Decompositions of surface flows

Abstract

We construct a complete invariant for surface flows of finite type. In fact, although the set of topological equivalence classes of minimal flows (resp. Denjoy flows) on a torus is uncountable, we enumerate the set of topological equivalence classes of flows with at most finitely many limit cycles but without Q-sets and non-degenerate singular points on a compact surface using finite labelled graphs. To enumerate such flows, we describe properties of border points of a flow without degenerate singular points on a compact surface. In particular, we show that each connected component of the complement of the "saddle connection diagram" is either an open disk, an open annulus, a torus, a Klein bottle, an open Möbius band, or an open essential subset. Moreover, we generalize the Poincaré-Bendixson theorem for a flow with arbitrarily many singular points on a compact surface. In fact, the ω-limit set of any nonclosed orbit is either a nowhere dense subset of singular points, a limit cycle, a limit "quasi-circuit", a locally dense Q-set, or a "quasi-Q-set". In addition, for such a surface flow, we characterize the necessary and sufficient conditions for the closure of the union of closed orbits corresponding to the non-wandering set.

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

“…Then S = S − (Sing(v) ∪ D(v)). The union Bd 0 (v) defined in [31] is the union of singular points and separatrices between multi-saddles. This means that Bd 0 (v) is the union of centers and the multi-saddle connection diagram D(v).…”

“…On the other hand, any area-preserving flows are non-wandering flows. Non-wandering flow on surfaces are classified and decomposed into elementary cells under finite existence of singular points, and are topologically characterized, and the topological invariants are constructed [8,21,22,[30][31][32].…”

Flows with time-reversal symmetric limit sets on surfaces

“…Thus we may assume that there are no limit cycles. By [10,Lemma 3.4], the nonexistence of limit cycles implies that Per(v) is open. Then P(v) ⊆ Sing(v) ⊔ P(v).…”

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

“…For a flow of weakly finite type on a surface, denote by BD + (v) the union of singular points, limit cycles, one-sided periodic orbits, and virtually border separatrices. The invariant subset BD + (v) corresponds to the original one for a flow of weakly finite type on a compact surface because of [18,Lemma 7.8]. Moreover, we have the following description.…”

“…( Proof. By original definitions of BD + (v) and BD + (v end ) in [18], since any singular points of v are sectored, we have BD + (v) = BD + (v end ). By [18,Lemma 7.8], the invariant subset BD + (v) = BD + (v end ) is the finite union of singular points, limit cycles, one-sided periodic orbits, and border separatrices.…”