Seminar
"Morse graphs" and "abstract orbits" of dynamical systems, topological spaces, and decompositions

Hold Date 
20220520 15:30～20220520 17:00 


Place 
IMI Conference Room (D414) 


Object person 



Speaker 
Tomoo Yokoyama (Gifu University) 

Abstract:
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 timeone 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 groupactions 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 hypergraphs of topological spaces and decompositions, arXiv:2112.13992, preprint.