Asymptoticity, Automorphism Groups and Strong Orbit Equivalence

Haritha Cheriyath, Sebastián Donoso

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, within any given strong orbit equivalence (SOE) class of a minimal Cantor system, the existence of subshifts with exactly k (any finite k), countably infinite, and uncountably many asymptotic components via explicit S-adic constructions for the finite and countable cases (Theorem 4.10 and Theorem 5.3), and an entropy obstruction for the uncountable case (Theorem 1.1 + Corollary 3.1) . The candidate solution follows the same structure and uses the same key ingredients (SOE via Bratteli/dimension-group invariants, S-adic models, and the zero-entropy constraint for countably many components), reaching the same conclusions for the three cases. The only discrepancies are minor overstatements (claiming realizations for all κ ≤ continuum and “arbitrarily prescribed finite entropy” for subshifts), which are not needed for the stated (i)–(iii) and are not asserted by the paper’s theorems.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper gives constructive and conceptually clear results on controlling asymptotic components within SOE classes, coupling explicit S-adic designs (finite/countable cases) with a clean entropy obstruction (uncountable case). The approach is technically sound and well-aligned with the Bratteli–Vershik framework. Minor clarifications about the exact uncountable cardinality and a more explicit checklist for conjugacy conditions would further improve readability.