McMullen’s and geometric pressures and approximating the Hausdorff dimension of Julia sets from below

• Autorzy: Przytycki Feliks

Identyfikatory

DOI

10.4064/ba211224-4-3

• Języki publikacji: EN

Abstrakty

EN

We introduce new variants of the notion of geometric pressure for rational functions on the Riemann sphere, including non-hyperbolic functions, in the hope that some of them will turn out useful to achieve fast approximation from below of the hyperbolic Hausdorff dimension of Julia sets.

Słowa kluczowe

EN

iteration of rational maps; Julia sets; Hausdorff dimension; geometric pressure; McMullen’s pressure; thermodynamic formalism

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2021
• Tom: Vol. 69, no. 2
• Strony: 115--137
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

Przytycki Feliks

• Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland

Bibliografia

• [1] [Bish] Ch. Bishop, Minkowski dimension and the Poincaré exponent, Michigan Math. J. 43 (1996), 231-246.
• [2] [Bow] R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. IHES 50 (1976), 11-26.
• [3]. [DS] A. Dudko and S. Sutherland, On the Lebesgue measure of the Feigenbaum Julia set, Invent. Math. 221 (2020), 16-202.
• [4] [DGT] A. Dudko, I. Gorbovickis and W. Tucker, in preparation.
• [5] [KvS] O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. London Math. Soc. (3) 99 (2009), 275-296.
• [6] [Lyu] M. Lyubich, Analytic low-dimensional dynamics: from dimension one to two, in: Proc. Int. Congress of Mathematicians - Seoul 2014, Vol. 1, Kyung Moon Sa, Seoul, 2014, 443-474.
• [7] [McM] C. McMullen, Hausdorff dimension and conformal dynamics. III. Computation of dimension, Amer. J. Math. 120 (1998), 691-721.
• [8] [Ore] O. Ore, Theory of Graphs, Colloq. Publ. 38, Amer. Math. Soc., 1962.
• [9] [P1] F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 (1993), 309-317.
• [10] [P2] F. Przytycki, Conical limit set and Poincaré exponent for iterations of rational functions, Trans. Amer. Math. Soc. 351 (1999), 2081-2099.
• [11] [P3] F. Przytycki, Thermodynamic formalism methods in one-dimensional real and complex dynamics, in: Proc. Int. Congress of Mathematicians - Rio de Janeiro 2018, Vol. III, World Sci., Singapore, 2081-2106.
• [12] [P4] F. Przytycki, On the Perron-Frobenius-Ruelle operator for rational maps on the Riemann sphere and for Hölder continuous functions, Bull. Brazil. Math. Soc. 20 (1990), 95-125.
• [13] [PRS1] F. Przytycki, J. Rivera-Letelier, S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math. 151 (2003), 29--63.
• [14] [PRS2] F. Przytycki, J. Rivera-Letelier and S. Smirnov, Equality of pressures for rational functions, Ergodic Theory Dynam. Systems 24 (2004), 891-914.
• [15. [PU] F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, Lon don Math. Soc. Lecture Note Ser. 371, Cambridge Univ. Press, 2010.
• [16] [Wal] P. Walters, An Introduction to Ergodic Theory, Springer, 1982.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).