G. Follo

Some Remarks on Fractals Generated by a Sequence of Finite Systems of Contractions

Given a complete metric space \ $X$ \ and a sequence \ $(\mathfrak{F}_n)$ \ of finite systems of contractions on \ $X$ \ we prove the existence of a unique compact set \ $K\subseteq X$ \ that is the limit in the Hausdorff metric of the sequence \ $((\mathfrak{F}_1\circ\mathfrak{F}_2\circ\cdots\circ\mathfrak{F}_n)(C))$ \ for every \ $C\subseteq X$ \ non-empty closed and bounded. If \ $X$ \ is also separable, given a sequence \ $(l^{(n)})=(l^{(n)}_1,l^{(n)}_2,dots,l^{(n)}_{m_n})$ \ of weights we construct a measure supported on \ $K$ \ which is the week limit of the sequences \ $(\nu_n)$ \ of measures defined as image measures of a generic borel regular measure \ $\nu$ , with bounded support, under \ $((\mathfrak{F}_1\circ\mathfrak{F}_2\circ\cdots\circ\mathfrak{F}_n)(C))$, weighted by \ $l^{(1)},l^{(2)},\dots,l^{(n)}$ .