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Perturbed billiard systems II. Bernoulli properties

Published online by Cambridge University Press:  22 January 2016

Izumi Kubo  and
Hiroshi Murata
Izumi Kubo
Affiliation:
Nagoya University Hiroshima University
Hiroshi Murata
Affiliation:
Nagoya University Hiroshima University
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One of the authors has shown the ergodicity of the perturbed billiard system which can describe the motion of a particle in a potential field of a special type [5], [6]. Since then, some development has been made, and we are now able to show the Bernoulli property of the system in this article. We hope, the result gives a new progress in statistical mechanics. Our method of the proof is inspired by the idea of D. S. Ornstein and B. Weiss [9], which has been used by G. Gallavotti and D. S. Ornstein [3] for a Sinai billiard system.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 81 , March 1981 , pp. 1 - 25
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

References

[ 1 ] Abramov, L. M., On the entropy of a flow, Dokl. Akad. Nauk SSSR 128 (1959), 873–876.=Amer. Math. Soc. Transl. (2) 49 (1966), 167–170.Google Scholar
[ 2 ] Friedman, N. A. and Ornstein, D. S., On isomorphism of weak Bernoulli transformations, Advances in Math. 5 (1970), 365–394.Google Scholar
[ 3 ] Gallavotti, G. and Ornstein, D. S., Billiards and Bernoulli schemes, Commun, math. Phys. 38 (1974), 83–101.Google Scholar
[ 4 ] Ito, Sh., Murata, H. and Totoki, H., Remarks on the isomorphism theorems for weak Bernoulli transformations in general case, Publ. RIMS, Kyoto Univ. 7 (1972), 541–580.Google Scholar
[ 5 ] Kubo, I., Perturbed billiard systems I. The ergodicity of the motion of a particle in a compound central field, Nagoya Math. J. 61 (1976), 1–57.CrossRefGoogle Scholar
[ 6 ] Kubo, I., The ergodicity of the motion of a particle in a potential field, International symposium on mathematical problems in theoretical physics, Jan. 1975 Kyoto Japan, 141–148.Google Scholar
[ 7 ] Ornstein, D. S., Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math. 5 (1970), 339–348.CrossRefGoogle Scholar
[ 8 ] Ornstein, D. S., Imbedding Bernoulli shifts in flows, Lecture Notes in Math. 160 [Sucheston, L. (editor) : Contribution to ergodic theory and probability] Springer-Verlag (1970), 178–218.Google Scholar
[ 9 ] Ornstein, D. S. and Weiss, B., Geodesic flows are Bernoullian, Israel J. Math. 14 (1973),184–198.CrossRefGoogle Scholar
[10] Rohlin, V. A. and Ya Sinai, G., Construction and properties of invariant measurable partitions, Dokl. Akad. Nauk SSSR 141 (1961), 1038–1041. = Soviet Math. Dokl. 2 (1961), 1611–1614.Google Scholar
[11] Ya. Sinai, G., On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics, Dokl. Akad. Nauk SSSR 153 (1963), 1261–1264. = Soviet Math. Dokl. 4 (1963), 1818–1822.Google Scholar
[12] Sinai, Ya. G., Dynamical systems with elastic reflections, Uspehi Math. Nauk 25 (1970), 141–192. = Russian Math. Surveys 25 (1970), 137–189.Google Scholar