# Mixing properties of tree-shifts

## TL;DR

This paper explores the properties of tree-shifts in symbolic dynamics, addressing key questions about their structure, and introduces block gluing tree-shifts to identify conditions for positive topological entropy.

## Contribution

It provides new results on the decidability and structural properties of tree-shifts, including the introduction of block gluing tree-shifts and conditions for positive entropy.

## Key findings

- Decidability of emptiness problem in tree-shifts
- Existence of aperiodic and nonextensible local patterns in tree-shifts
- Block gluing tree-shifts exhibit positive topological entropy under certain conditions

## Abstract

The uncertainty of multidimensional shift spaces draws attracted attention of researchers. For example, the emptiness problem is undecidable; there exist aperiodic shifts of finite type; there is a nonempty shift of finite type exhibiting nonextensible local patterns. This paper investigates symbolic dynamics on Cayley trees and gives affirmative answers to the above questions in tree-shifts. Beyond that, with introducing block gluing tree-shift, a sufficient condition of exhibiting positive topological entropy is revealed.