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Isolated elliptic fixed points for smooth Hamiltonians

Bassam Fayad, Maria Saprykina

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 à la} Anosov-Katok.

Isolated elliptic fixed points for smooth Hamiltonians

Abstract

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

Paper Structure

This paper contains 8 sections, 10 theorems, 47 equations, 1 figure.

Table of Contents

  1. Notations
  2. Orbits accumulating the axis and diffusive orbits
  3. Two degrees of freedom.
  4. Four degrees of freedom and higher.
  5. Three degrees of freedom
  6. Proof for the case $d=2$
  7. Proofs for the case $d=4$
  8. Proofs for the case $d=3$.

Key Result

Theorem A

For any ${\omega}_0 \in {\mathbb R}^d, d \geq 4$, there exists $H \in C^{\infty} (\mathbb R^{2d})$ as in $(*)$, such that $\Phi^t_H$ has no invariant torus of dimension $d$. More precisely, the manifolds $\{r_i=0\}$ for $i\leq d$, are foliated by invariant tori of dimension $\leq d-1$ and all other

Figures (1)

  • Figure 1: The diffusion lines are unbounded in the case $\omega_1 \omega_2<0$(the figure to the left) and bounded in the case $\omega_1 \omega_2>0$.

Theorems & Definitions (27)

  • Definition 1
  • Theorem A
  • Theorem B
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Remark 5
  • Proposition 1
  • ...and 17 more