Lyapunov characteristic exponents are nonnegative
HTML articles powered by AMS MathViewer
- 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.References
- A. M. Blokh and M. Yu. Lyubich, Nonexistence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems. II. The smooth case, Ergodic Theory Dynam. Systems 9 (1989), no. 4, 751–758. MR 1036906, DOI 10.1017/S0143385700005319
- Pierre Collet and Jean-Pierre Eckmann, Iterated maps on the interval as dynamical systems, Progress in Physics, vol. 1, Birkhäuser, Boston, Mass., 1980. MR 613981
- M. Denker and M. Urbański, On Sullivan’s conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), no. 2, 365–384. MR 1107011
- 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), no. 4, 431–444. Hyperbolic behaviour of dynamical systems (Paris, 1990). MR 1096102
- Michael-R. Herman, Exemples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann, Bull. Soc. Math. France 112 (1984), no. 1, 93–142 (French, with English summary). MR 771920 R. Mañé, On a theorem of Fatou, preprint, 1991. M. Martens, W. de Melo, and S. van Strien, Julia-Fatou-Sullivan theory for real one dimensional dynamics, preprint, Delft, 1988.
- W. de Melo and S. van Strien, A structure theorem in one-dimensional dynamics, Ann. of Math. (2) 129 (1989), no. 3, 519–546. MR 997312, DOI 10.2307/1971516
- Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287 (Russian). MR 0466791 S. van Strien, Hyperbolic and invariant measures for general ${C^2}$ interval maps satisfying the Misiurewicz condition, preprint, Delft, 1987.
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