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SRB Measures and Pesin’s Entropy Formula for Endomorphisms

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1978)

Abstract

A celebrated result conjectured by Ruelle and proved by Ledrappier, Strelcyn and Young in the smooth ergodic theory of diffeomorphisms is that an invariant measure of a C 2 diffeomorphism satisfies Pesin’s entropy formula if and only if it has absolutely continuous conditional measures on unstable manifolds [41, 42]. The later property is known as SRB property of the invariant measure. This result was also successfully generalized to random dynamical systems of diffeomorphisms [51].

In this chapter, we present a formulation of the SRB property for invariant measures of C 2 endomorphisms of a compact manifold via their inverse limit spaces, and then prove that this property is sufficient and necessary for the entropy formula. This is a non-invertible version of the main theorem of [42]. As a nontrivial corollary of this result, an invariantmeasure of a C 2 endomorphismhas this SRB property if it is absolutely continuous with respect to the Lebesgue measure of the manifold. Invariantmeasures having this SRB property also exist on Axiom A attractors of C 2 endomorphisms (see Corollary VII.1.1.2).

Keywords

  • Invariant Measure
  • Borel Subset
  • Entropy Formula
  • Conditional Measure
  • Random Dynamical System

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2009 Springer-Verlag Berlin Heidelberg

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QUIAN, M., XIE, JS., ZHU, S. (2009). SRB Measures and Pesin’s Entropy Formula for Endomorphisms. In: Smooth Ergodic Theory for Endomorphisms. Lecture Notes in Mathematics(), vol 1978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01954-8_7

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