Ruelle Operator for Infinite Conformal IFS

TL;DR

This paper extends the concept of Ruelle operators to infinite conformal iterated function systems (IFS), exploring their properties and how they relate to separation conditions within these systems.

Contribution

It introduces the Ruelle operator framework for infinite conformal IFSs and investigates their separation properties, expanding understanding beyond finite systems.

Findings

Established the existence of Ruelle operators for infinite conformal IFSs

Analyzed separation properties using the Ruelle operator framework

Extended finite IFS results to infinite conformal cases

Abstract

Let $(X, \{w_j \}_{j=1}^m, \{p_j \}_{j=1}^m)$ ($2 \leq m < \infty$) be a contractive iterated function system (IFS), where $X$ is a compact subset of ${\Bbb{R}}^d$. It is well known that there exists a unique nonempty compact set $K$ such that $K=\bigcup_{j=1}^m w_j(K)$. Moreover, the Ruelle operator on $C(K)$ determined by the IFS $(X, \{w_j \}_{j=1}^m, \{p_j \}_{j=1}^m)$ ($2 \leq m < \infty$) has been introduced in \cite{FL}. In the present paper, the Ruelle operators determined by the infinite conformal IFSs are discussed. Some separation properties for the infinite conformal IFSs are investigated by using the Ruelle operator.