GETCO 2022 / P. J. Chocano / Approximating Discrete Dynamical Systems скачать в хорошем качестве

GETCO 2022 / P. J. Chocano / Approximating Discrete Dynamical Systems

Discrete dynamical systems have been proved to be a very useful tool to model different situations, but a direct study of them may be difficult. For this reason, it is important to develop computational methods to get some of their relevant information. The goal of this contributed talk is to present topological methods to study fixed points. Finite topological spaces, that are combinatorial objects (partially ordered sets), have the same homology and homotopy groups of polyhedra and can be used to reconstruct them. Therefore, the idea is to use finite spaces to approximate discrete dynamical systems given by homeomorphisms f : K → K where K is a compact polyhedron. For this purpose, we first discuss the notion of dynamical system in this combinatorial setting and then introduce a class of multivalued maps inducing morphisms in homology groups. From this, we deduce a Lefschetz fixed point theorem. Finally, we use the theory developed to study fixed points of discrete dynamical systems defined on polyhedra and give some lines of future work.