Prevalence of exponential stability among nearly integrable Hamiltonian systems

Abstract

In the 1970s, 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. Recently, Guzzo has given examples of exponentially stable integrable Hamiltonians that are non-steep but satisfy a weak condition of transversality which involves only the affine subspaces spanned by integer vectors. We generalize this notion for an arbitrary integrable Hamiltonian and prove Nekhorochev's estimates in this setting. The point in this refinement lies in the fact that it allows one to exhibit a generic class of real analytic integrable Hamiltonians which are exponentially stable with fixed exponents. Genericity is proved in the sense of measure since we exhibit a prevalent set of integrable Hamiltonians which satisfy the latter property. This is obtained by an application of a quantitative Sard theorem given by Yomdin.