On $\mathbb{P}$-Weakly Hyperbolic Iterated Function Systems

TL;DR

This paper introduces the concept of $\\mathbb{P}$-weakly hyperbolic iterated function systems on compact metric spaces, establishing their invariant measures and proving an ergodic theorem, thus generalizing previous notions of weak hyperbolicity.

Contribution

It generalizes the concept of weakly hyperbolic IFS to a broader setting with a compact parameter space and proves key properties like existence, uniqueness, and ergodicity of invariant measures.

Findings

Existence and uniqueness of invariant measure for $\\mathbb{P}$-weakly hyperbolic IFS.

Proved an ergodic theorem for these systems.

Generalized previous weak hyperbolicity concepts to a more comprehensive setting.

Abstract

In this paper we will consider the concept of $\mathbb{P}$-weakly hyperbolic iterated function systems on compact metric spaces that generalizes the concept of weakly hyperbolic iterated function systems, as defined by Edalat in \cite{E} and by Arbieto, Santiago and Junqueira in \cite{ASJ} for a more general setting where the parameter space is a compact metric space. We prove the existence and uniqueness of the invariant measure of a $\mathbb{P}$-weakly hyperbolic IFS. Furthermore, we prove an ergodic theorem for $\mathbb{P}$-weakly hyperbolic IFS with compact parameter space.