LYAPUNOV CHARACTERISTIC EXPONENTS ARE NONNEGATIVE

We prove that, for an arbitrary rational map 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'\ 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 pp in the proof of Denker and Urbanski'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.