Abstract
An important class of fractal sets is given by the attractors of iterated function systems which are defined as the fixed points of the associated fractal operators. In the study of such an attractor, an important place is taken by the canonical projection between the shift space associated with the system and the attractor. In this paper, by using different fixed point theorems, we present the canonical projection as the fixed point of a certain operator defined on the space of continuous functions from the shift space on the metric space associated with the system.
Highlights
It is well known that some important mathematical objects, such as Cantor set, Sierpinsky gasket and carpet, Menger cube and the graph of Weierstrass function, are fractal sets.Most of them could be obtained as attractors of iterated function systems or of infinite iterated function systems, that is, they can be seen as fixed points of the fractal operator associated to the corresponding iterated function system
The second one uses different types of fixed point theorems to define the attractor of an IFS or IIFS
Let us mention the case of the connectivity of the attractor and the very important concept associated with an IFS with probabilities, namely that of Hutchinson measure
Summary
It is well known that some important mathematical objects, such as Cantor set, Sierpinsky gasket and carpet, Menger cube and the graph of Weierstrass function, are fractal sets. The second one uses different types of fixed point theorems to define the attractor of an IFS or IIFS (see [ ]). Let us mention the case of the connectivity of the attractor (see [ , ]) and the very important concept associated with an IFS with probabilities, namely that of Hutchinson measure. By using different fixed point theorems, we present the canonical projection between the shift space and the attractor of an IIFS as a fixed point of an operator on the space of continuous functions from the shift space on the metric space associated with the system. The second one contains some preliminaries concerning fixed point theorems and classical results concerning IIFS, the third is dedicated to the main results, the fourth deals with the case of ε-chainable metric spaces and the last one contains some remarks and examples
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