A NOTE ON THE DIFFERENTIABILITY OF PALMER’S TOPOLOGICAL EQUIVALENCE FOR DISCRETE SYSTEMS

The paper proves Z+-topological equivalence under (d0)–(d5) and upgrades to C1-topological equivalence under (d6)–(d7) by (i) representing G(k,·)=H(k,·)^{-1} via the Green operator, (ii) proving G is C1 (Lemma 3.1), and (iii) invoking Plastock’s global inversion corollary to conclude G is a global C1-diffeomorphism, hence H is C1 with DH=(DG∘H)^{-1} (Theorem 3.2). All key steps are present and correctly cited in the paper. By contrast, the model reproduces the Lyapunov–Perron construction and correctly shows L=G is C1, but then asserts H is C1 merely as the inverse of a C1 bijection without proving the necessary nonsingularity of DL everywhere or appealing to a global inverse theorem; this leaves a critical gap in the C1 conclusion for H.