Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-12-04T09:21:10.029Z Has data issue: false hasContentIssue false
  • English
  • Français

A proof of the estimation from below in Pesin's entropy formula

Published online by Cambridge University Press:  19 September 2008

Fran¸ois Ledrappier  and
Jean-Marie Strelcyn
Fran¸ois Ledrappier
Affiliation:
Laboratoire de Probabilités, Université Paris VI, 4 Place Jussieu, 75230 Paris Cedex 05
Jean-Marie Strelcyn
Affiliation:
Département de Mathématiques, Centre Scientifique et Poly technique, Université Paris-Nord, 93 Villetaneuse, France
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 give a proof of Pesin entropy formula in a very general setting.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 2 , Issue 2 , June 1982 , pp. 203 - 219
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Anosov, D. V.. Geodesic flow on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. of Math. 90 (1967) (in Russian),Google Scholar
Amer. Math. Soc., Providence (1969) (English translation).Google Scholar
[2]Anosov, D. V. & Sinai, Ya. G.. Certain smooth ergodic systems. Russ. Math. Surveys. 22, No. 5 (1967), 103–167.CrossRefGoogle Scholar
[3]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Math. no. 470, Springer: Berlin, 1975.CrossRefGoogle Scholar
[4]Bowen, R. & Ruelle, D.. The ergodic theory of Axiom A flows. Invent. Math. 29 (1975), 181–202.CrossRefGoogle Scholar
[5]Bunimovich, L. A. & Sinai, Ya. G.. Stochasticity of Attractor in Lorenz Model. Nonlinear Waves, pp. 212–226. Nauka: Moscow, 1978 (in Russian).Google Scholar
[6]Katok, A. & Strelcyn, J.-M.. Invariant manifolds for smooth maps with singularities, Part I: Existence (Preprint, 1980)Google Scholar
[7]Katok, A. & Strelcyn, J.-M.. Invariant manifolds for smooth maps with singularities, Part II: Absolute continuity. (Preprint, 1981)Google Scholar
[8]Katok, A. & Strelcyn, J.-M.. The estimation of entropy from above for differentiable maps with singularities. (Preprint, 1981)Google Scholar
[9]Ledrappier, F.. Some properties of absolutely continuous invariant measures on an interval. Ergod. Th. & Dynam. Sys. 1 (1981), 77–93.CrossRefGoogle Scholar
[10]Livsic, A. N. & Sinai, Ya. G.. On invariant measures compatible with the smooth structure for transitive U-systems. Soviet. Math. Dokl, 13 (1972), 1656–1659.Google Scholar
[11]Mané, R.. A proof of Pesin's formula. Ergod. Th. & Dynam. Sys. 1 (1981), 95–102.CrossRefGoogle Scholar
[12]Oseledec, V. I.. The multiplicative ergodic theorem, the Lyapunov characteristic numbers of dynamical systems, Trans. Mosc. Math. Soc. 19 (1968), 197–221.Google Scholar
[13]Pesin, Ya. B., Lyapunov characteristic exponents and smooth ergodic theory. Russ. Math. Surveys 32, No. 4 (1977), 55–114.CrossRefGoogle Scholar
[14]Pesin, Ya. B.. Description of τ-partition of a diffeomorphism with invariant measure. Math. Notes USSR Acad. Sci. 22, No. 1 (1977), 506–515.Google Scholar
[15]Pesin, Ya. B.. Families of invariant manifolds corresponding to non-zero characteristic exponents. Math. of the USSR-Izvestija 10, No. 6 (1978), 1261–1305.CrossRefGoogle Scholar
[16]Rochlin, V. A.. Lectures on the theory of entropy of transformations with invariant measure. Russ. Math. Surveys. 22, No. 5 (1967), 1–52.CrossRefGoogle Scholar
[17]Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Math. 9 (1978), 83–87.CrossRefGoogle Scholar
[18]Ruelle, D.. Ergodic theory of differentiable dynamical systems. Publ. Math. IHES 50 (1979), 27–58.CrossRefGoogle Scholar
[19]Ruelle, D.. A measure associated with Axiom A attractor. Amer. J. Math. 98 (1976), 619–654.CrossRefGoogle Scholar
[20]Ruelle, D.. Sensitive dependence on initial conditions and turbulent behavior of dynamical systems. Ann. N. Y. Acad. Sci. 316 (1978), 408–416.CrossRefGoogle Scholar
[21]Sinai, Ya. G.. Classical systems with countable Lebesgue spectrum II, Izv. Akad. Nauk. SSSR Ser. Math. 30, No. 1 (1966), 15–68 (in Russian);Google Scholar
Amer. Math. Soc. Transl. 68 (1968), 34–88 (English translation).Google Scholar
[22]Sinai, Ya. G.. Gibbs measure in ergodic theory. Russ. Math. Surveys. 27(4) (1972), 21–69.CrossRefGoogle Scholar
[23]Strelcyn, J.-M.. Plane billiards as smooth dynamical systems with singularities. (Preprint, 1981.)Google Scholar
[24]Ledrappier, F.. Propriétés ergodiques des mesures de Sinai. C.R. Acad. Sci. Paris. 294 (1982) 593–595.Google Scholar