Countable Contraction Maps in Metric Spaces: Invariant Sets and Measures

TL;DR

This paper investigates countable families of contractive maps in metric spaces, establishing the existence of minimal invariant sets, conditions for unique bounded invariant sets in affine cases, and characterizing invariant measures.

Contribution

It introduces the concept of the smallest invariant set for countable contractive maps and characterizes conditions for uniqueness of bounded invariant sets and measures.

Findings

Existence of a smallest invariant set for countable contractive maps.

Unique bounded invariant set exists if contraction ratios are uniformly less than 1.

Invariant measure support coincides with the smallest invariant set under certain conditions.

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.