Pesin’s Entropy Formula for Endomorphisms

Abstract

As we have seen in Chapters III and IV, Pesin’s entropy formula plays an important role in the smooth ergodic theory of dynamical systems. There are extensive results concerning Pesin’s entropy formula in both deterministic dynamical systems and random dynamical systems of diffeomorphisms. In [63], Pesin showed that if an invariant measure of a C ² diffeomorphism of a compact manifold is absolutely continuous with respect to the Lebesgue measure of the manifold, then it satisfies Pesin’s entropy formula. (See [55] for a simplified proof given by Mañé.) The above results were successfully generalized to random dynamical systems of diffeomorphisms [44, 36, 51].

The analogous result for non-invertible transformations was first considered for expanding maps [27, 57], as we have seen in Chapter III, and then obtained for generic C ^1+α maps [90]. Afterwards, Liu [46] provided a simpler proof using stable foliations.

In this chapter, we extend Pesin’s result to non-invertible C ² endomorphisms along the line of [46].