billiards; solutions of Tabachnikov's Commuting Billiard Conjecture and the 4-reflective case of Plakhov's Invisibility Conjecture (both in two dimensions; the boundary is required to be piecewise C 4 -smooth). We provide a survey and a small technical result concerning higher number of complex reflections.

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Note on the periodic points of the billiard

Cited by 33 publications

References 2 publications

On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards

On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards

On Odd-periodic Orbits in Complex Planar Billiards

On 4-Reflective Complex Analytic Planar Billiards

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