On the Uniqueness of the Density for the Invariant Measure in an Infinite Hyperbolic Iterated Function System

We consider a regular infinite hyperbolic iterated function satisfying a property which guarantees that the associated Frobenius-Perron operator ℒ is almost periodic. For such a system there is a unique invariant probablility measure μ supported on J, the limit set of the system and which is equivalent to the conformal measure m of the system. In this note we will demonstrate two properties of dμ/dm. Firstly, we show that there is a unique positive continuous function on X, ρ such that ℒρ = ρ and ∫ ρdm = 1. This function is the density of μ with respect to m. Secondly, we show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\{ \mathcal{L}(\coprod _X )\} _{^{n = 1} }^\infty$$ \end{document} converges uniformly to ρ on X.