Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T21:36:28.609Z Has data issue: false hasContentIssue false
  • English
  • Français

The entropy of C2 surface diffeomorphisms in terms of Hausdorff dimension and a Lyapunov exponent

Published online by Cambridge University Press:  19 September 2008

Leonardo Mendoza
Leonardo Mendoza
Affiliation:
Escuela de Ciencias, U.C.O.L.A., Apdo. 400 Barquisimeto, Venezuela
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.

In this paper we prove that if the entropy of an ergodic measure preserved by a C2 surface diffeomorphism is positive then it is equal to the product of the Hausdorff dimension of the quotient measure defined by the family of stable manifolds and the positive Lyapunov exponent.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 2 , June 1985 , pp. 273 - 283
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Bowen, R.. Topological entropy for non-compact sets. Trans. Amer. Math. Soc. 184 (1973), 125–136.CrossRefGoogle Scholar
[2]Denker, M., Grillenberger, C. & Sigmund, K.. Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics 527. Springer: Berlin, 1976.CrossRefGoogle Scholar
[3]Katok, A.. Lyapunov Exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. I.H.E.S. 51 (1980), 137–174.CrossRefGoogle Scholar
[4]Kingman, J. F. C.. Subadditive processes. In Springer Lecture Notes in Math. 539 (1976).CrossRefGoogle Scholar
[5]Mañé, R.. A proof of Pesin's formula. Ergod. Th. & Dynam. Sys. 1 (1981), 95–102.CrossRefGoogle Scholar
[6]Manning, A.. A relation between Lyapunov exponent, Hausdorff dimension and entropy. Ergod. Th. & Dynam. Sys. 1 (1981), 451–460.CrossRefGoogle Scholar
[7]D'Paola, L. Mendoza. Entropy of diffeomorphisms of surfaces. Ph.D. Thesis, University of Warwick (1983).Google Scholar
[8]Pesin, Ja.. Families of invariant manifolds corresponding to non-zero characteristic exponents. Math. U.S.S.R. Izvestija 10 (1976), 1261–1305.CrossRefGoogle Scholar
[9]Pesin, Ja.. Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surveys 32 (1977) 4, 55–114.CrossRefGoogle Scholar
[10]Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Math. 9 (1978), 83–87.CrossRefGoogle Scholar
[11]Ruelle, D.. Ergodic theory of differentiable dynamical systems. Publ. Math. I.H.E.S. 50 (1979), 27–58.CrossRefGoogle Scholar
[12]Young, L. S.. Dimension, entropy and Lyapunov exponents. Ergod. Th. & Dynam. Sys. 2 (1982), 109–124.CrossRefGoogle Scholar