Global behaviors of weak KAM solutions for exact symplectic twist maps

Abstract We investigated several global behaviors of the weak KAM solutions u c ( x , t ) parametrized by c ∈ H 1 ( 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 u c ( x , t ) parametrized by c ( σ ) ∈ H 1 ( T , R ) with c ( σ ) continuous and monotonic and ∂ t u c + H ( x , ∂ x u c + c , t ) = α ( c ) , a.e. ( x , t ) ∈ T 2 , such that sequence of weak KAM solutions { u c } c ∈ H 1 ( T , R ) is 1/2-Holder continuity of parameter σ ∈ R . Moreover, for each generalized characteristic (no matter regular or singular) solving { x ˙ ( s ) ∈ co [ ∂ p H ( x ( s ) , c + D + u c ( x ( s ) , s + t ) , s + t ) ] , x ( 0 ) = x 0 , ( x 0 , t ) ∈ T 2 , we evaluate it by a uniquely identified rotational number ω ( c ) ∈ H 1 ( 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.