Turbulent Closed Relations

Judy Kennedy, Christopher Mouron, Van Nall

correcthigh confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves the finite-case equivalence via Theorem 3.1 (CR-turbulence iff a shift iterate is separated turbulent) and prior results (Theorem 4.1 of [3]) to link loops, positive entropy, and uncountability, culminating in Theorem 4.2 . The model independently reproves the same five-way equivalence for finite relations by recasting X_F^+ as a one-step SFT, using cylinder pasting for (1)⇒(2), a standard horseshoe/cover-entropy argument for (2)⇒(4), spectral-radius reasoning for (4)⇒(3), and a combinatorial analysis for (3)⇔(5). The reasoning is correct and complete for the stated five items, though the paper additionally establishes (6)–(7), which the model does not address.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript generalizes turbulence to closed relations and provides a crisp equivalence theorem in the finite case, along with instructive examples and counterexamples. The finite-case equivalence is solid and well-motivated; however, a brief self-contained proof of certain standard implications (notably (2)⇒(4) and (4)⇒(3) in the finite case) would improve accessibility. The exposition is clear, and the results should interest researchers in topological dynamics and continuum theory.