Relations among Hamiltonian, area-preserving, and non-wandering flows on surfaces

Abstract

Area-preserving flows on compact surfaces are one of the classic examples of dynamical systems, also known as multi-valued Hamiltonian flows. Though Hamiltonian, area-preserving, and non-wandering properties for flows are distinct, there are some equivalence relations among them in the low-dimensional cases. In this paper, we describe equivalence and difference for continuous flows among Hamiltonian, divergence-free, and non-wandering properties topologically. More precisely, let v be a continuous flow with finitely many singular points on a compact surface. We show that v is Hamiltonian if and only if v is a non-wandering flow without locally dense orbits whose extended orbit space is a directed graph without directed cycles. Moreover, non-wandering, area-preserving, and divergence-free properties for v are equivalent to each other.

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

“…Notice that the extended orbit space of v is a quotient space of the orbit space of v. The extended orbit space of a flow of weakly finite type on a surface with finite genus and finite ends is a generalization of the extended orbit space on a compact surface. In particular, the extended orbit space of a Hamiltonian flows with finitely many singular points on a compact surface in the sense of this paper corresponds to the extended orbit space in the sense of one in [19]. Indeed, the extended orbit space S/v ex is the quotient space S/ ∼ ms , where x ∼ ms y if there is either an orbit or a multi-saddle connection that contains x and y.…”

“…In addition, the generic intermediate flows between structurally stable Hamiltonian flows on compact surfaces and non-compact punctured spheres are also characterized, and their topological complete invariants are constructed [16]. Furthermore, Hamiltonian flows with finitely many singular points on compact surfaces are topologically characterized using the non-wandering property [19]. In fact, the difference and equivalence among Hamiltonian, area-preserving, and non-wandering properties for flows with finitely many singular points on compact surfaces are topologically characterized.…”