UNIFORM APPROXIMATION PROBLEMS OF EXPANDING MARKOV MAPS

Yubin He, Lingmin Liao

correcthigh confidence

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

Audit review

The paper proves the zero–one law and the Hausdorff-dimension formulas for U_κ(x) and B_κ(x) via hitting-time/local-dimension relations and multifractal analysis, with the main technical novelty being the upper bound for dim_H U_κ(x) (Lemma 5.7). The candidate solution follows the same overall architecture (Gibbs coding, hitting times R(x,y), μ_max ∼ λ, multifractal spectrum), with minor imprecision (it states U_κ(x) ⇔ R(x,y) ≤ 1/κ without the small O+(x) caveat) but no substantive contradiction. Items (1)–(4) match Theorem 1.2 in the paper, and the key ingredients (Lemma 4.1, Corollary 4.4, Lemmas 5.1, 5.5–5.7) are used in spirit. The paper’s statements and proofs are correct, and the model’s outline is essentially correct and aligned with them. See Theorem 1.2 for the statements and Sections 4–5 for the proofs and lemmas cited .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper presents a rigorous and essentially optimal characterization of uniform approximation sets for expanding Markov maps under Gibbs measures, with a clear threshold and multifractal law. The key technical advancement is the upper bound for the Hausdorff dimension of U\_κ(x) below the α\_max threshold, built from careful covering/mixing lemmas. Minor exposition tweaks—for example, highlighting the O+(x) caveat in the U\_κ–R inclusions and adding a brief road map in Section 5—would further enhance clarity.