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

Abstract

AbstractIt is known that a Diophantine quasi-periodic torus with frequency $$\omega \in \Omega _{\tau }^d$$ ω ∈ Ω τ d of a $$C^{l}$$ C l 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 $$1+(l-2)/(\tau +1)$$ 1 + ( l - 2 ) / ( τ + 1 ) . It is also known that a Diophantine quasi-periodic torus of a Gevrey Hamiltonian $$H\in G^{\alpha ,L}$$ H ∈ G α , L is effectively stable for an exponentially long time on the inverse of the distance to the torus with exponent $$1/(\alpha (1+\tau ))$$ 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 $$T(r)\ge Cr^{-1}$$ T ( r ) ≥ C r - 1 is optimal in terms of the exponent.