Topological structure and entropy of mixing graph maps

TL;DR

This paper investigates the minimal topological entropy of mixing graph maps, providing bounds based on graph structure and calculating exact values for certain graphs, with implications for dynamical properties.

Contribution

It establishes a lower bound for entropy of mixing graph maps and computes exact infimum values for specific graph families, enhancing understanding of their dynamical complexity.

Findings

Lower bound for entropy: log(3)/Lambda(G)

Exact infimum values for some graphs

New proofs of known results like Blokh's theorem

Abstract

Let $\mathcal{P}_G$ be the family of all topologically mixing, but not exact self-maps of a topological graph $G$. It is proved that the infimum of topological entropies of maps from $\mathcal{P}_G$ is bounded from below by $(\log 3/ \Lambda(G))$, where $\Lambda(G)$ is a constant depending on the combinatorial structure of $G$. The exact value of the infimum on $\mathcal{P}_G$ is calculated for some families of graphs. The main tool is a refined version of the structure theorem for mixing graph maps. It also yields new proofs of some known results, including Blokh's theorem (topological mixing implies specification property for maps on graphs).