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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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  • [BL] A. M. Blokh and M. Yu. Lyubich, Non-existence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems II. The smooth case, Ergodic Theory Dynamical Systems 9 (1988), 751-758. MR 1036906 (91e:58101)
  • [CE] P. Collet and J.-P. Eckmann, Iterated maps on the interval as dynamical systems, Birkhäuser, Basel, Boston, and Stuttgart, 1980. MR 613981 (82j:58078)
  • [DU] M. Denker and M. Urbański, On Sullivan's conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), 365-384. MR 1107011 (92f:58097)
  • [GPS] P. Grzegorczyk, F. Przytycki, and W. Szlenk, On iterations of Misiurewicz's rational maps on the Riemann sphere, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), 431-444. MR 1096102 (92d:30017)
  • [H] M. Herman, Exemples de fractions rationnelles ayant une orbit dense sur la sphère de Riemann, Bull. Soc. Math. France 112 (1984), 93-142. MR 771920 (86d:58055)
  • [M] R. Mañé, On a theorem of Fatou, preprint, 1991.
  • [MMS] M. Martens, W. de Melo, and S. van Strien, Julia-Fatou-Sullivan theory for real one dimensional dynamics, preprint, Delft, 1988.
  • [MS] W. de Melo and S. van Strien, A structure theorem in one-dimensional dynamics, Ann. of Math. (2) 129 (1989), 519-546. MR 997312 (90m:58106)
  • [P] Ja. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys 32 (1977), 55-114. MR 0466791 (57:6667)
  • [S] S. van Strien, Hyperbolic and invariant measures for general $ {C^2}$ interval maps satisfying the Misiurewicz condition, preprint, Delft, 1987.

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Article copyright: © Copyright 1993 American Mathematical Society

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