Invariant idempotent $\ast$-measures for generalized iterated function systems

TL;DR

This paper extends the theory of invariant measures to generalized invariant function systems of $ ext{ extasterisk}$-measures on compact spaces, establishing their existence and uniqueness.

Contribution

It introduces the concept of invariant $ ext{ extasterisk}$-measures for GIFSs and proves their existence and uniqueness, generalizing previous IFS measure results.

Findings

Existence of invariant $ ext{ extasterisk}$-measures for GIFSs.

Uniqueness of these invariant measures.

Extension of Hutchinson-Barnsley theory to $ ext{ extasterisk}$-measures.

Abstract

The notion of $\ast$-measure on a compact Hausdorff space can be defined for arbitrary continuous triangular norm $\ast$. The well-known Hutchinson-Barnsley theory deals with the iterated function systems (IFSs) of probability measures and establishes existence and uniqueness of invariant measures. In the previous paper, IFSs of $\ast$-measures were considered. In the present paper we deal with generalized invariant function systems (GIFSs) of $\ast$-measures, which are counterparts of GIFSs in the sense of Mihail and Miculescu. The notion of invariant $\ast$-measure is introduced for such GIFSs and we prove existence and uniqueness of such elements.