Examples of nearly integrable systems on ${\mathbb{A}}^3$ with asymptotically dense projected orbits

Abstract

Given an integer $\kappa \ge 2$, we introduce a class of nearly integrable systems on ${\mathbb{A}}^3$, of the form $$H_n(\theta ,r)=\frac{1}{2}\Vert r{\Vert }^2+\frac{1}{n}U({\theta }_2,{\theta }_3)+f_n(\theta ,r)$$ where $U\in C^{\kappa }({\mathbb{T}}^2)$ is a generic potential function and $f_n$ a $C^{\kappa -1}$ additional perturbation such that $\Vert f_n{\Vert }_{C^{\kappa -1}({\mathbb{A}}^3)}\le \frac{1}{n}$, so that $H_n$ is a perturbation of the completely integrable system $h(r)=\frac{1}{2}\Vert r{\Vert }^2$. Let $\Pi :{\mathbb{A}}^3\to {\mathbb{R}}^3$ be the canonical projection. We prove that for each $\delta >0$, there exists $n_0$ such that for $n\ge n_0$, the system $H_n$ admits an orbit ${\Gamma }_n$ at energy $\frac{1}{2}$ whose projection $\Pi ({\Gamma }_n)$ is $\delta$-dense in $\Pi (H_n^{-1}(\frac{1}{2}))$, in the sense that the $\delta$-neighborhood of $\Pi ({\Gamma }_n)$ in ${\mathbb{R}}^3$ covers $\Pi (H_n^{-1}(\frac{1}{2}))$.

24 pages