Possibly infinite generalized iterated function systems comprising phy-max contractions

Abstract

One way to generalize the concept of iterated function system was
proposed by R. Miculescu and A. Mihail under the name of generalized iterated
function system (for short GIFS). More precisely, given m 2 N and a metric
space (X; d), a generalized iterated function system of order m is a nite family of
functions f1; :::; fn : Xm ! X satisfying certain contractive conditions. Another
generalization of the notion of iterated function system, due to F. Georgescu, R.
Miculescu and A. Mihail, is given by those systems consisting of '-max contrac-
tions. Combining these two lines of research, we prove that the fractal operator
associated to a possibly innite generalized iterated function system comprising
'-max contractions is a Picard operator (whose xed point is called the attractor
of the system). We associate to each possibly innite generalized iterated function
system comprising phy'-max contractions F (of order m) an operator HF : Cm ! C,
where C stands for the space of continuous and bounded functions from the shift
space on the metric space corresponding to the system. We prove that HF is a
Picard operator whose xed point is the canonical projection associated to F.