Lyapunov characteristic exponents are nonnegative

Abstract

We prove that, for an arbitrary rational map f f on the Riemann sphere and an arbitrary probability invariant measure on the Julia set, Lyapunov characteristic exponents are nonnegative a.e. In particular log ⁡ | f ′ | \log |f’| is integrable. An analogous theorem is proved for smooth maps of an interval with all critical points being nonflat. This allows us to fill a gap in the proof of Denker and Urbański’s theorem that there exists a probability conformal measure on the Julia set with exponent equal to the supremum of the Hausdorff dimensions of probability invariant measures with positive entropy.