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.
Categories: math.DS