Abstract
We study the mechanism of hyperbolicity in high-dimensional Hamiltonian systems. Especially we consider ergodic billiards with focusing components in dimensions d≥3. In this case astigmatism serves as an obstacle to hyperbolicity in billiards with large focusing components. The notion of absolutely focusing mirrors is extended to the dimensions d≥3 and the first classes of ergodic billiards with both focusing and dispersing components are constructed in d≥3.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Dynamical Systems II (2nd edn.), Ya. G. Sinai, ed. (Springer-Verlag, New York, 2000).
Ya. G. Sinai, Dynamical systems with elastic reflections, Russ. Math. Surv. 25:137–89 (1976).
L. A. Bunimovich, On ergodic properties of certain billiards, Funct. Anal. Appl. 8:254–55 (1974).
L. A. Bunimovich and J. Rehacek, How high-dimensional stadia look like, Comm. Math. Phys. 197:277–01 (1998); Nowhere dispersing 3D billiards with non-vanishing Lyapunov exponents, ibid. 189:729–57 (1997).
L. A. Bunimovich and J. Rehacek, On the ergodicity of many-dimensional focusing billiards, Ann. Inst. H. Poincaré 68:421–48 (1998).
L. A. Bunimovich, Many-dimensional nowhere dispersing billiards with chaotic behavior, Physica D 33:58–64 (1988).
M. P. Wojtkowski, Linearly stable orbits in 3-dimensional billiards, Comm. Math. Phys. 129:319–327 (1990).
H. Coddington, Tretise on Reflection and Retraction of Light (Simpkin & Marshall, London, 1829).
L. A. Bunimovich, On absolutely focusing mirrors, in Ergodic Theory and Related Topics, U. Krengel, ed., Lect. Notes Math. 1514 (Springer-Verlag, New York, 1992), pp. 62–82.
L. A. Bunimovich, Conditions of stochasticity of 2-dimensional billiards, Chaos 1:187–193 (1992).
M. P. Wojtkowski, Principles for the design of billiards with non-vanishing Lyapunov exponents, Comm. Math. Phys. 105:391–414 (1988).
R. Markarian, Billiards with Pesin region of measure one, Comm. Math. Phys. 118:87–97 (1988).
L. A. Bunimovich, A theorem on ergodicity of two-dimensional hyperbolic billiards, Comm. Math. Phys. 130:599–621 (1990).
V. J. Donnay, Using integrability to produce chaos: billards with positive entropy, Comm. Math. Phys. 141:225–257 (1991).
T. Papenbrock, Lyapunov exponents and Kolmogorov-Sinai entropy for a high-dimensional convex billiard (1999), Preprint.
L. A. Bunimovich, G. Casati, and I. Guarnery, Chaotic focusing billiards in higher dimensions, Phys. Rev. Lett. 77:2941–2944 (1996).
K. Burns and M. Gerber, Ergodic geodesic flows on product manifolds with low-dimensional factors, J. Reine Angew. Math. 450:1–35 (1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bunimovich, L.A. Hyperbolicity and Astigmatism. Journal of Statistical Physics 101, 373–384 (2000). https://doi.org/10.1023/A:1026405920274
Issue Date:
DOI: https://doi.org/10.1023/A:1026405920274