Isolated elliptic fixed points for smooth Hamiltonians

TL;DR

This paper constructs smooth Hamiltonian systems in higher dimensions with isolated elliptic fixed points that are not surrounded by invariant tori, challenging typical KAM theory expectations.

Contribution

It introduces a novel method to create Hamiltonians with isolated elliptic fixed points lacking invariant tori, extending previous constructions to higher dimensions.

Findings

Existence of Hamiltonians with isolated elliptic fixed points in any dimension ≥3.

Construction of Hamiltonians without invariant tori of certain dimensions in higher dimensions.

Application of a conjugation scheme inspired by Anosov-Katok methods.

Abstract

We construct on $\R^{2d}$, for any $d \geq 3$, smooth Hamiltonians having an elliptic equilibrium with an arbitrary frequency, that is not accumulated by a positive measure set of invariant tori. For $d\geq 4$, the Hamiltonians we construct have not any invariant torus of dimension $d$. Our examples are obtained by a version of the successive conjugation scheme {\it \`a la} Anosov-Katok.