The Dynamics and Geometry of Semi-Hyperbolic Rational Semigroups

TL;DR

This paper develops a thermodynamic formalism for semi-hyperbolic rational semigroups, establishing equilibrium states, statistical laws, and fractal geometry properties of Julia sets, extending classical dynamics and iterated function systems.

Contribution

It introduces new techniques for semi-hyperbolic rational semigroups, including nice families and equilibrium states, advancing the understanding of their geometric and statistical properties.

Findings

Existence and uniqueness of equilibrium states for broad classes of potentials.

Proven Hausdorff dimension of fiber Julia sets is smaller than that of the global Julia set.

Established multifractal analysis of Lyapunov exponents for these systems.

Abstract

We study skew-product dynamics for a large class of finitely-generated semi--hyperbolic semigroups of rational maps acting on the Riemann sphere, which generalizes both the theory of iteration of a single rational map of a single complex variable complex/holomorphic dynamics) and the theory of countable alphabet conformal iterated function systems (CIFSs). We construct the thermodynamic formalism for such dynamical systems and geometric potentials by developing the notion of nice families that extend to the case of our highly disconnected skew product phase space the powerful notion of nice sets due to Rivera--Letelier and Przytycki, and the allied earlier notion of $K(V)$ sets due to Denker and the last named author. We leverage out techniques to prove the existence and uniqueness of equilibrium states for a wide class of H\"older potentials, and concomitant statistical laws: central limit theorem, law of iterated logarithm, and exponential decay of correlations. We devote lots of space and effort to control (non-recurrent) critical points which is a notoriously challenging task even for a single rational function; more generators add qualitatively new challenges. Beyond dynamics, but still with dynamical methods, we advance the study of finer fractal geometrical properties of the intricate Julia sets associated to such systems and, in particular, via equilibrium states, we perform a multifractal analysis of Lyapunov exponents. We use the Nice Open Set Condition (NOSC) introduced by the last two authors, and apply our new techniques to settle a long-standing problem in the theory of rational semigroups by proving that for our class of semigroups the Hausdorff dimension of each fiber Julia set is strictly smaller than the Hausdorff dimension of the global Julia set of the semigroup.