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

For an exact twist map f, we introduce an inherent Lipschitz dynamics Σ₊ given by the discrete forward Lax-Oleinik semigroup. We investigate several properties of Σ₊ and show that for any discrete weak K.A.M. solution u, the non-differentiable points of u are globally propagated and forward invariant by Σ₊. In particular, such propagating dynamics possesses the same rotation number as the associated Aubry-Mather set with respect to u. A detailed exposition of the corresponding Arnaud's observation {Arnaud_2011} is then provided via Σ₊. Furthermore, we construct and analyze the dynamics on the pseudo-graphs of discrete weak K.A.M. solutions.