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A proof of Pesin's formula

Published online by Cambridge University Press:  19 September 2008

Ricardo Mañé
Ricardo Mañé*
Affiliation:
Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
*
†Dr R. Mañé, Instituto de Matemática Pura e Aplicada, Rua Luiz de Camões 68, 20060 Rio de Janeiro, RJ, Brazil.
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Abstract

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Let f: M → M be a diffeomorphism of a compact manifold M and let χ:M → R be defined by putting χ(x) equal to the sum of the non-negative characteristic exponents of f at x, each being counted with its multiplicity. If μ is an f-invariant probability of M which is absolutely continuous relative to Lebesgue measure, then Pesin has proved the entropy, hμ(f), is given by We prove this formula without using the theory of stable manifolds.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 1 , Issue 1 , March 1981 , pp. 95 - 102
Copyright
Copyright © Cambridge University Press 1981

References

REFERENCES

[1]Pesin, Y.. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys 32(1977), 55–114.Google Scholar
[2]Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9 (1978), 83–87.Google Scholar