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We investigate the dynamics of semigroups of rational maps on the Riemannsphere. To establish a fractal theory of the Julia sets of infinitely generatedsemigroups of rational maps, we introduce a new class of semigroups which wecall nicely expanding rational semigroups. More precisely, we prove Bowen'sformula for the Hausdorff dimension of the pre-Julia sets, which we alsointroduce in this paper. We apply our results to the study of the Julia sets ofnon-hyperbolic rational semigroups. For these results, we do not assume thecone condition, which has been assumed in the study of infinite contractingiterated function systems. Similarly, we show that Bowen's formula holds forthe limit set of a contracting conformal iterated function system without thecone condition.

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