Aubry-Mather theory for contact Hamiltonian systems III

By exploiting the contact Hamiltonian dynamics $(T^*M\times\mathbb R,\Phi_t)$ around the Aubry set of contact Hamiltonian systems, we provide a relation among the Mather set, the $\Phi_t$-recurrent set, the strongly static set, the Aubry set, the Ma\~{n}\'{e} set and the $\Phi_t$-non-wandering set. Moreover, we consider the strongly static set, as a new flow-invariant set between the Mather set and the Aubry set, in the strictly increasing case. We show that this set plays an essential role in the representation of certain minimal forward weak KAM solution and the existence of transitive orbits around the Aubry set.