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 Tr. Mat. Inst. Steklova, 2015, Volume 289, Pages 309–317 (Mi tm3621)

On a conjugacy problem in billiard dynamics

D. V. Treschev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: We study symmetric billiard tables for which the billiard map is locally (near an elliptic periodic orbit of period $2$) conjugate to a rigid rotation. In the previous paper (Physica D 255, 31–34 (2013)), we obtained an equation (called below the conjugacy equation) for such tables and proved that if $\alpha$, the rotation angle, is rationally incommensurable with $\pi$, then the conjugacy equation has a solution in the category of formal series. In the same paper there is also numerical evidence that for “good” rotation angles the series have positive radii of convergence. In the present paper we carry out a further study (both analytic and numerical) of the conjugacy equation. We discuss its symmetries, dependence of the convergence radius on $\alpha$, and other aspects.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.1134/S037196851502017X

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English version:
Proceedings of the Steklov Institute of Mathematics, 2015, 289, 291–299

Bibliographic databases:

UDC: 531.01

Citation: D. V. Treschev, “On a conjugacy problem in billiard dynamics”, Selected issues of mathematics and mechanics, Collected papers. In commemoration of the 150th anniversary of Academician Vladimir Andreevich Steklov, Tr. Mat. Inst. Steklova, 289, MAIK Nauka/Interperiodica, Moscow, 2015, 309–317; Proc. Steklov Inst. Math., 289 (2015), 291–299

Citation in format AMSBIB
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\bookinfo Collected papers. In commemoration of the 150th anniversary of Academician Vladimir Andreevich Steklov
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\vol 289
\pages 309--317
\publ MAIK Nauka/Interperiodica
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• http://mi.mathnet.ru/eng/tm3621
• https://doi.org/10.1134/S037196851502017X
• http://mi.mathnet.ru/eng/tm/v289/p309

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. V. Kozlov, “Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas”, Russian Math. Surveys, 71:2 (2016), 253–290
2. A. P. Markeev, “Ob ustoichivosti dvukhzvennoi traektorii paraboloidnogo bilyarda Birkgofa”, Nelineinaya dinam., 12:1 (2016), 75–90
3. M. Bialy, A. E. Mironov, “On fourth-degree polynomial integrals of the Birkhoff billiard”, Proc. Steklov Inst. Math., 295 (2016), 27–32
4. A. P. Markeev, “On the stability of periodic trajectories of a planar Birkhoff billiard”, Proc. Steklov Inst. Math., 295 (2016), 190–201
5. A. P. Markeev, “The stability of two-link trajectories of a Birkhoff billiard”, Pmm-J. Appl. Math. Mech., 80:4 (2016), 280–289
6. A. Plakhov, S. Tabachnikov, D. Treschev, “Billiard transformations of parallel flows: a periscope theorem”, J. Geom. Phys., 115 (2017), 157–166
7. D. Treschev, “A locally integrable multi-dimensional billiard system”, Discret. Contin. Dyn. Syst., 37:10 (2017), 5271–5284
8. M. Bialy, A. E. Mironov, “A survey on polynomial in momenta integrals for billiard problems”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2131 (2018), 20170418
9. A. Glutsyuk, E. Shustin, “On polynomially integrable planar outer billiards and curves with symmetry property”, Math. Ann., 372:3-4 (2018), 1481–1501
10. M. Bialy, A. E. Mironov, “Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane”, Russian Math. Surveys, 74:2 (2019), 187–209
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