Hyperbolic periodic orbits and optimal $1/t$ convergence of the Lax--Oleinik semigroup

Let $M$ be a closed manifold and $H:T^*M\to\mathbb{R}$ a $C^k$ ($k\geq2$) Tonelli Hamiltonian whose Aubry set consists of finitely many hyperbolic orbits, each of which is either a periodic orbit or a fixed point. We construct a $C^k$ critical subsolution that is quadratically strict outside the projected Aubry set. We then use this subsolution to establish an $O(1/t)$ rate of convergence for the associated Lax--Oleinik semigroup and the minimal action functions. This rate is optimal whenever at least one of these hyperbolic orbits is periodic.