On the code space and Hutchinson measure for countable iterated function system consisting of cyclic [formula omitted]-contractions

Abstract

The notion of Iterated Function System (IFS) plays a vital role in the theory of fractals. The investigation of the code space, code map, and invariant measure associated with an IFS is an integral part of the IFS theory. During the last few decades, there has been significant interest in studying various generalizations and extensions of the classical IFS theory. A countable IFS consisting of cyclic φ-contractions (Pasupathi et al. 2020) is a recent addition to the IFS theory. However, the aforementioned study confines to itself the existence of a set attractor (fractal) associated with the IFS consisting of cyclic φ-contractions. This work aims to supplement the deliberations on the aforementioned IFS. We study the code space associated with the countable cyclic φ-contraction IFS and prove that the results are analogous to the classical IFS setting. Our main finding is that a unique invariant measure (Hutchinson measure) corresponding to the countable cyclic φ-contraction IFS with probabilities exists.