Nearly-Optimal Effective Stability Estimates Around Diophantine Tori of Hölder Hamiltonians

By making use of adapted analytic smoothing techniques, we prove that the solutions of Hölder-differentiable Hamiltonian systems, associated to initial conditions in a small ball of radius \(\rho >0\) $\rho >0$ around a Lagrangian, \((\gamma ,\tau )\) $(\gamma ,\tau )$ -Diophantine, quasi-periodic torus, are stable over a time \(t^{\text {stab}}\simeq 1/(|\rho |^{1+\frac{\ell -1}{\tau +1}}|\ln \rho |^{\ell -1})\) $t^{\text{stab}}\simeq 1/{(|\rho |}^{1+\frac{\ell -1}{\tau +1}}{|ln\rho |}^{\ell -1})$ , where \(\ell >2d+1, \ell \in \mathbb {R}\) $\ell >2d+1,\ell \in \mathbb{R}$ , is the regularity, and d is the number of degrees of freedom. In the finitely differentiable case (for integer \(\ell \) $\ell$ ), 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 \(\varepsilon >0\) $\epsilon >0$ there exists a \(C^\ell \) $C^{\ell }$ -Hamiltonian, with \( \ell \ge 3\) $\ell \ge 3$ , admitting a sequence of solutions starting at distance \(\rho _n \rightarrow 0\) ${\rho }_n\to 0$ from a \((\gamma ,\tau )\) $(\gamma ,\tau )$ -Diophantine torus that diffuse in a time of order \(t^{\text {diff}}_n\simeq 1/(|\rho _n|^{1+\frac{\ell -1}{\tau +1}+\varepsilon })\) $t_n^{\text{diff}}\simeq 1/(|{\rho }_n{|}^{1+\frac{\ell -1}{\tau +1}+\epsilon })$ . Therefore, for \(\ell >2d+1\) $\ell >2d+1$ , the stability estimates that we show are optimal up to an arbitrarily small polynomial correction.