Singular dynamics for discrete weak K.A.M. solutions of exact twist maps

For any exact twist map $f$ and any cohomology class $c\in\mathbb{R}$, let $u_c$ be any associated discrete weak K.A.M. solution, and we introduce an inherent Lipschitz dynamics $\Sigma_+$ given by the discrete forward Lax-Oleinik semigroup. We investigate several properties of $\Sigma_+$ and show that the non-differentiable points of $u_c$ are globally propagated and forward invariant by $\Sigma_+$. In particular, such propagating dynamics possesses the same rotation number $\alpha'(c)$ as the associated Aubry-Mather set at cohomology class $c$. As applications, we provide via $\Sigma_+$ {a discrete analogue of Bernard's regularization theorem \cite{Ber07} and} a detailed exposition of Arnaud's observation \cite{Arnaud_2011}. Furthermore, we construct and analyze the corresponding dynamics on the full pseudo-graphs of discrete weak K.A.M. solutions.