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Lyapunov characteristic exponents are nonnegative

Author: Feliks Przytycki
Journal: Proc. Amer. Math. Soc. 119 (1993), 309-317
MSC: Primary 58F23; Secondary 30D05
MathSciNet review: 1186141
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Abstract: 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 \vert f'\vert$ 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.

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