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 $cin H^1(mathbb T,mathbb R)$. 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)inmathbb T^2, ] such that sequence of weak KAM solutions ${u_c}_{cin H^1(mathbb T,mathbb R)}$ is $1/2-$Holder continuity of parameter $sigmain mathbb{R}$. Moreover, for each generalized characteristic (no matter regular or singular) solving [ left{ begin{aligned} &dot{x}(s)in text{co} Big[partial_pHBig(x(s),c+D^+u_cbig(x(s),s+tbig),s+tBig)Big], & &x(0)=x_0,quad (x_0,t)inmathbb 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.