Characterization and Topological Behavior of Homomorphism Tree-Shifts

TL;DR

This paper explores the properties of homomorphism tree-shifts, demonstrating their equivalence to classical shift spaces in terms of finite type and mixing properties, and establishing connections between their dynamical behaviors.

Contribution

It establishes the equivalence of shift of finite type and mixing properties between one-sided shifts and their associated hom tree-shifts, extending classical results to tree-shift contexts.

Findings

Equivalence of shift of finite type between one-sided shifts and hom tree-shifts.

Coincidence of mixing properties across different definitions for tree-shifts.

Correspondence between irreducibility and mixing properties of $X_A$ and $ ext{T}_A$.

Abstract

The purpose of this article is twofold. On one hand, we reveal the equivalence of shift of finite type between a one-sided shift $X$ and its associated hom tree-shift $\mathcal{T}_{X}$, as well as the equivalence in the sofic shift. On the other hand, we investigate the interrelationship among the comparable mixing properties on tree-shifts as those on multidimensional shift spaces. They include irreducibility, topologically mixing, block gluing, and strong irreducibility, all of which are defined in the spirit of classical multidimensional shift, complete prefix code (CPC), and uniform CPC. In summary, the mixing properties defined in all three manners coincide for $\mathcal{T}_{X}$. Furthermore, an equivalence between irreducibility on $\mathcal{T}_{A}$ and irreducibility on $X_A$ are seen, and so is one between topologically mixing on $\mathcal{T}_{A}$ and mixing property on $X_A$, where $X_A$ is the one-sided shift space induced by the matrix $A$ and $T_A$ is the associated tree-shift. These equivalences are consistent with the mixing properties on $X$ or $X_A$ when viewed as a degenerate tree-shift.