Billiards: A Genetic Introduction to the
                    Dynamics of Systems with Impacts

Козлов, В. В.; Трещев, Дмитрий Валерьевич

doi:10.1090/mmono/089

Cited by 323 publications

(266 citation statements)

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“…We briefly introduce some other notation and terminology, referring to Section 2 and to [KT91] and [PS92] for further background and definitions regarding billiards. By Lsp.…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…The linear Poincaré map P of is the derivative at .0/ of the first return map to a transversal toˆt at .0/. By a nondegenerate periodic reflecting ray , we mean one whose linear Poincaré map P has no eigenvalue equal to one; see [PS92], [KT91]. The following relates P and the Hessian of the length functional in angular coordinates:…”

Section: Billiards and The Length Functionalmentioning

confidence: 99%

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Inverse spectral problem for analytic domains, II: ℤ₂-symmetric domains

Zelditch

2009

Ann. Math.

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This paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit. The algorithm is based on a new expression for the localized wave trace as a special multiple oscillatory integral over the boundary, and on a Feynman diagrammatic analysis of the stationary phase expansion of the oscillatory integral. The algorithm is particularly effective for Euclidean plane domains possessing a ‫ޚ‬ 2 symmetry which reverses the orientation of a bouncing ball orbit. It is also very effective for domains with dihedral symmetries. For simply connected analytic Euclidean plane domains in either symmetry class, we prove that the domain is determined within the class by either its Dirichlet or Neumann spectrum. This improves and generalizes the best prior inverse result that simply connected analytic plane domains with two symmetries are spectrally determined within that class.

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“…We briefly introduce some other notation and terminology, referring to Section 2 and to [KT91] and [PS92] for further background and definitions regarding billiards. By Lsp.…”

Section: Statement Of Resultsmentioning

confidence: 99%

Section: Billiards and The Length Functionalmentioning

confidence: 99%

Inverse spectral problem for analytic domains, II: ℤ₂-symmetric domains

Zelditch

2009

Ann. Math.

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“…Indeed, in the last part of this paper we demonstrate how these tools may be used to instantly extend novel results which were obtained for billiards to the steep potential setting; It is well known that the billiard map is integrable inside an ellipsoid [20]. Moreover, Birkhoff-Poritski conjecture claims that in 2 dimensions among all the convex smooth concave billiard tables only ellipses are integrable [34].…”

Section: Introductionmentioning

confidence: 58%

Approximating Multi-Dimensional Hamiltonian Flows by Billiards

Rapoport

Rom‐Kedar

Turaev

2007

Commun. Math. Phys.

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Consider a family of smooth potentials V ε , which, in the limit ε → 0, become a singular hard-wall potential of a multi-dimensional billiard. We define auxiliary billiard domains that asymptote, as ε → 0 to the original billiard, and provide asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the C r norm and an iteration scheme for improving this approximation. Applying this theory to smooth potentials which limit to the multi-dimensional close to ellipsoidal billiards, we predict when the separatrix splitting persists for various types of potentials.

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“…A long-standing conjecture by Birkhoff states that among all billiards inside smooth convex curves, ellipses are characterized by integrability of the billiard map [8]. On the other hand, examples of their ergodic counterparts (for example, Bunimovich stadium [9], dispersive Sinai billards [10]) are equally well-known [11].…”

Section: Introductionmentioning

confidence: 99%

A difference-equation formalism for the nodal domains of separable billiards

2016

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Recently, the nodal domain counts of planar, integrable billiards with Dirichlet boundary conditions were shown to satisfy certain difference equations in [Ann. Phys. 351, 1 (2014)]. The exact solutions of these equations give the number of domains explicitly. For complete generality, we demonstrate this novel formulation for three additional separable systems and thus extend the statement to all integrable billiards.

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Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts

Cited by 323 publications

References 0 publications

Inverse spectral problem for analytic domains, II: ℤ₂-symmetric domains

Inverse spectral problem for analytic domains, II: ℤ₂-symmetric domains

Approximating Multi-Dimensional Hamiltonian Flows by Billiards

A difference-equation formalism for the nodal domains of separable billiards

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Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts

Cited by 323 publications

References 0 publications

Inverse spectral problem for analytic domains, II: ℤ2-symmetric domains

Inverse spectral problem for analytic domains, II: ℤ2-symmetric domains

Approximating Multi-Dimensional Hamiltonian Flows by Billiards

A difference-equation formalism for the nodal domains of separable billiards

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Product

Resources

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Inverse spectral problem for analytic domains, II: ℤ₂-symmetric domains

Inverse spectral problem for analytic domains, II: ℤ₂-symmetric domains