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Dynamics of infinitely generated nicely expanding rational semigroups and the inducing method


Authors: Johannes Jaerisch and Hiroki Sumi
Journal: Trans. Amer. Math. Soc. 369 (2017), 6147-6187
MSC (2010): Primary 30D05, 37F15
DOI: https://doi.org/10.1090/tran/6862
Published electronically: May 11, 2017
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Abstract: We investigate the dynamics of semigroups of rational maps on the Riemann sphere. To establish a fractal theory of the Julia sets of infinitely generated semigroups of rational maps, we introduce a new class of semigroups which we call nicely expanding rational semigroups. More precisely, we prove Bowen's formula for the Hausdorff dimension of the pre-Julia sets, which we also introduce in this paper. We apply our results to the study of the Julia sets of non-hyperbolic rational semigroups. For these results, we do not assume the cone condition, which has been assumed in the study of infinite contracting iterated function systems. Similarly, we show that Bowen's formula holds for the limit set of a contracting conformal iterated function system without the cone condition.


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Johannes Jaerisch
Affiliation: Department of Mathematics, Faculty of Science and Engineering, Shimane University, Nishikawatsu 1060, Matsue, Shimane, 690-8504 Japan
Email: jaerisch@riko.shimane-u.ac.jp

Hiroki Sumi
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043 Japan
Address at time of publication: Course of Mathematical Science, Department of Human Coexistence, Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto 606-8501, Japan
Email: sumi@math.h.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/tran/6862
Keywords: Complex dynamical systems, rational semigroups, expanding semigroups, Julia set, Hausdorff dimension, Bowen's formula, random complex dynamics, random iteration, iterated function systems, self-similar sets
Received by editor(s): January 27, 2015
Received by editor(s) in revised form: September 14, 2015
Published electronically: May 11, 2017
Article copyright: © Copyright 2017 American Mathematical Society