In this talk, we consider when the time-one map reconstructs the topology of the original flow. We show that a quotient space of the orbit space, called abstract weak orbit space, of Hamiltonian
flow with finitely many singular points on a compact surface is homeomorphic to the abstract weak orbit space of the time-one map up to arbitrarily small reparametrization, and that the abstract weak
orbit space of a Morse flow on a compact manifold and the time-one map is homeomorphic.
Moreover, the abstract weak orbit spaces are topological invariants which 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. In addition, we show that the abstract weak orbit spaces are finite for several kinds of flows on manifolds, and we state non-triviality of the abstract weak orbit spaces using several examples (e.g. Hamiltonian flows on surfaces, reducible chain-recurrent homeomorphisms, pseudo-Anosov homeomorphisms, chaotic flows in the sense of Devaney, non-identical non-minimal non-pointwise-periodic volume-preserving flows.).

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