Communications in Mathematical Physics

, Volume 174, Issue 3, pp 661–679 | Cite as

Chaotic properties of the elliptical stadium

  • Roberto Markarian
  • Sylvie Oliffson Kamphorst
  • Sônia Pinto de Carvalho


The elliptical stadium is a curve constructed by joining two half-ellipses, with half axesa>1 andb=1, by two straight segments of equal length 2h.

Donnay [6] has shown that if 1 <a <\(\sqrt 2\) and ifh is big enough, then the corresponding billiard map has a positive Lyapunov exponent almost everywhere; moreover,h→∞ asa\(\sqrt 2\)

In this work we prove that if\(1< a< \sqrt {4 - 2\sqrt 2 }\), then\(h > 2a^2 \sqrt {a^2 - 1}\) assures the positiveness of a Lyapunov exponent. And we conclude that, for these values ofa andh, the elliptical stadium billiard mapping is ergodic and has theK-property.


Neural Network Statistical Physic Assure Complex System Nonlinear Dynamics 
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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Roberto Markarian
    • 1
  • Sylvie Oliffson Kamphorst
    • 2
  • Sônia Pinto de Carvalho
    • 2
  1. 1.Instituto de Matemática y Estadística “Prof. Ing. Rafael LanguardiaFacultad de IngenieríaMontevideoUruguay
  2. 2.Departmento de MatemáticaICEx, UFMGBelo HorizonteBrasil

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