Sharp convergence rates for the vanishing discount problem with hyperbolic Aubry sets

Let $H\in C^2(T^{\ast }M)$ be a Tonelli Hamiltonian on a closed connected manifold and let $u_{\lambda }$ solve $$\lambda u_{\lambda }+H(x,D{u}_{\lambda })=c(H)\text{in }M\mathrm{.}$$ We study the convergence rate of $u_{\lambda }$ to the selected critical solution $u_0$. Assume that the lifted Aubry set is a finite union $\tilde{A}={\mathrm{\Gamma }}_1\bigsqcup \cdots \bigsqcup {\mathrm{\Gamma }}_N$, where each ${\mathrm{\Gamma }}_i$ is either a hyperbolic equilibrium or a periodic orbit hyperbolic in the critical energy level. We prove $$-C\lambda \le u_{\lambda }-u_0\le C\lambda \mathrm{\mid }\log \lambda \mathrm{\mid }\mathrm{.}$$ Let ${\mu }_i$ be the projected Mather measure associated with ${\mathrm{\Gamma }}_i$. If $${\int }_M{u}_0\mathrm{d}{\mu }_i=0\text{for every }i,$$ then $$\mathrm{\parallel }{u}_{\lambda }-u_0{\mathrm{\parallel }}_{\mathrm{\infty }}\le C\lambda \mathrm{.}$$ In particular, if the lifted Aubry set consists of a single hyperbolic equilibrium or a single hyperbolic periodic orbit, the convergence rate is $O(\lambda )$. We give examples showing that both convergence rates $O(\lambda )$ and $O(\lambda \mathrm{\mid }\log \lambda \mathrm{\mid })$ are optimal. Without hyperbolicity, finite-order degenerate examples give lower bounds of order ${\lambda }^{1\mathrm{/}(2r-1)}$ with $r\ge 2$. We also construct infinite-order degenerate examples with arbitrarily slow convergence. Taken together, these results provide, to our knowledge, the first systematic quantitative theory for the vanishing discount problem in the Tonelli setting.