Countable contraction mappings in metric spaces: invariant sets and measure

Abstract

We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {Fi: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps Fi are of the form Fi(x) = rix + bi on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supiri is strictly smaller than 1.Further, if ρ = {ρk}k∈ℕ is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set.