Mean Li–Yorke chaos and multifractal analysis on subshifts

The paper proves, for any irreducible one‑sided SFT (Σ_A, σ) and any ω∈Σ_A, that h_B(ML_ω)=0, h_P(ML_ω)=h(Σ_A), and h_B(LY_ω)=h(Σ_A). These are stated as Theorem 1.2 and Theorem 1.1, respectively, and are established via a general multifractal framework plus a bi‑Lipschitz transfer argument to pass from almost‑everywhere ω to every ω . The model’s results match the statements, but its proof contains key errors. In part (a), the central combinatorial step incorrectly deduces a small Hamming radius from a bound on the number of bad length‑ℓ windows; the correct direction yields |S|≤bℓ, not |S|≲b/ℓ, so the subsequent entropy estimate collapses. In parts (b)–(c), the construction relies on choosing free blocks whose first symbol differs from the corresponding symbol of ω, which is generally impossible for loops based at a fixed vertex in an SFT, and the periodic/aperiodic residue‑class issue is not handled. The paper’s proofs avoid these pitfalls by working in the generalized multifractal setting, using packing‑entropy tools and explicit bi‑Lipschitz maps ϕ_{ω,ω′} (and their block-structured versions ϕ^M_{ω,ω′}) to transfer results across base points and across cyclic components of Σ_A .