On limit sets of monotone maps on regular curves

Aymen Daghar, Habib Marzougui

correctmedium confidence

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

Audit review

The paper proves exactly the three claims at issue: (1) every ω-limit set of a monotone self-map on a regular curve is minimal (Theorem 3.1), with a two-case boundary-counting argument using finite-boundary neighborhoods; (2) a limsup minimality principle for null families under monotone maps on regular curves (Theorem 5.5), from which (3) the α-limit of any negative orbit is minimal, and αf(x) is minimal for x∈X∞\P(f) (Theorem 5.7). The candidate solution reproduces the same architecture: the finite-boundary/null-family device, the two-case proof for ω-sets, and a limsup argument for α-sets. Minor gaps (e.g., explicitly stating that {f−n(x)} are pairwise disjoint for nonperiodic x) do not affect correctness. See the abstract and the statements/proofs of Theorems 3.1, 5.5, 5.7, and Proposition 2.2 in the paper for the matching results and methods .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript cleanly extends known results about minimality of limit sets to monotone maps on regular curves, employing a coherent toolkit (finite-boundary neighborhoods, null families, and hyperspace arguments). The main theorems are correct and the proofs are largely clear. A few minor steps could be stated explicitly (e.g., disjointness of {f\^{-n}(x)} for nonperiodic x), and indexing could be streamlined for readability. Overall, the paper is a solid contribution for specialists in continuum dynamics.