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

TL;DR

This paper develops a fractal theory for Julia sets of infinitely generated rational semigroups, introducing nicely expanding classes and proving Bowen's formula without the cone condition.

Contribution

It introduces nicely expanding rational semigroups and proves Bowen's formula for their Julia sets without relying on the cone condition.

Findings

Proved Bowen's formula for pre-Julia sets of these semigroups.

Extended fractal dimension results to non-hyperbolic rational semigroups.

Established Bowen's formula for limit sets of contracting conformal iterated function systems.

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.