Abstract
We study the dimension spectrum for Lyapunov exponents for rational maps on the Riemann sphere.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barreira L., Pesin Ya., Schmeling J.: On a general concept of multifractality. Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity. Chaos 7, 27–38 (1997)
Barreira L., Saussol B., Schmeling J.: Distribution of frequencies of digits via multifractal analysis. J. Number Theory 97, 410–438 (2002)
Besicovitch A.: On the sum of digits of real numbers represented in the dyadic system. Math. Ann. 110, 321–330 (1934)
Collet P., Lebowitz J., Porzio A.: The dimension spectrum of some dynamical systems. J. Stat. Phys. 47, 609–644 (1987)
Denker M., Mauldin R.D., Nitecki Z., Urbański M.: Conformal measures for rational functions revisited. Fund. Math. 157, 161–173 (1998)
Denker M., Przytycki F., Urbanski M.: On the transfer operator for rational functions on the Riemann sphere. Ergodic Theory Dynam. Syst. 16, 255–266 (1996)
Gelfert K., Rams M.: The Lyapunov spectrum of some parabolic systems. Ergodic Theory Dynam. Syst. 29, 919–940 (2009)
Graczyk J., Smirnov S.: Non-uniform hyperbolicity in complex dynamics. Invent. Math. 175, 335–415 (2009)
Makarov N., Smirnov S.: On “thermodynamics” of rational maps. I. Negative spectrum. Commun. Math. Phys. 211, 705–743 (2000)
Mañé, R.: The Hausdorff dimension of invariant probabilities of rational maps. In: Dynamical Systems, Valparaiso 1986, pp. 86–117. Lecture Notes in Math. 1331, Springer, 1988
Mauldin, D., Urbański, M. (eds): Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets. Cambridge University Press, Cambridge (2003)
Pesin Y: Dimension Theory in Dynamical Systems: Contemporary Views and Applications. The University of Chicago Press, Chicago (1997)
Pliss V.: On a conjecture due to Smale. Differ. Uravn. 8, 262–268 (1972)
Prado, E.: Teichmüller distance for some polynomial-like maps, preprint, arXiv:math/9601214v1
Przytycki F.: On the Perron-Frobenius-Ruelle operator for rational maps on the Riemann sphere and for Hölder continuous functions. Bol. Soc. Brasil. Mat. (N. S.) 20, 95–125 (1990)
Przytycki F.: Lyapunov characteristic exponents are nonnegative. Proc. Am. Math. Soc. 119, 309–317 (1993)
Przytycki F.: Conical limit set and Poincaré exponent for iterations of rational functions. Trans. Am. Math. Soc. 351, 2081–2099 (1999)
Przytycki, F., Rivera-Letelier, J.: Nice inducing schemes and the thermodynamics of rational maps, preprint, arXiv:0806.4385v2
Przytycki F., Rivera-Letelier J., Smirnov S.: Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps. Invent. Math. 151, 29–63 (2003)
Przytycki F., Rivera-Letelier J., Smirnov S.: Equality of pressures for rational functions. Ergodic Theory Dynam. Syst. 24, 891–914 (2004)
Przytycki, F., Urbański, M.: Conformal Fractals: Ergodic Theory Methods, London Mathematical Society Lecture Note Series 371, Cambridge University Press, 2010
Schmeling J.: On the completeness of multifractal spectra. Ergodic Theory Dynam. Syst. 19, 1595–1616 (1999)
Urbański M.: Measures and dimensions in conformal dynamics. Bull. Am. Math. Soc. 40, 281–321 (2003)
Walters P: An Introduction to Ergodic Theory. Springer, Berlin (1982)
Weiss H.: The Lyapunov spectrum for conformal expanding maps and axiom-A surface diffeomorphisms. J. Stat. Phys. 95, 615–632 (1999)
Wijsman R.: Convergence of sequence of convex sets, cones, and functions. II. Trans. Am. Math. Soc. 123, 32–45 (1966)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gelfert, K., Przytycki, F. & Rams, M. On the Lyapunov spectrum for rational maps. Math. Ann. 348, 965–1004 (2010). https://doi.org/10.1007/s00208-010-0508-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-010-0508-4