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 ρ>0 around a Lagrangian, (γ,τ)-Diophantine, quasi-periodic torus, are stable over a time tstab≃1/(|ρ|1+ℓ-1τ+1|lnρ|ℓ-1), where ℓ>2d+1,ℓ∈R, is the regularity, and d is the number of degrees of freedom. In the finitely differentiable case (for integer ℓ), 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 ε>0 there exists a Cℓ-Hamiltonian, with ℓ≥3, admitting a sequence of solutions starting at distance ρn→0 from a (γ,τ)-Diophantine torus that diffuse in a time of order tndiff≃1/(|ρn|1+ℓ-1τ+1+ε). Therefore, for ℓ>2d+1, the stability estimates that we show are optimal up to an arbitrarily small polynomial correction.