Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics
Dynamical Systems
2017-06-28 v1 Algebraic Topology
Combinatorics
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 the `windows' into cubical complexes, and we assign to the vertices of the cubes labels determined by the dynamics. In this way we encode the dynamics information into a combinatorial structure. We use a version of the Sperner Lemma saying that if the labeling satisfies certain conditions, then there exist fixed points/periodic orbits/orbits with prescribed itineraries. Our arguments are elementary.
Keywords
Cite
@article{arxiv.1706.08960,
title = {Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics},
author = {Marian Gidea and Yitzchak Shmalo},
journal= {arXiv preprint arXiv:1706.08960},
year = {2017}
}