Minimal orbits close to periodic frequencies

Abstract

Let $\mathcal{L}(Q,\dot{Q})=\frac{1}{2}|\dot{Q}{|}^2+h(Q,\dot{Q})$ with h analytic of small norm. The problem of Arnold's diffusion consists in finding conditions on h which guarantee the existence of orbits Q of $\mathcal{L}$ with $\dot{Q}$ connecting two arbitrary points of frequency space. Recently, J. N. Mather has found a sufficient condition for Arnold's diffusion; this condition is not read on h itself, but on the set of all action-minimizing orbits of $\mathcal{L}$. In this paper we try to characterize those action-minimizing orbits whose mean frequency is close to periodic.