分野別セミナー

In this talk, we discuss a topological invariant of flows, called abstract orbit space, which is a refinement of Morse graphs of flows on compact metric spaces, Reeb graphs of Hamiltonian flows with finitely many singular points on surfaces, and the CW decompositions which consist of the unstable manifolds of singular points for Morse flows on closed manifolds. In particular, the topological invariant is generically applicable to a reconstruction problem: when the time-one map reconstructs the topology of the original flow?
Moreover, interpreting the recurrence of dynamical systems into one for topological spaces, we introduce "Morse graph" and "abstract orbit" for topological spaces and for decompositions on topological spaces. In particular, the "Morse graphs" and the abstract orbit spaces for any group-actions and foliated spaces are defined. We show that similar statements for dynamical systems hold for topological spaces and decompositions using these concepts. For instance, the abstract orbit space is a unified concept of abstract cell complexes, Morse decompositions, and Reeb graphs of any Morse function. The first part of this talk is based on [1] and the second part on [2].

[1] T. Yokoyama, Refinements of topological invariants of flows. Discrete & Continuous Dynamical Systems, 42(5):2295–2331, 2022.

[2] T. Yokoyama, Morse hyper-graphs of topological spaces and decompositions, arXiv:2112.13992, preprint.