Quotient spaces and topological invariants of flows.

We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces, to describe both gradient dynamics and recurrent dynamics. In particular, the abstract weak orbit spaces of flows on topological spaces are generalizations of both Morse graphs of flows on compact metric spaces and Reeb graphs of Hamiltonian flows with finitely many singular points on surfaces. Moreover, we show that the abstract weak orbit spaces are complete and finite for several kinds of flows on manifolds, and we state several examples whose Morse graphs are singletons but whose abstract weak orbit spaces are not singletons. In addition, we consider when the time-one map reconstructs the topology of the original flow. Therefore we show that the orbit space of a Hamiltonian flow with finitely many singular points on a compact surface is homeomorphic to the abstract weak orbit space of the time-one map by taking an arbitrarily small reparametrization, and that the abstract weak orbit spaces of a Morse flow on a compact manifold and the time-one map are homeomorphic. Furthermore, the abstract weak orbit space of a Morse flow on a closed manifold is a refinement of the CW decomposition which consists of the unstable manifolds of singular points. Though the CW decomposition of a Morse flow on a closed manifold is finite, the intersection of the unstable manifold and the stable manifold of saddles of a Morse-Smale flow on a closed manifold need not consist of finitely many connected components (or equivalently need not consist of finitely many abstract weak orbits). Therefore we study the finiteness of abstract weak orbit spaces of Morse(-Smale) flow on compact manifolds.