Chains of compact cylinders for cusp-generic nearly integrable convex systems on \mathbbA³

2016/02/07 by Marco, Jean-Pierre
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1602.02399

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 \mathbbA³=\mathbbT³× ℝ³. We consider systems of the form H(θ,r)=h(r)+f(θ,r), where h is a C^κ strictly convex and superlinear function on ℝ³ and f∈ C^κ(\mathbbA³), κ≥2. Given e>\textrmMin h and a finite family of arbitrary open sets O_i in ℝ³ intersecting h^-1(e), a diffusion orbit associated with these data is an orbit of H which intersects each open set \widehat O_i=\mathbbT³× O_i⊂\mathbbA³. 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 \mathbbA²=\mathbbT²×ℝ², which is a crucial step to prove Theorem I.

Related

• Arnold diffusion for cusp-generic nearly integrable convex systems on \mathbb A³
• Diffusion along chains of normally hyperbolic cylinders
• Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders
• Arnold diffusion in nearly integrable Hamiltonian systems
• Uniform hyperbolicity of invariant cylinder