Isolated elliptic fixed points for smooth Hamiltonians

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.