2212.01987
On the fractal dimensions for deterministic and random substitution graph systems
Ziyu Li
incompletemedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
Deterministic part: the paper’s definitions and proofs leading to dim_B(Ξ)=dim_H(Ξ)=log ρ(M)/log ρ_min(D) are essentially sound and align with the model (via uniform shortest paths, a minimization over products of D∈D, and PF-type growth for Δ(Ξ_t)) . Random part: the paper equates E(log||·||) with log||E(·)|| to identify Lyapunov exponents (Theorem 5.9), which is invalid; its almost-sure claims then lack justification . The model treats both cases correctly using Perron–Frobenius, the nonlinear Collatz–Wielandt theory, and Kingman/Furstenberg–Kesten for random products.
Referee report (LaTeX)
\textbf{Recommendation:} major revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The deterministic theory is well organized and, using uniform shortest paths and spectral estimates, convincingly yields Δ(Ξ\_t) ≍ ρ\_min(D)\^t and dim\_B = dim\_H = log ρ(M)/log ρ\_min(D). However, the stochastic theory hinges on an incorrect use of expectations and logarithms to deduce Lyapunov exponents; this undermines the correctness of the random claims as written. The paper would merit publication after replacing the random proofs with rigorous Kingman/Furstenberg–Kesten arguments and clarifying a few technical assumptions in the deterministic covering arguments.