SELF-SIMILARITY AND SPECTRAL THEORY: ON THE SPECTRUM OF SUBSTITUTIONS

Alexander I. Bufetov, Boris Solomyak

correctmedium confidence

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

Audit review

The uploaded paper (a 2021 survey by Bufetov–Solomyak) states and sketches the criterion that if the top Lyapunov exponent χζ of the spectral cocycle satisfies χζ < (1/2) log θ (under irreducibility of the characteristic polynomial), then the substitution Z-action has pure singular spectrum; this is Theorem 4.10 in the survey, derived via Corollary 4.8/4.9 and Host’s equidistribution on the diagonal of the torus, and relying on the spectral cocycle and matrix Riesz-product machinery . The candidate solution reproduces this same line of reasoning. Apart from a minor imprecision in the stated local-dimension bound (the survey gives a precise formula for flows), the conclusion and proof strategy agree.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

This is a careful synthesis of known results: the spectral cocycle framework, dimension estimates for spectral measures, and an equidistribution step yield a clean sufficient condition for pure singular spectrum in substitution Z-actions. The contribution as presented in the survey is primarily expository, consolidating results (with a brief proof sketch) and directing the reader to full proofs elsewhere. Clarity is high overall, but the flow-to-map transfer and the precise role of irreducibility could be emphasized further, and the dimension statements could be synchronized across contexts for maximal transparency.