SRB Measures and Pesin’s Entropy Formula for Endomorphisms

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).