Aubry-Mather theory for contact Hamiltonian systems II

In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems \begin{document}$ H(x,u,p) $\end{document} with certain dependence on the contact variable \begin{document}$ u $\end{document}. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set \begin{document}$ \tilde{\mathcal{S}}_s $\end{document} consists of strongly static orbits, which coincides with the Aubry set \begin{document}$ \tilde{\mathcal{A}} $\end{document} in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show \begin{document}$ \tilde{\mathcal{S}}_s\subsetneqq\tilde{\mathcal{A}} $\end{document} in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of \begin{document}$ H $\end{document} on \begin{document}$ u $\end{document} fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the minimal viscosity solution and non-minimal ones.