Analytic families of holomorphic iterated function systems

TL;DR

This paper studies analytic families of holomorphic iterated function systems, proving the real analyticity of the pressure function and classifying these systems based on their continuous dependence on parameters, enhancing understanding of their global structure.

Contribution

The paper establishes the real analyticity of the pressure function and provides a classification theorem for analytic families of holomorphic IFSs under the λ-topology, extending previous geometric results.

Findings

Proved the pressure function is real analytic.

Classified analytic families of holomorphic IFSs based on λ-topology.

Enhanced understanding of the global structure of conformal IFSs.

Abstract

This paper deals with analytic families of holomorphic iterated function systems. Using real analyticity of the pressure function (which we prove), we establish a classification theorem for analytic families of holomorphic iterated function systems which depend continuously on a parameter when the space of holomorphic iterated function systems is endowed with the "$\lambda$-topology". This classification theorem allows us to generalize some geometric results from the paper ("Lambda-Topology vs. Pointwise Topology", to appear in Ergod. Th. & Dynam. Sys.) of the authors, and gives us a better and clearer understanding of the global structure of the space of conformal IFSs.