Representation of Global Viscosity Solutions for Tonelli Hamiltonians

We consider the Lax-Oleinik operator $\mathcal{T}$ associated with the non-stationary Hamilton-Jacobi equation $\partial_tu + H(t,x,\partial_xu) = \alpha_0$ for a Tonelli Hamiltonian $H$ and its \Mane critical value $\alpha_0$. It is known from the work of A. Fathi and J.N. Mather \cite{MR1792479} that the convergence of this semigroup fails in the non-autonomous framework. In this context, we study the action of $\mathcal{T}$ on its non-wandering set $\Omega(\mathcal{T})$. First, we show that $\mathcal{T}$ acts as an isometry on this set, and then we characterize $\Omega(\mathcal{T})$ as the set of global viscosity solutions of the Hamilton-Jacobi equation, i.e. solutions that are defined for all real times. Next, we introduce a generalized Peierls barrier $\underline{k}$ and a set of generalized static classes $\underline{\mathbb{M}}$ within the Mather set. Using these, we represent elements $u$ of $\Omega(\mathcal{T})$ as \begin{equation*} u(x) = \inf_{y \in \underline{\mathbb{M}}} \{ u(y) + \underline{k}(y,x) \} \end{equation*} We apply this representation formula to prove Fathi's convergence theorem for autonomous systems and provide a representation formula for $n$-periodic viscosity solutions. Additionally, we establish that the dynamics of non-wandering viscosity solutions are governed by the Lagrangian flow on the Mather set. Specifically, we show that if the Mather set consists solely of $N$-periodic orbits for some integer $N$, then all non-wandering viscosity solutions are $N$-periodic. Furthermore, we show that if the restriction of the Lagrangian flow to the Mather set is uniformly recurrent for a time sequence $p_n$, then all non-wandering viscosity solutions are uniformly recurrent for the same time sequence $p_n$.