Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics

Abstract

We present a combinatorial approach to rigorously show the existence of fixed points, periodic orbits, and symbolic dynamics in discrete-time dynamical systems, as well as to find numerical approximations of such objects. Our approach relies on the method of 'correctly aligned windows'. We subdivide 'windows' into cubical complexes, and we assign to the vertices of the cubes labels determined by the dynamics. In this way, we encode the information on the dynamics into combinatorial structure. We use a version of Sperner's Lemma to infer that, if the labeling satisfies certain conditions, then there exist fixed points/periodic orbits/orbits with prescribed itineraries. The method developed here does not require the computation of algebraic topologytype invariants, as only combinatorial information is needed; our arguments are elementary.

Combinatorial Proof of Kakutani's Fixed Point Theorem

“…Thus, Theorem 3.2 can probably be used to provide a combinatorial proof that the fixed point index works. In [GS18] such a proof was given for the case of correctly aligned windows, but the proof can be extended to various cases where the Brouwer degree of g does not equal zero. It makes sense that such an index should also work for upper semicontinuous multivalued mappings, where each point maps to convex set.…”