Hamiltonian stability and subanalytic geometry

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 compact set in R is steep if and only if its restriction to any proper affine subspace of R admits only isolated critical points. Moreover, we obtain sharp results of exponential stability under the previous assumption. We also state a necessary condition for exponential stability, which is close to steepness. Finally, we give methods to compute the steepness indices for an arbitrary steep function.