Dynamics of infinitely generated nicely expanding rational semigroups and the inducing method
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 preJulia sets, which we also introduce in this paper. We apply our results to the study of the Julia sets of nonhyperbolic 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2015
 DOI:
 10.48550/arXiv.1501.06772
 arXiv:
 arXiv:1501.06772
 Bibcode:
 2015arXiv150106772J
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Complex Variables;
 Mathematics  Probability;
 30D05;
 37F15
 EPrint:
 to appear in Trans. Amer. Math. Soc