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

## TL;DR

This paper introduces a combinatorial method using correctly aligned windows and Sperner's Lemma to rigorously detect fixed points, periodic orbits, and symbolic dynamics in discrete-time systems, providing both theoretical guarantees and numerical approximations.

## Contribution

It develops an elementary combinatorial framework that encodes dynamical information into labeled cubical complexes for detecting key dynamical objects.

## Key findings

- Provides a rigorous method for fixed point detection
- Enables approximation of periodic orbits and symbolic dynamics
- Uses combinatorial labeling and Sperner's Lemma for guarantees

## 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.