DIMENSION OF LIMIT SETS IN VARIABLE CURVATURE

Daniel Pizarro, Felipe Riquelme, Sebastián Villarroel

correctmedium confidence

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

Audit review

The paper proves HD(Λ) = max{HD(Λ^r), HD(Λ^l)} in pinched negative curvature (Theorem 1.2) by a careful reduction to weak recurrence (Proposition 3.2) and a Falk–Stratmann dimension estimate (Proposition 3.3), together with the fact HD(Λ^r)=δ_Γ (Bishop–Jones) . The candidate solution incorrectly asserts that a fixed-radius family of shadows {O_o(γ·o,R)} covers the full limit set Λ, which generally fails (it covers radial points), and from this deduces HD(Λ)≤δ_Γ; this contradicts the paper’s d-group regime where HD(Λ)>δ_Γ (δ⋆>0) . It also conflates HD(Λ) with HD(Λ^r), which the paper explicitly separates. Hence the model’s proof is not correct.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper gives a clean and general dimension formula for limit sets in variable negative curvature, reconciling the roles of recurrent (radial) and linear-escape behavior. The approach is efficient and uses well-chosen tools (Bishop–Jones, a sharp geometric inclusion, and a Falk–Stratmann-type bound). I found no substantive gaps. Minor editorial improvements would further aid readability, but the core arguments are correct and of interest to the dynamics/geometry community.