Invariant measures for place dependent idempotent iterated function systems

TL;DR

This paper characterizes invariant measures in place-dependent idempotent iterated function systems using dynamical systems concepts, offering new formulas for attractors in fuzzy systems.

Contribution

It provides a complete characterization of densities of invariant idempotent probabilities and introduces an alternative formula for attractors in fuzzy iterated function systems.

Findings

Complete characterization of densities of invariant idempotent probabilities.

Application of dynamical systems tools like Mañé potential and Aubry set.

New formula for attractors in fuzzy iterated function systems.

Abstract

We study the set of invariant idempotent probabilities for place dependent idempotent iterated function systems defined in compact metric spaces. Using well-known ideas from dynamical systems, such as the Ma\~{n}\'{e} potential and the Aubry set, we provide a complete characterization of the densities of such idempotent probabilities. As an application, we provide an alternative formula for the attractor of a class of fuzzy iterated function systems.