A remark on omega limit sets for non-expansive dynamics

The paper proves that for a C^1, forward-complete flow on R^n, nonexpansive w.r.t. a strictly convex norm and with at least one bounded trajectory, (i) all trajectories are bounded and (ii) every ω-limit set coincides (up to translation) with an ω-limit set of a single conserved linear system ẋ=Bx and is a torus. It does so by showing: isometry on the global attractor A (via Dini’s theorem), strict convexity implies convexity and backward invariance of A, existence of an equilibrium, Mankiewicz extension to linear isometries on span(A), a one-parameter group e^{tB}, and torus structure from the closure of e^{tB} (compact abelian Lie group). The candidate solution follows the same structure and lemmas, with minor presentational differences (e.g., a quick argument for boundedness via a bounded-tube estimate around a bounded trajectory). Key steps match the paper’s Lemma/Corollary chain and Theorem 1. See the paper’s theorem statement and lemmas on isometry, convexity/backward invariance of A, the Mankiewicz extension and the torus conclusion .