Diffusion times and stability exponents for nearly integrable analytic systems

Abstract

For a positive integer n and R>0, we set$B_R^n = \left\{ {x \in \mathbb{R}^n |\left\| x \right\|_\infty< R} \right\}$. Given R>1 and n≥4 we construct a sequence of analytic perturbations (H _j ) of the completely integrable Hamiltonian$h\left( r \right) = \tfrac{1}{2}r_1^2 + ...\tfrac{1}{2}r_{n - 1}^2 + r_n$ on$\mathbb{T}^n \times B_R^n$, with unstable orbits for which we can estimate the time of drift in the action space. These functions H _j are analytic on a fixed complex neighborhood V of$\mathbb{T}^n \times B_R^n$, and setting$\varepsilon _j : = \left\| {h - H_j } \right\|_{C^0 (V)}$ the time of drift of these orbits is smaller than (C(1/ɛ _j )^1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface, the result is therefore almost optimal since the stability exponent for such orbits is 1/2(n−2). An analogous result for Hamiltonian diffeomorphisms is also proved. Two main ingredients are used in order to deal with the analytic setting: a version of Sternberg's conjugacy theorem in a neighborhood of a normally hyperbolic manifold in a symplectic system, for which we give a complete (and seemingly new) proof; and Easton windowing method that allow us to approximately localize the wandering orbits and estimate their speed of drift.