Perturbation estimates of weak KAM solutions and minimal invariant sets for nearly integrable Hamiltonian systems

Abstract:

For nearly integrable and Tonelli system \[ H_{\epsilon }=H_0(p)+\epsilon H_1(q,p,t). \quad (q,p,t)\in \mathbb {T}^n\times \mathbb {R}^n\times \mathbb {T},\] we give the perturbation estimates of weak KAM solution $u_{\epsilon }$ with respect to parameter $\epsilon$ and prove the stability of the Mather set $\tilde {\mathcal {M}}_\epsilon$, Aubry set $\tilde {\mathcal {A}}_\epsilon$, Mañé set $\tilde {\mathcal {N}}_\epsilon$ and even the backward (forward) calibrated curves under the perturbation.