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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 $c(σ)\in H^1(\mathbb T,\mathbb R)$ with $c(σ)$ continuous and monotonic and \[ \partial_tu_c+H(x,\partial_x u_c+c,t)=α(c),\quad \text{a.e.\ } (x,t)\in\mathbb T^2, \] such that sequence of weak KAM solutions $\{u_c\}_{c\in H^1(\mathbb T,\mathbb R)}$ is $1/2-$Hölder continuity of parameter $σ\in \mathbb{R}$. 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 $ω(c)\in H_1(\mathbb T,\mathbb R)$. 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.