2020/04/05 by Zhang, Jianlu
#37E40 #37E45 #37J40 #37J45 #37J50 #49L25 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2004.02078
We investigated several global behaviors of the weak KAM solutions uc(x,t) parametrized by c∈ H1(\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 uc(x,t) parametrized by c(σ)∈ H1(\mathbb T,\mathbb R) with c(σ) continuous and monotonic and ∂tuc+H(x,∂x uc+c,t)=α(c), a.e. (x,t)∈\mathbb T2, such that sequence of weak KAM solutions \uc\c∈ H1(\mathbb T,\mathbb R) is 1/2-Hölder continuity of parameter σ∈ ℝ. Moreover, for each generalized characteristic (no matter regular or singular) solving \ \beginaligned amp;x(s)∈ co [∂pH(x(s),c+D+uc(x(s),s+t),s+t)], amp;
amp;x(0)=x0, (x0,t)∈\mathbb T2,amp; \endaligned . we evaluate it by a uniquely identified rotational number ω(c)∈ H1(\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.