Fuzzy-set approach to invariant idempotent measures

Rudnei D. da Cunha, Elismar R. Oliveira, Filip Strobin

correcthigh confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves conjugacy Θ∘MS = ZS∘Θ via densities (Lemma 5.5) and a scale θ (Theorem 5.4), then transfers contractivity from the fuzzy Hutchinson operator ZS (Theorem 4.7) to MS to get uniqueness, Picard convergence, a Banach-Lipschitz bound, and support equals the set attractor AS (Theorems 6.1–6.2). The candidate solution reproduces these steps: it derives the density formula, establishes the same conjugacy, uses α-cuts and Hausdorff estimates to show ZS is Matkowski contractive, then lifts this to MS via Θ to reach the same fixed-point and convergence conclusions, including the Lipschitz case and support equality. No substantive discrepancies were found; the approaches are essentially the same.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper introduces a clear and effective conjugacy between the idempotent Markov operator and a fuzzy Hutchinson operator, inducing a natural metric on idempotent measures and providing a contraction-based proof of invariant idempotent measures with algorithmic implications. The narrative is correct and well grounded in the literature. Minor clarifications would further strengthen readability and self-containment.