On the conformal preimage decay exponent of the Julia sets of rational graph-directed Markov systems

TL;DR

This paper introduces and studies the conformal preimage decay exponent for Julia sets of rational graph-directed Markov systems, linking it to topological entropy and pressure in the associated skew product map.

Contribution

It defines the conformal preimage decay exponent and establishes its equality with the difference between topological entropy and a generalized topological pressure.

Findings

The decay exponent equals the entropy-pressure difference.

The exponent coincides with a generalized topological pressure.

Provides new insights into Julia set dynamics and thermodynamic formalism.

Abstract

We define and investigate the conformal preimage decay exponent of the Julia sets of rational graph-directed Markov systems. We show that this exponent coincides with the difference between the topological entropy and upper sequential capacity topological pressure for the rational skew product map associated with the system $S$. Here, upper sequential capacity topological pressure is a slight generalisation of upper capacity topological pressure given in \cite{MR969568}.