Communications in Mathematical Physics

, Volume 129, Issue 3, pp 535–560 | Cite as

A “Transversal” Fundamental Theorem for semi-dispersing billiards

  • A. Krámli
  • N. Simányi
  • D. Szász


For billiards with a hyperbolic behavior, Fundamental Theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds. A “Transversal” Fundamental Theorem has recently been suggested by the present authors to proveglobal ergodicity (and then, as an easy consequence, the K-property) of semidispersing billiards, in particular, the global ergodicity of systems ofN≧3 elastic hard balls conjectured by the celebratedBoltzmann-Sinai ergodic hypothesis. (In fact, the suggested “Transversal” Fundamental Theorem has been successfully applied by the authors in the casesN=3 and 4.) The theorem generalizes the Fundamental Theorem of Chernov and Sinai that was really the fundamental tool to obtainlocal ergodicity of semi-dispersing billiards. Our theorem, however, is stronger even in their case, too, since its conditions are simpler and weaker. Moreover, a complete set of conditions is formulated under which the Fundamental Theorem and its consequences like the Zig-zag theorem are valid for general semi-dispersing billiards beyond the utmost interesting case of systems of elastic hard balls. As an application, we also give conditions for the ergodicity (and, consequently, the K-property) of dispersing-billiards. “Transversality” means the following: instead of the stable and unstable foliations occurring in the Chernov-Sinai formulation of the stable version of the Fundamental Theorem, we use the stable foliation and an arbitrary nice one transversal to the stable one.


Neural Network Manifold Statistical Physic Complex System Nonlinear Dynamics 
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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • A. Krámli
    • 1
  • N. Simányi
    • 2
  • D. Szász
    • 2
  1. 1.Computer and Automation Institute of the Hungarian Academy of SciencesBudapestHungary
  2. 2.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

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