Minimal orbits close to periodic frequencies

Abstract.

Let ${\cal L}(Q,\dot Q)={1\over2}\vert\dot{Q}\vert^2+h(Q,\dot Q)$ 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 ${\cal L}$ 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 ${\cal L}$. In this paper we try to characterize those action-minimizing orbits whose mean frequency is close to periodic.