Smooth Ergodic Theory for Endomorphisms pp 15-26 | Cite as

# Expanding Maps

## Abstract

In the previous chapter, we showed the Margulis-Ruelle inequality, which says that the measure-theoretic entropy is bounded above by the sum of positive Lyapunov exponents.

In 1977, Pesin [63] showed that for a *C* ^{2} diffeomorphism *f* if its invariant Borel probability measure µ is absolutely continuous with respect to the Lebesgue measure on the manifold, then the equality in (II.II.1) holds, i.e.

Now identity (PEF) is known as Pesin’s entropy formula.

In this chapter, we consider a simple case that this identity holds, namely expanding maps. It is simpler than the case of diffeomorphisms as the local unstable manifold of an expanding map at a point is a neighborhood of the point. The results of this chapter are from [27].

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