Countable contraction mappings in metric spaces: Invariant Sets and Measures

BARROZO, MARIA FERNANDA; MOLTER, URSULA M.

CENTRAL EUROPEAN JOURNAL OF MATHEMATICS - (Print)

VERSITA

Lugar: Varsovia; Año: 2014 vol. 12 p. 593 - 602

1895-1074

We consider a complete metric space $(X,d)$ and a countable number of \contraction~on $X$, $\FF=\{F_i:i\in\N\}$. We show the existence of a {\em smallest} invariant set (with respect to inclusion) for $\FF$. If the maps $F_i$ are of the form $F_i(\x) = r_i \x + b_i$ on $X=\R^d$, we can prove a converse of the classic result on \contraction. 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 $\rho = \{\rho_k\}_{k\in \N}$ is a probability sequence, we show that if there exists an invariant measure for the system $(\FF,\rho)$, 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.