Topological Entropy Dimension on Subsets for Nonautonomous Dynamical Systems

The paper’s statements (Theorem 4.8 and Corollary 4.9) assert that when 0 < D(f_{1,∞},K,1) ≤ h_top(f_{1,∞},K) < ∞ (with K invariant) the classical and Pesin entropy dimensions both equal 1, and also claim D(f_{1,∞},K)=1 when 0 < h_top(f_{1,∞},K) < ∞. Those claims align with the candidate’s goals. However, two gaps appear. First, in the paper’s proof of Theorem 4.4(3) the step M(·,s,·) ≤ M(·,1,·)·M(·,s−1,·) is not justified by the displayed inequality over a single covering, since infimums of products are not controlled by products of infimums in the way used; this weakens the argument that D(f_{1,∞},K) ≤ 1 via the Pesin-side route (though D ≤ 1 still follows from D ≤ DK and DK ≤ 1 under finite h_top). Second, the paper does not explicitly prove that D(f_{1,∞},K,1)>0 from h_top(f_{1,∞},K)>0 (Lemma 4.6 only states D(f_{1,∞},K,1)=Ph(f_{1,∞},K) ≤ h_top(f_{1,∞},K)), yet Corollary 4.9(2) implicitly needs this positivity jump. The model’s solution is otherwise strong and fixes the s>1 and s<1 cases directly via covering-number bounds, but it assumes Ph(f_{1,∞},K)=h_top(f_{1,∞},K) for NDS to pass from h_top>0 to D(·,1)>0; that equality is not established in the paper (only Ph ≤ h_top is stated). Hence: the paper’s presentation lacks some key justifications and the model relies on an external equality not substantiated here. Overall, both are incomplete.