Remarks on the Dynamic of the Ruelle Operator and invariant differentials |

P. M. Makienko |

2008, issue 2, Ñ. 180–205 |

Abstract |

Let $R$ be a rational map. We are interesting in the dynamic of the Ruelle operator of $R$ on suitable spaces of differential forms. In particular the necessary and sufficient conditions (in terms of convergence of sequences of measures) of existence of invariant measurable conformal structures on $J(R)$ are obtained. |

Ruelle Operator, Julia set, quasiconformal deformationKeywords: |

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## References |

[1] J. Aaronson, An Introduction to Infinite Ergodic Theory, Math. Surveys and Monographs, 50, 1997. [2] A. Avila, “Infinitesimal perturbations of rational maps”, Nonlinearity, 15 (2002), 695–704. [3] J. Birman, L.N.M., 1167, 1987, 24–46. [4] H. Bruin, S. van Strien, Expansion of derivatives in one-dimensional dynamics, Preprint, Sept 2000. [5] A. Epstein, Infinitesimal Thurston Rigidity and the Fatou-Shishikura Inequality, Preprint Stony Brook, 1999. [6] A. Eremenko, M. Lyubich, “Dynamical properties of some classes of entire functions”, Annales de l'institute Fourier, 42:4 (1992), 989–1020. [7] A. Douady, J. Hubbard, “A proof of Thurston's topological characterization of rational functions”, Acta Math., 171 (1993), 263–297. [8] F. Gardiner, N. Lakic, Quasiconformal Teichmuller Theory, Math. Surveys and Monographs, 76, Amer. Math. Soc., Providence, RI, 2000. [9] I. Kra, Automorphic forms and Kleinian groups, W. A. Benjamin, Inc, Massachusetts, 1972, 464 pp. [10] U. Krengel, Ergodic theorems, With a supplement by Antoine Brunel, de Gruyter Studies in Mathematics, 6, Walter de Gruyter & Co., Berlin – New York, 1985, 357 pp. [11] G. M. Levin, “On Analytic Approach to The Fatou Conjecture”, Fundamenta Mathematica, 171 (2002), 177–196. [12] M. Lyubich, “On typical behavior of the trajectories of a rational mapping of the sphere”, Soviet. Math. Dokl., 27 (1983), 22–25. [13] P. Makienko, “On measurable field compatible with some rational functions”, Proceedings of conference “Dynamical systems and related topics” (Japan), 1990. [14] P. Makienko, Remarks on Ruelle operator and invariant line fields problem, Preprint of FIM ETH Zurich, July, 2000, 25 pp. [15] P. Makienko, “Remarks on Ruelle Operator and Invariant Line Fields Problem. II”, Ergodic Theory and Dynamical Systems, 25:5 (2005), 1561–1581. [16] R. Mane, P. Sad and D. Sullivan, “On the dynamic of rational maps”, Ann. Sci. Ec. Norm. Sup., 16 (1983), 193–217. [17] C. McMullen, “Rational Maps and Kleinian Groups”, Proceeding of the International Congress of Mathematicians, Springer – Verlag, New York, 1990, 889–899. [18] C. McMullen, “Families of rational maps and iterative root-finding algorithm”, Ann. of Math., 125 (1987), 467–493. [19] C. McMullen and D. Sullivan, “Quasiconformal homeomorphisms and dynamics III: The Teichmuller space of a rational map”, Adv. Math., 135 (1998), 351–395. [20] W. de Melo, S. van Strien, One-Dimensional Dynamics, A Series of Modern Surveys in Math., Springer – Verlag, 1993. [21] J. Milnor, On Latte?s Maps, Preprint of IM, Stony Brook No 1, 2004. [22] F. Przytycki, “On Measure and Hausdorff dimension of Julia sets of holomorphic Collet-Eckmann maps”, International Conference on Dynamical Systems, Montevideo, 1995, 167–181. [23] D. Sullivan, “Quasiconformal homeomorphisms and dynamics I, II, III”, Ann. of Math., 2 (1985), 401–418. |