Existence and Uniqueness of Orbital Measures

TL;DR

This paper provides an elementary proof for the existence and uniqueness of a solution to a measure equation related to iterated function systems, involving a convex combination of a given measure and a transformed measure.

Contribution

It offers a simple proof of existence and uniqueness for solutions to a measure equation in the context of IFS, expanding understanding of measure invariance.

Findings

Proves existence and uniqueness of the measure solution

Applies to measures on topological spaces in IFS context

Simplifies previous proofs of measure solutions

Abstract

We note an elementary proof of the existence and uniqueness of a solution $% \mu \in \mathbb{P(X)}$ to the equation $\mu =p\mu_{0}+q\hat{F}\mu $. Here $\mathbb{X}$ is a topological space, $\mathbb{P(X)}$ is the set of Borel measures of unit mass on $\mathbb{X}$, $\mu_{0}\in $ $\mathbb{P(X)}$ is given, $p>0$, and $q\geq 0$ with $p+q=1$. The transformation $\hat{F}:% \mathbb{P(X)\to P(X)}$ is defined by $\hat{F}\upsilon =\tsum\limits_{n=1}^{N}p_{n}\upsilon \circ f_{n}^{-1}$ where $f_{n}:\mathbb{% X\to X}$ is continuous, $p_{n}>0$ for $n=1,2,...,N$, $N$ is a finite strictly positive integer, and $\tsum\limits_{n=1}^{N}p_{n}=1$. This problem occurs in connection with iterated function systems (IFS).