A convergence criterion for the unstable manifolds of the MacKay approximate renormalisation

We give an explicit arithmetical condition which guarantees the existence of the unstable manifold of the MacKay approximate renormalisation scheme for the breakup of invariant tori in one and a half degrees of freedom Hamiltonian systems, correcting earlier results. Furthermore, when our condition is violated, we give an example of points on which the unstable manifold does not converge. (c) 2022 Elsevier B.V. All rights reserved.