On the Lyapunov spectrum for rational maps. (English) Zbl 1206.37016
The authors study the dimension spectrum for Lyapunov exponents for rational maps on the Riemann sphere. Three main theorems are presented. Lower and upper bounds for the dimension are treated separately. The results on the spectrum for Lyapunov exponents are formulated in terms of the topological pressure.
Reviewer: Jan Andres (Olomouc)
MSC:
| 37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |
| 37C45 | Dimension theory of smooth dynamical systems |
| 28D99 | Measure-theoretic ergodic theory |
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
References:
| [1] | 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) · Zbl 0933.37002 · doi:10.1063/1.166232 |
| [2] | Barreira L., Saussol B., Schmeling J.: Distribution of frequencies of digits via multifractal analysis. J. Number Theory 97, 410–438 (2002) · Zbl 1027.11055 · doi:10.1016/S0022-314X(02)00003-3 |
| [3] | Besicovitch A.: On the sum of digits of real numbers represented in the dyadic system. Math. Ann. 110, 321–330 (1934) · Zbl 0009.39503 · doi:10.1007/BF01448030 |
| [4] | Collet P., Lebowitz J., Porzio A.: The dimension spectrum of some dynamical systems. J. Stat. Phys. 47, 609–644 (1987) · Zbl 0683.58023 · doi:10.1007/BF01206149 |
| [5] | Denker M., Mauldin R.D., Nitecki Z., Urbański M.: Conformal measures for rational functions revisited. Fund. Math. 157, 161–173 (1998) · Zbl 0915.58041 |
| [6] | 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) · Zbl 0852.46024 · doi:10.1017/S0143385700008804 |
| [7] | Gelfert K., Rams M.: The Lyapunov spectrum of some parabolic systems. Ergodic Theory Dynam. Syst. 29, 919–940 (2009) · Zbl 1180.37051 · doi:10.1017/S0143385708080462 |
| [8] | Graczyk J., Smirnov S.: Non-uniform hyperbolicity in complex dynamics. Invent. Math. 175, 335–415 (2009) · Zbl 1163.37008 · doi:10.1007/s00222-008-0152-8 |
| [9] | Makarov N., Smirnov S.: On ”thermodynamics” of rational maps. I. Negative spectrum. Commun. Math. Phys. 211, 705–743 (2000) · Zbl 0983.37033 · doi:10.1007/s002200050833 |
| [10] | 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 |
| [11] | Mauldin, D., Urbański, M. (eds): Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets. Cambridge University Press, Cambridge (2003) · Zbl 1033.37025 |
| [12] | Pesin Y: Dimension Theory in Dynamical Systems: Contemporary Views and Applications. The University of Chicago Press, Chicago (1997) |
| [13] | Pliss V.: On a conjecture due to Smale. Differ. Uravn. 8, 262–268 (1972) · Zbl 0243.34077 |
| [14] | Prado, E.: Teichmüller distance for some polynomial-like maps, preprint, arXiv:math/9601214v1 |
| [15] | 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) · Zbl 0723.58030 · doi:10.1007/BF02585438 |
| [16] | Przytycki F.: Lyapunov characteristic exponents are nonnegative. Proc. Am. Math. Soc. 119, 309–317 (1993) · Zbl 0787.58037 · doi:10.1090/S0002-9939-1993-1186141-9 |
| [17] | Przytycki F.: Conical limit set and Poincaré exponent for iterations of rational functions. Trans. Am. Math. Soc. 351, 2081–2099 (1999) · Zbl 0920.58037 · doi:10.1090/S0002-9947-99-02195-9 |
| [18] | Przytycki, F., Rivera-Letelier, J.: Nice inducing schemes and the thermodynamics of rational maps, preprint, arXiv:0806.4385v2 · Zbl 1211.37055 |
| [19] | 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) · Zbl 1038.37035 · doi:10.1007/s00222-002-0243-x |
| [20] | Przytycki F., Rivera-Letelier J., Smirnov S.: Equality of pressures for rational functions. Ergodic Theory Dynam. Syst. 24, 891–914 (2004) · Zbl 1058.37032 · doi:10.1017/S0143385703000385 |
| [21] | Przytycki, F., Urbański, M.: Conformal Fractals: Ergodic Theory Methods, London Mathematical Society Lecture Note Series 371, Cambridge University Press, 2010 · Zbl 1202.37001 |
| [22] | Schmeling J.: On the completeness of multifractal spectra. Ergodic Theory Dynam. Syst. 19, 1595–1616 (1999) · Zbl 0965.37022 · doi:10.1017/S0143385799151988 |
| [23] | Urbański M.: Measures and dimensions in conformal dynamics. Bull. Am. Math. Soc. 40, 281–321 (2003) · Zbl 1031.37041 · doi:10.1090/S0273-0979-03-00985-6 |
| [24] | Walters P: An Introduction to Ergodic Theory. Springer, Berlin (1982) · Zbl 0475.28009 |
| [25] | Weiss H.: The Lyapunov spectrum for conformal expanding maps and axiom-A surface diffeomorphisms. J. Stat. Phys. 95, 615–632 (1999) · Zbl 0948.37031 · doi:10.1023/A:1004591209134 |
| [26] | Wijsman R.: Convergence of sequence of convex sets, cones, and functions. II. Trans. Am. Math. Soc. 123, 32–45 (1966) · Zbl 0146.18204 · doi:10.1090/S0002-9947-1966-0196599-8 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
