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

Abstract

We associate to each iterated function system consisting of ϕmax-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.

Invariant measures of Markov operators associated to iterated function systems consisting of phi-max-contractions with probabilities

“…Since in one of our previous works we introduced a new kind of iterated function systems, namely those consisting of ϕ-max-contractions, and we prove the existence and uniqueness of their attractor (see [6]), along the lines of research previously mentioned, the next step, -which is accomplished in the present paper-is to study the Markov operators associated to such systems with probabilities. We prove that each such operator has a unique invariant measure whose support is the attractor of the system.…”

“…We say that the ϕ-max-IFS S has attractor if F S is a Picard operator (with respect to the Hausdorff-Pompeiu metric) and the fixed point of F S is called the attractor of the system S and it is denoted by A S . Theorem 3.2 (see Theorem 3.2 from [6]). Each ϕ-max-IFS has attractor.…”