Nearly-optimal effective stability estimates around Diophantine tori of Hölder Hamiltonians

Abstract

We prove that the solutions of Hölder-differentiable Hamiltonian systems, associated to initial conditions in a small ball of radius $\rho >0$ around a Lagrangian, $(\gamma ,\tau )-$Diophantine, quasi-periodic torus, are stable over a time $t^{\text{stab}}\simeq 1/(|\rho {|}^{1+\frac{\ell -1}{\tau +1}}|\ln \rho {|}^{\ell -1})$, where $\ell >2d+1,\ell \in \mathbb{R}$, is the regularity, and $d$ is the number of degrees of freedom. In the finitely differentiable case (for integer $\ell$), this result improves the previously known effective stability bounds around Diophantine tori. Moreover, by a previous work based on the Anosov-Katok construction, it is known that for any $\epsilon >0$ there exists a $C^{\ell }$-Hamiltonian, with $\ell \ge 3$, admitting a sequence of solutions starting at distance ${\rho }_n\to 0$ from a $(\gamma ,\tau )$-Diophantine torus that diffuse in a time of order $t_n^{\text{diff}}\simeq 1/(|{\rho }_n{|}^{1+\frac{\ell -1}{\tau +1}+\epsilon })$. Therefore the stability estimates that we show are optimal up to an arbitrarily small polynomial correction.