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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Lyapunov characteristic exponents are nonnegative
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by Feliks Przytycki PDF
Proc. Amer. Math. Soc. 119 (1993), 309-317 Request permission

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 $\log |f’|$ 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.
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 119 (1993), 309-317
  • MSC: Primary 58F23; Secondary 30D05
  • DOI: https://doi.org/10.1090/S0002-9939-1993-1186141-9
  • MathSciNet review: 1186141