The recurrence spectrum for dynamical systems beyond specification

Hiroki Takahasi

correcthigh confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:57 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves Theorem 1.1: for subshifts admitting a language decomposition with G having (W')-specification and h(Cp∪Cs)<h_top(Σ), every recurrence set R_{σ,ξ}(a,b) has full Hausdorff dimension, i.e., dim_H R_{σ,ξ}(a,b)=dim_H Σ, with the proof (via seed sets, Moran fractals, and a mass distribution bound) completed for the one-sided case and stated as analogous for the two-sided case (inequality (1.2) and its conclusion) . The model’s solution is circular (it cites the paper’s main result as a dependency instead of proving it) and it misstates the Hausdorff-dimension/entropy relation by ignoring the two-sided factor: the paper makes clear that dim_H Σ equals h_top(Σ) for one-sided shifts and 2 h_top(Σ) for two-sided shifts in the metric used .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The main theorem extends full-dimension recurrence phenomena well beyond classical specification systems, using a robust symbolic framework and Moran fractal constructions. The one-sided case is proved in detail; the two-sided case is claimed analogous. A brief sketch of the two-sided modifications would improve completeness, but overall the work appears correct, clear, and impactful within ergodic theory and dimension theory of dynamical systems.