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

TL;DR

This paper establishes an explicit arithmetical criterion ensuring the existence of unstable manifolds in Hamiltonian systems, correcting previous results and providing counterexamples when the criterion is not met.

Contribution

It introduces a precise arithmetical condition for the convergence of unstable manifolds in the MacKay renormalisation scheme, improving understanding of invariant tori breakup.

Findings

Explicit arithmetical condition for unstable manifold existence

Counterexamples where the condition fails and convergence does not occur

Correction of earlier theoretical results

Abstract

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.