Hostname: page-component-cb9f654ff-mnl9s Total loading time: 0 Render date: 2025-08-20T16:00:37.147Z Has data issue: false hasContentIssue false
  • English
  • Français

Periodic orbits of a dynamical system in a compound central field and a perturbed billiards system

Published online by Cambridge University Press:  19 September 2008

Takehiko Morita
Takehiko Morita
Affiliation:
Department of Mathematics, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a compound central field in the Euclidean plane which is governed by a finite number of bell-shaped potential functions with finite range. The study of the qualitative behavior of the Hamilton flow in such a potential field can be reduced to that of the so-called perturbed billiards system. The main result in this paper is the construction of a symbolic dynamics of the Hamilton flow by using the perturbed billiards system provided that the energy E > 0 is small enough. We also try to show an analogue of the prime number theorem for the closed orbits of the flow following ideas presented by Morita and Parry and Pollicott.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 3 , September 1994 , pp. 599 - 619
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[A]Arnold, V. I.. Mathematical Methods of Classical Mechanics. 2nd edn.Springer: New York, Berlin, Heideberg, 1989.CrossRefGoogle Scholar
[G]Gutzwiller, M.C.Chaos in Classical and Quantum Mechanics. Springer: New York, Berlin, Heideberg, 1990.CrossRefGoogle Scholar
[K]Kubo, I.. Perturbed billiard systems I. Nagoya Math. J. 61 (1976), 1–57.CrossRefGoogle Scholar
[M]Morita, T.. The symbolic representation of billiards without boundary condition. Trans. Amer. Math. Soc. 325 (1991), 819–828.CrossRefGoogle Scholar
[P.P]Parry, W. and Pollicott, M.. An analogue of the prime number theorem for closed orbits of axiom A flows. Ann. Math. 118 (1983), 573–591.CrossRefGoogle Scholar
[S]Sinai, Ya. G.. Dynamical systems with elastic reflections. Russian Math. Surv. 25 (1970), 137–189.CrossRefGoogle Scholar