Hostname: page-component-f554764f5-fr72s Total loading time: 0 Render date: 2025-04-20T07:02:41.990Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant families of cones and Lyapunov exponents

Published online by Cambridge University Press:  19 September 2008

Maciej Wojtkowski
Maciej Wojtkowski
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that in several cases preservation of cones leads to non-vanishing of (some) Lyapunov exponents. It gives simple and effective criteria for nonvanishing of the exponents, which is demonstrated on the example of the billiards studied by Bunimovich. It is also shown that geodesic flows on manifolds of non-positive sectional curvature can be treated from this point of view.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 1 , March 1985 , pp. 145 - 161
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Alekseev, V. M.. Quasi random dynamical systems. Mat. USSR Sbornik 7 (1969), 1–43.CrossRefGoogle Scholar
[2]Ballmann, W. & Brin, M.. On the ergodicity of geodesic flows. Ergod. Th. & Dynam. Sys. 2 (1982), 311–315.CrossRefGoogle Scholar
[3]Benettin, G., Galgani, I., Giorgilli, A. & Strelcyn, J.-M.. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: theory. Meccanica (1980), 9–20.CrossRefGoogle Scholar
[4]Bunimovich, L. A.. On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65 (1979), 295–312.CrossRefGoogle Scholar
[5]Burns, K.. Hyperbolic behaviour of geodesic flows on manifolds with no focal points. Ergod. Th. & Dynam. Sys. 3 (1983), 1–12.CrossRefGoogle Scholar
[6]Cornelis, E. & Wojtkowski, M.. A criterion for the positivity of the Lyapunov characteristic exponent. Preprint (1983).Google Scholar
[7]Gallavotti, G.. Lectures on billiards. In Lecture Notes in Physics, 38, Springer (1975).Google Scholar
[8]Katok, A.. Bernoulli diffeomorphisms on surfaces. Ann. Math. 110, (1979), 529–547.CrossRefGoogle Scholar
[9]Katok, A. & Strelcyn, J.-M.. Invariant manifolds for smooth maps with singularities, I, II. Preprint (1981).Google Scholar
[10]Oseledec, V. I.. The multiplicative ergodic theorem. The Lyapunov characteristic numbers of dynamical systems. Trans. Mosc. Math. Soc. 19 (1968), 197–231.Google Scholar
[11]Pesin, Ya. B.. Lyapunov characteristic exponents and smooth ergodic theory. Russ. Math. Surveys 32 (1977), 55–114.CrossRefGoogle Scholar
[12]Przytycki, F.. Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviours. Ergod. Th. & Dynam. Sys. 2, (1982), 439–463.CrossRefGoogle Scholar
[13]Ruelle, D.. Ergodic theory of differentiable dynamical systems. Publ. Math. IHES 50 (1979), 27–58.CrossRefGoogle Scholar