Vertex maps on graphs -- Perron-Frobenius Theory

TL;DR

This paper explores how Perron-Frobenius theory applies to vertex maps on graphs, providing insights into periodic orbits, transitivity, mixing, horseshoes, and entropy in dynamical systems.

Contribution

It establishes new connections between Perron-Frobenius theory and dynamical properties of vertex maps on graphs.

Findings

Characterization of periods of periodic orbits

Results on transitivity and topological mixing

Analysis of horseshoes and topological entropy

Abstract

The goal of this paper is to describe the connections between Perron-Frobenius theory and vertex maps on graphs. In particular, it is shown how Perron-Frobenius theory gives results about the sets of integers that can arise as periods of periodic orbits, about the concepts of transitivity and topological mixing, and about horseshoes and topological entropy. This is a preprint. The final version will appear in the Journal of Difference Equations and Applications.