Products of Exact Dynamical Systems and Lorentzian Continued Fractions

Brandon G. Barreto-Rosa, Jean-Philippe Burelle, Anton Lukyanenko, Martha Richey

correctmedium confidence

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

Audit review

The paper proves that the product of systems satisfying Rokhlin’s Exactness Theorem is exact via a direct covering-by-rectangles argument that adapts Rokhlin’s proof to the product setting. The candidate solution gives a different, correct argument: first deducing each coordinate map is exact from the Rokhlin/Rényi hypotheses, then using reverse martingales and coordinatewise conditional expectations to show the product tail σ-algebra is trivial, hence the product is exact. The proofs are distinct but compatible.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The main product-exactness theorem under Rokhlin/Rényi hypotheses is correct and practically useful for applications to continued fractions. The proof is based on standard but deftly combined tools (product-measure coverings and Rokhlin’s argument). The paper would benefit from a couple of clarifications in the product proof to make the transfer of coordinate Rényi bounds to the product level entirely explicit. Overall, the contribution is technically sound and of solid interest to specialists in ergodic theory and dynamical aspects of continued fractions.