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We construct topological invariants, called abstract weak orbit spaces, offlows and homeomorphisms on topological spaces, to describe both gradientdynamics and recurrent dynamics. In particular, the abstract weak orbit spacesof flows on topological spaces are generalizations of both Morse graphs offlows on compact metric spaces and Reeb graphs of Hamiltonian flows withfinitely many singular points on surfaces. Moreover, we show that the abstractweak orbit spaces are complete and finite for several kinds of flows onmanifolds, and we state several examples whose Morse graphs are singletons butwhose abstract weak orbit spaces are not singletons. In addition, we considerwhen the time-one map reconstructs the topology of the original flow. Thereforewe show that the orbit space of a Hamiltonian flow with finitely many singularpoints on a compact surface is homeomorphic to the abstract weak orbit space ofthe time-one map by taking an arbitrarily small reparametrization, and that theabstract weak orbit spaces of a Morse flow on a compact manifold and thetime-one map are homeomorphic. Furthermore, the abstract weak orbit space of aMorse flow on a closed manifold is a refinement of the CW decomposition whichconsists of the unstable manifolds of singular points. Though the CWdecomposition of a Morse flow on a closed manifold is finite, the intersectionof the unstable manifold and the stable manifold of saddles of a Morse-Smaleflow on a closed manifold need not consist of finitely many connectedcomponents (or equivalently need not consist of finitely many abstract weakorbits). Therefore we study the finiteness of abstract weak orbit spaces ofMorse(-Smale) flow on compact manifolds.

Quotient spaces and topological invariants of flows