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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 weakKAM solutions $u_c(x,t)$ parametrized by $c(\sigma)\in H^1(\mathbb T,\mathbbR)$ 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 $\{u_c\}_{c\inH^1(\mathbb T,\mathbb R)}$ is $1/2-$H\"older continuity of parameter $\sigma\in\mathbb{R}$. Moreover, for each generalized characteristic (no matter regularor 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 itby a uniquely identified rotational number $\omega(c)\in H_1(\mathbb T,\mathbbR)$. This property leads to a certain topological obstruction in the phasespace and causes local transitive phenomenon of trajectories. Besides, wediscussed this applies to high-dimensional cases.

Global Behaviors of weak KAM Solutions for exact symplectic Twist Maps