On a conjugacy problem in billiard dynamics

  • D. V. Treschev


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 α, the rotation angle, is rationally incommensurable with π, 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 α, and other aspects.


STEKLOV Institute Periodic Point Rigid Rotation CONJUGACY Problem Convergence Radius 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M.-C. Arnaud, “C 1-generic billiard tables have a dense set of periodic points,” Regular Chaotic Dyn. 18(6), 697–702 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    G. D. Birkhoff, Dynamical Systems (Am. Math. Soc., New York, 1927), Colloq. Publ. 9.zbMATHGoogle Scholar
  3. 3.
    S. V. Bolotin, “Integrable Birkhoff billiards,” Vestn. Mosk. Univ., Ser 1: Mat. Mekh., No. 2, 33–36 (1990) [Moscow Univ. Mech. Bull. 45 (2), 10–13 (1990)].Google Scholar
  4. 4.
    S. V. Bolotin, “Integrable billiards on surfaces of constant curvature,” Mat. Zametki 51(2), 20–28 (1992) [Math. Notes 51, 117–123 (1992)].MathSciNetGoogle Scholar
  5. 5.
    V. V. Kozlov and D. V. Treshchev, Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts (Am. Math. Soc., Providence, RI, 1991), Transl. Math. Monogr. 89.Google Scholar
  6. 6.
    D. Treschev, “Billiard map and rigid rotation,” Physica D 255, 31–34 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    H. Whitney, “Analytic extensions of differentiable functions defined in closed sets,” Trans. Am. Math. Soc. 36, 63–89 (1934).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations