On quadrilateral orbits in planar billiards

Glutsyuk, Alexey; Kudryashov, Yury

doi:10.1134/s1064562411030343

Cited by 20 publications

(27 citation statements)

References 3 publications

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“…For k = 3 this was proved in [2,22,23,27,29]. For k = 4 in dimension two this was proved in [9,10]. Remark 1.…”

mentioning

confidence: 89%

“…One of the main ideas of the proof of Theorem 1.7 is similar to that from [9,10,7]: to study the degenerate orbit set U \ U 0 . This idea itself together with basic algebraic geometry allowed to treat the algebraic case in [7].…”

Section: Remark 12mentioning

confidence: 99%

“…Without loss of generality we consider that each mirror is transcendental, since otherwise, a = c, by Theorem 1. 9. Let U be the 4-reflective set.…”

Section: Intersected Neighbor Mirrors Proof Of Corollary 35mentioning

confidence: 99%

“…As it was shown by V.Ya.Ivrii [14], his conjecture implies the famous H.Weyl's conjecture on the two-term asymptotics of the spectrum of Laplacian [28]. A brief historical survey of both conjectures with references is presented in [9,10]. 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.…”

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confidence: 99%

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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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“…For k = 3 this was proved in [2,22,23,27,29]. For k = 4 in dimension two this was proved in [9,10]. Remark 1.…”

mentioning

confidence: 89%

Section: Remark 12mentioning

confidence: 99%

“…Without loss of generality we consider that each mirror is transcendental, since otherwise, a = c, by Theorem 1. 9. Let U be the 4-reflective set.…”

Section: Intersected Neighbor Mirrors Proof Of Corollary 35mentioning

confidence: 99%

mentioning

confidence: 99%

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On 4-Reflective Complex Analytic Planar Billiards

Glutsyuk

2016

J Geom Anal

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“…For a detailed review of the history of Ivrii's conjecture see [29] or [26]. The conjecture was only proved in some special cases, in particular in the case of triangular orbits [71,74,85,81] and, quite recently, in the much more complicated case of rectangular orbits in planar billiards [25,26].…”

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confidence: 99%

Exterior Billiards

Plakhov¹

2013

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The book contains an account of results obtained by the author and his collaborators on billiards in the complement of bounded domains and their applications in aerodynamics and geometrical optics.We consider several problems related to aerodynamics of bodies in highly rarefied media. It is assumed that the medium particles do not interact with each other and are elastically reflected when colliding with the body boundary; these assumptions drastically simplify the aerodynamics and allow to reduce it to a number of purely mathematical problems.First we examine problems of minimal resistance in the case of translational motion of bodies. These problems generalize the Newton problem of least resistance; the difference is that the bodies are generally nonconvex in our case and therefore the particles can make multiple reflections from the body surface. It is proved that typically the infimum of resistance equals zero; thus, there exist 'almost perfectly streamlined' bodies.Next we consider the generalization of Newton's problem on minimal resistance of convex axisymmetric bodies to the case of media with thermal motion of particles. Two kinds of solutions are found: first, Newton-like bodies and second, shapes obtained by gluing together two Newton-like bodies along their rear ends.Further, we state results on characterization of billiard scattering by nonconvex and rough bodies; next we solve some special problems of optimal mass transportation. These two groups of results are applied to problems of minimal and maximal resistance for bodies that move forward and at the same time slowly rotate. It is found, in particular, that the resistance of a three-dimensional convex body can be increased at most twice and decreased at most by 3.05% by roughening its surface.Next, we consider a rapidly rotating rough disc moving in a rarefied medium on the plane. It is shown that the force acting on the disc is not generally parallel to the direction of the disc motion, that is, has a nonzero transversal component. This phenomenon is called Magnus effect (proper or inverse, depending on the direction of the transversal component). We show that the kind of Magnus effect depends on the kind of disc roughness, and study this dependence. The problem of finding all admissible values of the force acting on the disc is formulated in terms of a vector-valued problem of optimal mass transportation.Finally, we describe bodies that have zero resistance when translating through a medium, and state results on existence or non-existence of bodies with mirror surface invisible in one or several directions. We also consider the problem of constructing retroreflectors: bodies with specular surface that reverse the direction of any incident beam of light.

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Problem of Minimum Resistance to Translational Motion of Bodies

Plakhov

2012

Exterior Billiards

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On quadrilateral orbits in planar billiards

Cited by 20 publications

References 3 publications

On 4-Reflective Complex Analytic Planar Billiards

On 4-Reflective Complex Analytic Planar Billiards

Exterior Billiards

Problem of Minimum Resistance to Translational Motion of Bodies

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