Minimal orbits close to periodic frequencies (Q1281749)

Summary: Let \({\mathcal L}(Q,\dot Q)={1\over 2}|\dot Q|^2+ h(Q,\dot Q)\) 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.