Global Behaviors of weak KAM Solutions for exact symplectic Twist Maps

We investigated several global behaviors of the weak KAM solutions $u_c(x,t)$ parametrized by $c\in H^1(\mathbb{T},\mathbb{R})$. For the suspended Hamiltonian $H(x,p,t)$ of the exact symplectic twist map, we could find a family of weak KAM solutions $u_c(x,t)$ parametrized by $c(\sigma)\in H^1(\mathbb T,\mathbb R)$with$c(\sigma)$ continuous and monotonic and $${\mathrm{\partial }}_t{u}_c+H(x,{\mathrm{\partial }}_x{u}_c+c,t)=\alpha (c),\text{a.e. }(x,t)\in {\mathbb{T}}^2,$$ such that sequence of weak KAM solutions ${u_c}_{c\in H^1(\mathbb T,\mathbb R)}$is$1/2-$H\ddot{o}ldercontinuityofparameter$\sigma\in $\mathbb{R}$$. Moreover, for each generalized characteristic (no matter regular or singular) solving $$\{\begin{array}{rlr}{\displaystyle }& {\displaystyle \dot{x}(s)\in \text{co}\left[{\mathrm{\partial }}_{p}H\right(x(s),c+D^{+}{u}_c(x(s),s+t),s+t\left)\right],}& {\displaystyle }\\ {\displaystyle }& {\displaystyle x(0)=x_0,(x_0,t)\in {\mathbb{T}}^2,}& {\displaystyle }\end{array}$$ we evaluate it by a uniquely identified rotational number $\omega©\in H_1(\mathbb T,\mathbb R)$. This property leads to a certain topological obstruction in the phase space and causes local transitive phenomenon of trajectories. Besides, we discussed this applies to high-dimensional cases.