  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.
