We investigated several global behaviors of the weak KAM solutions uc(x,t) parametrized by c∈H1(T,R). For the suspended Hamiltonian H(x,p,t) of the exact symplectic twist map, we could find a family of weak KAM solutions uc(x,t) parametrized by c(σ)∈H1(T,R) with c(σ) continuous and monotonic and∂tuc+H(x,∂xuc+c,t)=α(c),a.e.(x,t)∈T2, such that sequence of weak KAM solutions {uc}c∈H1(T,R) is 1/2-Hölder continuity of parameter σ∈R. Moreover, for each generalized characteristic (no matter regular or singular) solving{x˙(s)∈co[∂pH(x(s),c+D+uc(x(s),s+t),s+t)],x(0)=x0,(x0,t)∈T2, we evaluate it by a uniquely identified rotational number ω(c)∈H1(T,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.