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Focusing Components in Typical Chaotic Billiards Should be Absolutely Focusing

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Abstract

We demonstrate that the defocusing mechanism fails to work if not all focusing components of the boundary are absolutely focusing. More precisely, we construct billiard tables with arbitrary long free path away from a non-absolutely focusing component such that a nonlinearly stable periodic orbit exists. Therefore the only known standard procedure of constructing chaotic ergodic billiards works in general only if all focusing boundary components are absolutely focusing.

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References

  1. Arnol′d, V.I.: Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics. Vol. 60, second ed. New York: Springer-Verlag, 1989

  2. Boldrighini C., Keane M., Marchetti F.: Billiards in polygons. Ann. Probab. 6(4), 532–540 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bunimovich L.A.: On ergodic properties of certain billiards. Funk. Anal. i Priložen. 8(3), 73–74 (1974)

    Google Scholar 

  4. Bunimovich L.A.: On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65(3), 295–312 (1979)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Bunimovich L.A.: Many-dimensional nowhere dispersing billiards with chaotic behavior. Phys. D 33(1–3), 58–64 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bunimovich L.A.: Conditions of stochasticity of two-dimensional billiards. Chaos 1(2), 183–187 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  7. Bunimovich, L.A.: On absolutely focusing mirrors. In: Ergodic Theory and Related Topics, III (Güstrow, 1990), Lecture Notes in Math., Vol. 1514, Berlin: Springer, 1992, pp. 62–82

  8. Bunimovich L.A.: Absolute focusing and ergodicity of billiards. Regul. Chaotic Dyn. 8(1), 15–28 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Bunimovich L.A., Del Magno G.: Track billiards. Commun. Math. Phys. 288, 699–713 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. Bussolari L., Lenci M.: Hyperbolic billiards with nearly flat focusing boundaries, I. Physica D 237(18), 2272–2281 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Chernov, N., Markarian, R.: Chaotic billiards. In: Mathematical Surveys and Monographs. Vol. 127. Providence, RI: Amer. Math. Soc., 2006

  12. Del Magno, G., Markarian, R.: On the Bernoulli property of planar hyperbolic billiards, 2006, available at http://www.ma.utexas.edu/mp_arc/c/06/06-164.pdf

  13. Dias Carneiro M.J., Oliffson Kamphorst S., Pintode Carvalho S.: Elliptic islands in strictly convex billiards. Erg. Th. Dynam. Syst. 23(3), 799–812 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Donnay V.J.: Using integrability to produce chaos: billiards with positive entropy. Commun. Math. Phys. 141(2), 225–257 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Kamphorst S.O., Pinto-de Carvalho S.: The first Birkhoff coefficient and the stability of 2-periodic orbits on billiards. Exp. Math. 14(3), 299–306 (2005)

    MATH  MathSciNet  Google Scholar 

  16. Lazutkin V.F.: Existence of a continuum of closed invariant curves for a convex billiard. Usp. Mat. Nauk 2(3(165)), 201–202 (1972)

    MathSciNet  Google Scholar 

  17. Lazutkin V.F.: Existence of caustics for the billiard problem in a convex domain. Izv. Akad. Nauk SSSR Ser. Mat. 37, 186–216 (1973)

    MATH  MathSciNet  Google Scholar 

  18. Markarian R.: Billiards with Pesin region of measure one. Commun. Math. Phys. 118(1), 87–97 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Moser, J.: Stable and Random Motions in Dynamical Systems. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press, 2001, with special emphasis on celestial mechanics, Reprint of the 1973 original, with a foreword by Philip J. Holmes

  20. Sinaĭ Y.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Usp. Mat. Nauk 25(2 (152)), 141–192 (1970)

    MATH  Google Scholar 

  21. Wojtkowski M.: Principles for the design of billiards with nonvanishing Lyapunov exponents. Commun. Math. Phys. 105(3), 391–414 (1986)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. ABC Math Program, Georgia Institute of Technology, Atlanta, GA, 30332, U.S.A

    Leonid A. Bunimovich

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, U.S.A.

    Leonid A. Bunimovich & Alexander Grigo

Authors
  1. Leonid A. Bunimovich
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  2. Alexander Grigo
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Correspondence to Alexander Grigo.

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Communicated by G. Gallavotti

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Bunimovich, L.A., Grigo, A. Focusing Components in Typical Chaotic Billiards Should be Absolutely Focusing. Commun. Math. Phys. 293, 127–143 (2010). https://doi.org/10.1007/s00220-009-0927-9

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  • DOI: https://doi.org/10.1007/s00220-009-0927-9

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