The Hutchinson–Barnsley theory for iterated function system with bounded cyclic contractions

Abstract

The existence of an invariant set, which forms the introductory part of the classical Hutchinson–Barnsley theory of an Iterated Function System (IFS), has been recently established for an IFS consisting of continuous cyclic contractions (Pasupathi et al., 2020). The current work seeks to supplement the cited reference in two ways. One intriguing aspect that sets the fixed point theorems of cyclic maps apart from the classical Banach contraction principle is the lack of a continuity requirement. As a result, in contrast to the research study in the cited reference, it appears more natural to consider the cyclic IFS without making the additional continuity assumption on the cyclic contractions involved in the IFS. With this in mind, the first goal of the present note is to consider a type of cyclic IFS wherein the constituent maps need not be continuous. Second, we obtain the coding map and invariant measure corresponding to the cyclic IFS.