Iterated function systems consisting of phi-max-contractions have attractor

TL;DR

This paper establishes a connection between phi-max-contractions in iterated function systems and their attractors by defining a unique fixed point operator on the space of continuous functions, providing a new analytical approach.

Contribution

It introduces a novel operator framework for iterated function systems with phi-max-contractions, linking fixed points to attractors and canonical projections.

Findings

Existence of a unique fixed point operator for phi-max-contractions

The fixed point corresponds to the system's attractor

For convex contractions, the fixed point is the canonical projection

Abstract

We associate to each iterated function system consisting of phi-max-contractions an operator (on the space of continuous functions from the shift space on the metric space corresponding to the system) having a unique fixed point whose image turns out to be the attractor of the system. Moreover, we prove that the unique fixed point of the operator associated to an iterated function system consisting of convex contractions is the canonical projection from the shift space on the attractor of the system.