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 $$\partial_tu_c+H(x,\partial_x u_c+c,t)=\alpha(c),\quad \text{a.e.\ } (x,t)\in\mathbb T^2,$$ such that sequence of weak KAM solutions $\{u_c\}_{c\in H^1(\mathbb T,\mathbb R)}$ is $1/2-$H\"older continuity of parameter $\sigma\in \mathbb{R}$. Moreover, for each generalized characteristic (no matter regular or singular) solving $$\left\{ \begin{aligned} &\dot{x}(s)\in \text{co} \Big[\partial_pH\Big(x(s),c+D^+u_c\big(x(s),s+t\big),s+t\Big)\Big], & \\ &x(0)=x_0,\quad (x_0,t)\in\mathbb T^2,& \end{aligned} \right.$$ we evaluate it by a uniquely identified rotational number $\omega(c)\in H_1(\mathbb T,\mathbb R)$. 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.