Some remarks on the optimality of the Bruno-R{\"u}ssmann condition

TL;DR

This paper demonstrates that the Bruno-Rüssmann condition is the optimal criterion for preserving quasi-periodic invariant curves in analytic twist maps and extends similar results to certain Hamiltonian flows, establishing the condition's fundamental importance.

Contribution

It proves the optimality of the Bruno-Rüssmann condition for analytic invariant curve preservation and extends the analysis to specific Hamiltonian flows, improving previous results.

Findings

Bruno-Rüssmann condition is optimal for analytic invariant curve preservation.

Established a similar optimality result for 2-degree-of-freedom Hamiltonian flows.

Provided a weaker but improved result for higher-dimensional Hamiltonian systems.

Abstract

We prove that the Bruno-R{\"u}ssmann condition is optimal for the analytic preservation of a quasi-periodic invariant curve for an analytic twist map. The proof is based on Yoccoz's corresponding result for analytic circle diffeomorphisms and the uniqueness of invariant curves with a given irrational rotation number. We also prove a similar result for analytic Tonelli Hamiltonian flow with n = 2 degrees of freedom; for n $\ge$ 3 we only obtain a weaker result which recovers and slightly improves a theorem of Bessi.