In the 70’s, Nekhorochev proved that for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long time with respect to the size of the perturbation, provided that the unperturbed Hamiltonian satisfies some generic transversality condition known as steepness. Using theorems of real subanalytic geometry, we derive a geometric criterion for steepness: a numerical function $h$ which is real analytic around a...

We present a proof of Herman’s Last Geometric Theorem asserting that if $F$ is a smooth diffeomorphism of the annulus having the intersection property, then any given $F$-invariant smooth curve on which the rotation number of $F$ is Diophantine is accumulated by a positive measure set of smooth invariant curves on which $F$ is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable...