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

This article is cited in 10 scientific papers (total in 10 papers)

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.


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

Bibliographic databases:

UDC: 531.01
Received: January 15, 2015

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
\by D.~V.~Treschev
\paper On a conjugacy problem in billiard dynamics
\inbook Selected issues of mathematics and mechanics
\bookinfo Collected papers. In commemoration of the 150th anniversary of Academician Vladimir Andreevich Steklov
\serial Tr. Mat. Inst. Steklova
\yr 2015
\vol 289
\pages 309--317
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\jour Proc. Steklov Inst. Math.
\yr 2015
\vol 289
\pages 291--299

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    1. V. V. Kozlov, “Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas”, Russian Math. Surveys, 71:2 (2016), 253–290  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. P. Markeev, “Ob ustoichivosti dvukhzvennoi traektorii paraboloidnogo bilyarda Birkgofa”, Nelineinaya dinam., 12:1 (2016), 75–90  mathnet
    3. M. Bialy, A. E. Mironov, “On fourth-degree polynomial integrals of the Birkhoff billiard”, Proc. Steklov Inst. Math., 295 (2016), 27–32  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    4. A. P. Markeev, “On the stability of periodic trajectories of a planar Birkhoff billiard”, Proc. Steklov Inst. Math., 295 (2016), 190–201  mathnet  crossref  crossref  mathscinet  isi  elib
    5. A. P. Markeev, “The stability of two-link trajectories of a Birkhoff billiard”, Pmm-J. Appl. Math. Mech., 80:4 (2016), 280–289  crossref  mathscinet  isi  scopus
    6. A. Plakhov, S. Tabachnikov, D. Treschev, “Billiard transformations of parallel flows: a periscope theorem”, J. Geom. Phys., 115 (2017), 157–166  crossref  mathscinet  zmath  isi  scopus
    7. D. Treschev, “A locally integrable multi-dimensional billiard system”, Discret. Contin. Dyn. Syst., 37:10 (2017), 5271–5284  crossref  mathscinet  zmath  isi  scopus
    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  crossref  isi  scopus
    9. A. Glutsyuk, E. Shustin, “On polynomially integrable planar outer billiards and curves with symmetry property”, Math. Ann., 372:3-4 (2018), 1481–1501  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  adsnasa  isi  elib
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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