A minimum principle for Lyapunov exponents and a higher-dimensional version of a Theorem of Mane'

Abstract

We consider compact invariant sets \Lambda for C^{1} maps in arbitrary dimension. We prove that if \Lambda contains no critical points then there exists an invariant probability measure with a Lyapunov exponent \lambda which is the minimum of all Lyapunov exponents for all invariant measures supported on \Lambda. We apply this result to prove that \Lambda is uniformly expanding if every invariant probability measure supported on \Lambda is hyperbolic repelling. This generalizes a well known theorem of Mane' to the higher-dimensional setting.