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

Natalia Mazurenko, Mykhailo Zarichnyi

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.

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

Abstract

The notion of -measure on a compact Hausdorff space can be defined for arbitrary continuous triangular norm . 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 -measures were considered. In the present paper we deal with generalized invariant function systems (GIFSs) of -measures, which are counterparts of GIFSs in the sense of Mihail and Miculescu. The notion of invariant -measure is introduced for such GIFSs and we prove existence and uniqueness of such elements.

Paper Structure

This paper contains 9 sections, 4 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\ast$-measures
  4. Hyperspace representation
  5. Normal upper semicontinuous fuzzy sets
  6. $G$-symmetric powers
  7. Generalized Iterated Function Systems
  8. Results
  9. Remarks and open questions

Key Result

Lemma 2.1

Let $X,Y$ be compact metric spaces, $\mathrm{pr}_Y\colon X\times Y\to Y$ be the projections. Let $A,B\in\exp (X\times Y)$. If $\mathrm{pr}_Y(A)=\mathrm{pr}_Y(B)$, then $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (14)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 2.1
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 4 more