The canonical projection associated to certain possibly infinite generalized iterated function system as a fixed point

TL;DR

This paper introduces an operator associated with generalized iterated function systems (GIFS) and establishes conditions for its fixed point, which corresponds to the system's attractor, thus addressing an open problem in the field.

Contribution

It defines a new operator on continuous functions for GIFS and provides conditions for its fixed point to be unique and related to the attractor, extending previous work by Mihail.

Findings

Operator H can be continuous or contractive under certain conditions.

The fixed point of H corresponds to the attractor of the GIFS.

The paper partially solves an open problem regarding the canonical projection.

Abstract

In this paper, influenced by the ideas from A. Mihail, The canonical projection between the shift space of an IIFS and its attractor as a fixed point, Fixed Point Theory Appl., 2015, Paper No. 75, 15 p., we associate to every generalized iterated function system F (of order m) an operator H defined on C^m and taking values on C, where C stands for the space of continuous functions from the shift space on the metric space corresponding to the system. We provide sufficient conditions (on the constitutive functions of F) for the operator H to be continuous, contraction, phi-contraction, Meir-Keeler or contractive. We also give sufficient condition under which H has a unique fixed point. Moreover, we prove that, under these circumstances, the closer of the imagine of the fixed point is the attractor of F and that the fixed point is the canonical projection associated to F. In this way we give a partial answer to the open problem raised on the last paragraph of the above mentioned Mihail's paper.