Chains of compact cylinders for cusp-generic nearly integrable convex systems on $\mathbb{A}^3$

Abstract

This paper is the first of a series of three dedicated to a proof of the Arnold diffusion conjecture for perturbations of {convex} integrable Hamiltonian systems on $\mathbb{A}^3=\mathbb{T}^3\times \mathbb{R}^3$. We consider systems of the form $H(\theta,r)=h(r)+f(\theta,r)$, where $h$ is a $C^\kappa$ strictly convex and superlinear function on $\mathbb{R}^3$ and $f\in C^\kappa(\mathbb{A}^3)$, $\kappa\geq2$. Given $e>\textrm{{Min}}\,h$ and a finite family of arbitrary open sets $O_i$ in $\mathbb{R}^3$ intersecting $h^{-1}(e)$, a diffusion orbit associated with these data is an orbit of $H$ which intersects each open set $\widehat O_i=\mathbb{T}^3\times O_i\subset\mathbb{A}^3$. The first main result of this paper (Theorem I) states the existence (under cusp-generic conditions on $f$ in Mather's terminology) of "chains of compact and normally hyperbolic invariant $3$-dimensional cylinders" intersecting each $\widehat O_i$. Diffusion orbits drifting along these chains are then proved to exist in subsequent papers. The second main result (Theorem II) consists in a precise description of the hyperbolic features of classical systems (sum of a quadratic kinetic energy and a potential) on $\mathbb{A}^2=\mathbb{T}^2\times\mathbb{R}^2$, which is a crucial step to prove Theorem I.