Topological characterizations of Hamiltonian flows with finitely many singular points on unbounded surfaces

Under a regularity of singular points, a flow with an orientable surface with finite genus and finite ends is Hamiltonian if and only if it is a flow without limit circuits or non-closed recurrent points such that the extended orbit space is a finite directed graph without directed cycles. On the other hand, under finite volume assumption, any Hamiltonian flows with finitely many singular points on an orientable surface with finite genus and finite ends can be embedded in Hamiltonian flows on compact surfaces. Furthermore, the finite union of centers, multi-saddles, and virtually border separatrices of a Hamiltonian flow with sectored ends on an orientable surface with finite genus and finite ends as a surface graph is a topological complete invariant.