Pesin Entropy Formula for C1 Diffeomorphisms with Dominated Splitting

TL;DR

This paper extends Pesin's entropy formula to C1 diffeomorphisms with dominated splitting, establishing entropy bounds for measures describing typical orbit statistics and conditions for equality.

Contribution

It proves a lower bound for metric entropy in terms of Lyapunov exponents for a broad class of invariant measures in systems with dominated splitting.

Findings

Entropy is bounded below by sum of Lyapunov exponents on the dominating subbundle.

Measures with non negative exponents and non positive exponents on the dominated subbundle satisfy Pesin's formula.

Results apply to Lebesgue-almost all orbits in systems with dominated splitting.

Abstract

For any C1 diffeomorphism with dominated splitting we consider a nonempty set of invariant measures which describes the asymptotic statistics of Lebesgue-almost all orbits. They are the limits of convergent subsequences of averages of the Dirac delta measures supported on those orbits. We prove that the metric entropy of each of these measures is bounded from below by the sum of the Lyapunov exponents on the dominating subbundle. As a consequence, if those exponents are non negative, and if the exponents on the dominated subbundle are non positive, those measures satisfy the Pesin Entropy Formula.