CRM CAMP in Nonlinear Analysis

Computer-Assisted Mathematical Proofs in Nonlinear Analysis

June 22, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

Combinatorial Topological Dynamics

Seminar presented by Marian Mrozek (Jagiellonian University, Poland)

Since the publication in 1998 of the seminal work by Robin Forman on combinatorial Morse theory there has been growing interest in dynamical systems on finite spaces. The main motivation to study combinatorial dynamics comes from data science. But, they also provide very concise models of dynamical phenomena and show some potential in certain computer assisted proofs in dynamics.

In the talk I will present the basic ideas of Conley theory for combinatorial dynamical system, particularly for a combinatorial multivector field which is a generalization of combinatorial vector field introduced by Forman. The theory is based on concepts which are analogous to the concepts of classical theory: isolating neighborhood, isolated invariant set, index pair, Conley index, Morse decomposition, connection matrix. The concepts are analogous but in some cases surprisingly different in details. This may be explained by the non-Hausdorff nature of combinatorial topological spaces.

Despite the differences there seem to be strong formal ties between the combinatorial and classical dynamics on topological level. A Morse decomposition of a combinatorial vector field on an abstract simplicial complex induces a semiflow on the geometric realization of the complex with a Morse decomposition exhibiting the same Conley-Morse graph. Actually, this correspondence of Morse decompositions and Conley-Morse graphs applies to every semiflow which is transversal to the boundaries of top dimensional cells of a certain cellular decomposition of the phase space associated with the combinatorial vector field.

There is also a formal relation in the opposite direction. Given a smooth flow and a cellular decomposition of its phase space which is transversal to the flow, there is an induced combinatorial multivector field on the cellular structure of the phase space. Moreover, if the induced combinatorial multivector field admits a periodic trajectory with an appropriate Conley index, a periodic orbit exists also for the original smooth flow.

The formal ties seem to provide a natural framework for a rigorous global analysis of the dynamics of a flow: the decomposition into the gradient and recurrent part together with the computation of the Conley-Morse graph, connection matrix and revealing the internal structure of the recurrent part.

Based on joint work with J. Barmak, T. Dey, M. Juda, T. Kaczynski, T. Kapela, J. Kubica,  M. Lipiński  R. Slechta, R. Srzednicki, J. Thorpe and Th. Wanner.