Some questions concerning attractors for non-autonomous dynamical systems

The paper’s Theorem 3.11 states exactly the two claims: (1) any compact invariant set A whose points lie in the small (u,v)-block must be contained in M_ε, and (2) if A is a Lyapunov attractor relative to M_ε, then A is a Lyapunov attractor relative to Q×ℝ^d. The paper proves (1) immediately from Corollary 3.9, which asserts that any bi-infinite solution staying in the Δ-block lies on M_ε, and it sketches (2) by conjugating the dynamics on M_ε to a reduced local flow on Q×U and invoking Palmer’s Proposition 3 to lift Lyapunov attraction to the ambient phase space (Theorem 3.11 and its proof sketch; see and ). The candidate solution reaches the same conclusions by appealing to standard normally hyperbolic invariant manifold (NHIM) structure—local product neighborhood, strong-stable/unstable laminations, and local maximality—to get (1), and by saturating neighborhoods along strong-stable leaves to get (2). This is a different but sound route provided the usual NHIM hypotheses hold in the skew-product local-flow setting. Minor mismatches: the candidate adds a “projection condition” that is not part of the paper’s Lyapunov-attractor definition (Definition 2.1; ), and it assumes NHIM laminations/local maximality rather than using the paper’s explicit Corollary 3.9 lemma-based route (Lemma 3.8 and Corollary 3.9; , ). Overall, both are correct; the proofs are different.