Communications in Mathematical Physics

, Volume 129, Issue 3, pp 535–560

# A “Transversal” Fundamental Theorem for semi-dispersing billiards

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

## Abstract

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.

## Keywords

Neural Network Manifold Statistical Physic Complex System Nonlinear Dynamics
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.

## References

1. B-S (1973) Bunimovich, L.A., Sinai, Ya.G.: On the fundamental theorem of dispersing billiards. Math. Sb.90, 415–431 (1973)Google Scholar
2. C (1982) Chernov, N.I.: Construction of transversal fibers for multidimensional semi-dispersing billiards. Funkt. Anal. i. Pril.16, 35–46 (1982)Google Scholar
3. G (1975) Gallavotti, G.: Lectures on the billiard. Lecture Notes in Physics, vol. 38. Moser, J. (ed.), pp. 236–295. Berlin, Heidelberg New York: Springer 1975Google Scholar
4. G (1981) Galperin, G.: On systems of locally interacting and repelling particles moving in space. Trudy MMO43, 142–196 (1981)Google Scholar
5. I (1988) Illner, R.: On the number of collisions in a hard sphere particle system in all space. Technical Report (1988)Google Scholar
6. K-S (1986) Katok, A., Strelcyn, J.-M.: Invariant manifolds, entropy and billiards, smooth maps with singularities. Lecture Notes in Mathematics, vol. 1222. Berlin, Heidelberg New York: Springer 1986Google Scholar
7. K-S-F (1980) Kornfeld, I.P., Sinai, Ya.G., Fomin, S.V.: Ergodic theory. Moscow: Nauka 1980Google Scholar
8. K-S-Sz (1989-A) Krámli, A., Simányi, N., Szász, D.: Dispersing billiards without focal points on surfaces are ergodic. Commun. Math. Phys.125, 439–457 (1989)
9. K-S-Sz (1989-B) Krámli, A., Simányi, N., Szász, D.: Ergodic properties of semi-dispersing billiards. I. Two cylindric scatterers in the 3-D torus. Nonlinearity2, 311–326 (1989)
10. K-S-Sz (1989-C) Krámli, A., Simányi, N., Szász, D.: Three billiard balls on thev-dimensional torus is a K-flow (submitted to Ann. Math.)Google Scholar
11. P (1977) Pesin, Ya.B.: Lyapunov characteristic exponents and smooth ergodic theory. Usp. Mat. Nauk.32, 55–112 (1977)Google Scholar
12. S (1970) Sinai, Ya.G.: Dynamical systems with elastic reflections. Usp. Mat. Nauk.25, 141–192 (1970)Google Scholar
13. S (1979) Sinai, Ya.G.: Ergodic properties of the Lorentz gas. Funkcional. Anal. i. Pril.13, 46–59 (1979)Google Scholar
14. S-Ch (1987) Sinai, Ya.G., Chernov, N.I.: Ergodic properties of some systems of 2-D discs and 3-D spheres. Usp. Mat. Nauk.42, 153–174 (1987)Google Scholar
15. V (1982) Vetier, A.: Sinai billiard in a potential field (construction of stable and unstable fibers). Colloquia Soc. Math. J. Bolyai36, 1079–1146 (1982)Google Scholar

## 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