OPEN DYNAMICAL SYSTEMS WITH A MOVING HOLE

The paper’s Theorem 1.1 and its proof are correct and complete: it establishes dim_H K_ω = lower box = lim inf (log |Σ^ω_n|)/(n log b) and dim_P K_ω = upper box = lim sup (log |Σ^ω_n|)/(n log b), and it identifies |Σ^ω_{m-1+n}| with the entry-sum norm of A_{ω_0}⋯A_{ω_{n-1}} via a precise counting lemma (Lemma 2.3) . The candidate solution reaches the same conclusions but uses two incorrect steps: (i) it asserts N(K_ω,b^{-n}) = |Σ^ω_n| exactly, which need not hold because coverings of K_ω at scale b^{-n} need not cover the entire level-n cylinders and may combine adjacent parts near shared endpoints; the paper instead proves comparability indirectly to get the correct box-dimension limits . (ii) it claims H^s_{b^{-n}}(K_ω) = |Σ^ω_n| b^{-ns}, conflating coverings of the whole intervals I_d with coverings of the smaller sets K_ω ∩ I_d; the paper uses a careful covering argument (its (2.11)–(2.13)) and a lemma controlling |Σ^ω_{k+n}(c)| to obtain the correct Hausdorff lower bound without such an equality . Hence the model’s conclusions are right but the proof is flawed, while the paper’s argument is sound.