Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T04:32:15.940Z Has data issue: false hasContentIssue false
  • English
  • Français

Dimension, entropy and Lyapunov exponents

Published online by Cambridge University Press:  13 August 2009

Lai-Sang Young
Lai-Sang Young
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824
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 consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure μ. Define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure. We prove a formula relating HD (μ) to the entropy and Lyapunov exponents of the map. Other classical notions of fractional dimension such as capacity and Rényi dimension are discussed. They are shown to be equal to Hausdorff dimension in the present context.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 2 , Issue 1 , March 1982 , pp. 109 - 124
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Besicovitch, A. S.. Sets of fractional dimension II: On the sum of digits of real numbers represented in the dyadic system. Math. Ann. 110 (1934), 321–329.CrossRefGoogle Scholar
[2]Billingsley, P.. Ergodic Theory and Information. Wiley: New York, 1965.Google Scholar
[3]Bowen, R.. Some systems with unique equilibrium states. Math. Systems Theory. 8 (1974), 193–202.CrossRefGoogle Scholar
[4]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Math. no. 470. Springer: Berlin, 1975.CrossRefGoogle Scholar
[5]Brin, M. & Katok, A.. On local entropy. Preprint.Google Scholar
[6]Douady, A. & Oesterlé, J.. Dimension de Hausdorff des attracteurs. C.R. Acad. Sci. 24 (1980), 1135–1138.Google Scholar
[7]Eggleston, H. G.. The fractional dimension of a set defined by decimal properties. Quart. J. Math. Oxford Ser. 20 (1949), 31–36.CrossRefGoogle Scholar
[8]Frederickson, P., Kaplan, J., Yorke, E. & Yorke, J.. The Liapunov dimension of strange attractors. Preprint.Google Scholar
[9]Kaplan, J. & Yorke, J.. Chaotic Behavior of Multidimensional Difference Equations. Lecture Notes in Math. no. 730, pp. 228–237. Springer: Berlin, 1979.Google Scholar
[10]Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. I.H.E.S. 51 (1980), 137–174.Google Scholar
[11]Kolmogorov, A. N.. A new invariant for transitive dynamical systems. Dokl. Akad. Nauk SSSR. 119 (1958), 861–864.Google Scholar
[12]Ledrappier, F.. Some relations between dimension and Lyapunov exponents. Commun. Math. Phys. 81 (1981), 229–238.CrossRefGoogle Scholar
[13]Mañé, R.. A proof of Pesin's formula. Ergod. Th. & Dynam. Sys. 1 (1981), 95–102.CrossRefGoogle Scholar
[14]Manning, A.. A relation between Lyapunov exponents, Hausdorff dimension and entropy. Ergod. Th. & Dynam. Sys. 1 (1981), 451–459.CrossRefGoogle Scholar
[15]Nitecki, Z.. Differentiable Dynamics. MIT Press: Cambridge, Mass., 1971.Google Scholar
[16]Oseledec, V. I.. A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems. Trudy Moskov. Mat. Obšč. 19 (1968), 179–210, Engl. translation.Google Scholar
[17]Pesin, Ja.. Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surveys. 32 (1977), 55–114.CrossRefGoogle Scholar
[18]Rényi, A.. Dimension, entropy and information. Transactions of the second Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (1957) 545–556.Google Scholar
[19]Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9 (1978), 83–87.CrossRefGoogle Scholar
[20]Ruelle, D.. Ergodic theory of differentiable dynamical systems. Publ. Math. I.H.E.S. 50 (1979), 27–58.Google Scholar
[21]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[22]Young, L.. Capacity of attractors. Ergod. Th. & Dynam. Sys. 1 (1981), 381–388.CrossRefGoogle Scholar