Countable Contraction Maps in Metric Spaces: Invariant Sets and Measures

Abstract

We consider a complete metric space $(X,d)$ and a countable number of contractive mappings on $X$, $\mathcal{F}=\{F_i:i\in\mathbb N\}$. We show the existence of a {\em smallest} invariant set (with respect to inclusion) for $\mathcal{F}$. If the maps $F_i$ are of the form $F_i(\x) = r_i \boldmath{x} + b_i$ on $X=\mathbb{R}^d$, we can prove a converse of the classic result on contraction maps. Precisely, we can show that for that case, there exists a {\em unique} bounded invariant set if and only if $r = \sup_i r_i$ is strictly smaller than 1. Further, if $\rho = \{\rho_k\}_{k\in \mathbb N}$ is a probability sequence, we show that if there exists an invariant measure for the system $(\mathcal{F},\rho)$, then it's support must be precisely this smallest invariant set. If in addition there exists any {\em bounded} invariant set, this invariant measure is unique - even though there may be more than one invariant set.