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.

$$h_\mu(f) = \int_M {\sum\limits_i {\lambda _i (x)^ +m_i (x)d\mu(x).} }$$

((PEF))

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].