Fuzzy-set approach to invariant idempotent measures

TL;DR

This paper introduces a fuzzy-set based metric for invariant idempotent measures, establishing a bijection with fuzzy sets and demonstrating convergence properties of the associated Markov operator.

Contribution

It develops a new metrization of the space of idempotent measures via fuzzy sets, enabling analysis of invariant measures with stronger topology and convergence results.

Findings

Established a bijection between idempotent measures and fuzzy sets

Proved the Markov operator is a contraction under the new metric

Provided algorithms to visualize invariant measures as greyscale images

Abstract

We provide a new approach to the Hutchinson-Barnsley theory for idempotent measures first presented in N. Mazurenko, M. Zarichnyi, Invariant idempotent measures, Carpathian Math. Publ., 10 (2018), 1, 172--178. The main feature developed here is a metrization of the space of idempotent measures using the embedding of the space of idempotent measures to the space of fuzzy sets. The metric obtained induces a topology stronger than the canonical pointwise convergence topology. A key result is the existence of a bijection between idempotent measures and fuzzy sets and a conjugation between the Markov operator of an IFS on idempotent measures and the fuzzy fractal operator of the associated Fuzzy IFS. This allows to prove that the Markov operator for idempotent measures is a contraction w.r.t. the induced metric and, from this, to obtain a convergence theorem and algorithms that draw pictures of invariant measures as greyscale images.