On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians

Abstract

It is known that a Diophantine quasi-periodic torus with frequency ω∈Ωτd of a Cl Hamiltonian is effectively stable for a time T(r) that is polynomial on the inverse of the distance to the torus, that we denote by r, with exponent 1 + (l- 2) / (τ+ 1) . It is also known that a Diophantine quasi-periodic torus of a Gevrey Hamiltonian H∈ Gα,L is effectively stable for an exponentially long time on the inverse of the distance to the torus with exponent 1 / (α(1 + τ)) . In this note, we see that following the methods in [11] one can show the almost optimality of these exponents. We also show that, for a dense subset of non-resonant vectors, for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians, the naive lower bound T(r) ≥ Cr- 1 is optimal in terms of the exponent. © 2023, The Author(s).