Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T11:40:24.579Z Has data issue: false hasContentIssue false
  • English
  • Français

Elliptic islands in generalized Sinai billiards

Published online by Cambridge University Press:  14 October 2010

Victor J. Donnay
Victor J. Donnay
Affiliation:
Department of Mathematics, Bryn Mawr College, Bryn Mawr, Pa, 19010, USA
Get accessRights & Permissions [Opens in a new window]

Abstract

We find necessary conditions for a generalized Sinai billiard to be ergodic, independent of the geometry of the disks; i.e. if the conditions are not satisfied, then we can always construct special geometries for which the system is not ergodic. We prove non-ergodicity by constructing an elliptic periodic orbit and showing that this orbit satisfies the non-degeneracy condition of KAM theory. By a combination of theory and numerical calculation involving a computer algebra system (Mathematica), we show that the first Birkhoff invariant is non-zero. We apply this result to produce non-ergodic Hamiltonian flows that are generated by smooth potentials of finite range (repelling potentials and Lennard–Jones potentials) and by geodesic flows on the torus.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 5 , October 1996, pp. 975 - 1010
Copyright
Copyright © Cambridge University Press 1996

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

[Ba]Baldwin, P. R.. Soft billiard systems. Physica D 29 (1988), 321–342.Google Scholar
[BG]Burns, K. and Gerber, M.. Real analytic Bernoulli geodesic flows on S 2. Ergod. Th. & Dynam. Sys. 9 (1989), 27–45.CrossRefGoogle Scholar
[Bu1]Bunimovich, L. A.. Conditions of stochasticity of 2-dimensional billiards. Chaos 1 (1991), 187–193.CrossRefGoogle Scholar
[Bu2]Bunimovich, L. A.. On absolutely focusing mirrors. Ergodic Theory and Related Topics III (Lecture Notes in Mathematics 1514.) Eds Krengel, U. et al. Springer, 1992, pp. 62–82.Google Scholar
[D1]Donnay, V. J.. Geodesic flow on the two-sphere, part I: positive measure entropy. Ergod. Th. & Dynam. Sys. 8 (1988), 531–553.CrossRefGoogle Scholar
[D2]Donnay, V. J.. Geodesic flow on the two-sphere, part II: ergodicity. Dynamical Systems (Lecture Notes in Mathematics 1342). Springer, 1988, pp. 112–153.Google Scholar
[D3]Donnay, V. J.. Using integrability to produce chaos: billiards with positive entropy. Comm. Math. Phys. 141 (1991), 225–257.CrossRefGoogle Scholar
[DL]Donnay, V. J. and Liverani, C.. Potentials on the two-torus for which the Hamiltonian flow is ergodic. Commun. Math. Phys. 135 (1991), 267–302.CrossRefGoogle Scholar
[Du]Duarte, P.. Plenty of elliptic islands for the Standard family of area preserving maps. Ann. Inst. H. Poincare Anal. Non-Lineaire 11 (1994), 359–409.CrossRefGoogle Scholar
[H]Hopf, E.. Closed surfaces without conjugate points. Proc. Nat. Acad. Sci. 34 (1948), 47–51.CrossRefGoogle ScholarPubMed
[HDMS]Hayli, A., Dumont, T., Moulin-Ollagnier, J., and Stelcyn, J-M.. Quelques résultats nouveaux sur les billards de Robnik. J. Phys. A: Math. Gen. 20 (1987), 3237–3249.CrossRefGoogle Scholar
[K11]Klingenberg, W.. Lectures on Closed Geodesics. Springer, 1978.CrossRefGoogle Scholar
[K12]Klingenberg, W.. Riemannian Geometry, de Gruyter Studies in Mathematics, 1982.Google Scholar
[Kn1]Knauf, A.. Ergodic and topological properties of Coulombic periodic potentials. Comm. Math. Phys. 100 (1987), 89–112.CrossRefGoogle Scholar
[Kn2]Knauf, A.. On soft billiard systems. Physica D 36 (1989), 259–262.Google Scholar
[Ku]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
[KuM]Kubo, I. and Murata, H.. Perturbed billiard systems, II. Bernoulli properties. Nagoya Math. J. 81 (1981), 1–25.CrossRefGoogle Scholar
[Ma]Markarian, R.. Ergodic properties of plane billiards with symmetric potentials. Comm. Math. Phys. 145 (1992), 435–446.CrossRefGoogle Scholar
[Mc]McQuarrie, D. A.. Statistical Mechanics. Harper & Row, New York, 1976.Google Scholar
[Moe]Moeckel, R.. Generic bifurcations of the twist coefficient. Ergod. Th. & Dynam. Sys. 10 (1990), 185–195.CrossRefGoogle Scholar
[Mos]Moser, J.. Stable and Random Motions in Dynamical Systems (Ann. Math. Studies 77). Princeton University Press, 1973.Google Scholar
[P]Pesin, Ya. B.. Lyapunov characteristic exponents and smooth ergodic theory. Russ. Math. Surv. 32 (1977), 55–114.CrossRefGoogle Scholar
[Si]Sinai, Ya. G.. On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics. Dokl. Akad. Nauk. SSSR 153 (1963), 1261–1264.Google Scholar
[SM]Siegel, C. L. and Moser, J. K.. Lectures on Celestial Mechanics. Springer, 1971.CrossRefGoogle Scholar
[Sz]Szasz, D.. The K property of ‘orthogonal’ cylindric billiards. Comm. Math. Phys. 160 (1994), 581–597.CrossRefGoogle Scholar
[VI]Vetier, A.. Sinai billiard in potential field (construction of fibers). Ed. P. Révész. Coll. Math. Soc. J. Bolyai 36 (1982), 1079–1146.Google Scholar
[V2]Vetier, A.. Sinai billiard in potential field (absolute continuity). Proc. 3rd Pann. Symp. Eds Mogyoródi, J., Vincze, I. and Wertz, W.. Reidel, 1984, pp. 341–351.Google Scholar
[W]Wan, Y. H.. Computation of the stability condition for the Hopf bifurcation of diffeomorphisms on R 2. Siam J. Appl. Math. 34 (1978), 167–175.CrossRefGoogle Scholar