Iterated function systems based on the degree of nondensifiability

Abstract

In the present paper we introduce the concept of iterated function systems (IFS) having at least one $\varphi$-condensing mapping which belongs to the finite set of self-mappings that define the IFS. It is shown the existence of an invariant for those IFS. Whenever all the self-mappings are $\varphi$-condensing we prove that the invariant set is compact. We propose some applications of those IFS having $\varphi$-condensing self-mappings to the superposition operator defined on the Banach space $\mathcal{C}([0,1])$.