STERNBERG THEOREMS FOR COUPLED MAP LATTICES

The paper proves a Sternberg-type linearization for coupled map lattices with spatial decay by (i) constructing a finite-order normal form using Γ-spectrum/Sylvester-operator arguments and (ii) completing the conjugacy via a contraction mapping on an iterated/rescaled map F^m (operator G_m) under quantitative gap conditions derived from α, β and r0; see Theorem 1, Lemma 3 and the contraction estimate (17) as well as the normal-form step in Section 7 (Theorem 6) . The candidate’s solution also achieves linearization in the same setting but via a different proof: it solves the homogeneous cohomological equations up to order r0 using an explicit Neumann-series right inverse for L_M (enabled by α′>(β′)^s) and then solves the full conjugacy by a direct contraction on a weighted C^r_Γ space of maps vanishing to order r0+1. Both approaches rely on the Banach-algebra structure of L_Γ and smallness of B to ensure M, M^{-1} ∈ L_Γ (Proposition 3 and Proposition 4) . The assumptions align with the paper’s (H1)–(H2), and the model’s stronger modulus gap α′>(β′)^s implies the finite non-resonance used in the paper. Hence both are correct, but the proofs differ in the way the homological equations are solved and how the final contraction is set up.