Existence and Uniqueness of Orbital Measures

Abstract

We note an elementary proof of the existence and uniqueness of a solution to the equation $\mu =p{\mu }_0+q\hat{F}\mu$. Here $\mathbb{X}$ is a topological space, $\mathbb{P}(\mathbb{X})$ is the set of Borel measures of unit mass on $\mathbb{X}$, ${\mu }_0\in$ $\mathbb{P}(\mathbb{X})$ is given, $p>0$, and $q\ge 0$ with $p+q=1$. The transformation $\hat{F}:$ 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).