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

Seul Bee Lee, Stefano Marmi, Tanja I. Schindler

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.

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

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.
Paper Structure (1 section, 2 theorems, 22 equations)

This paper contains 1 section, 2 theorems, 22 equations.

Table of Contents

  1. Introduction and statement of result

Key Result

Proposition 1.1

The sum of the unstable manifold converges if and only if

Theorems & Definitions (6)

  • Proposition 1.1
  • Remark 1.2
  • Lemma 1.3
  • proof
  • proof : Proof of Proposition \ref{['prop: main prop']}
  • Example 1.4