Topological properties of cellular automata on trees

TL;DR

This paper explores the topological characteristics of cellular automata on k-ary trees, proving the non-existence of positively expansive automata and examining properties like permutivity and surjectivity.

Contribution

It establishes the non-existence of positively expansive cellular automata on trees and analyzes various topological properties and their interrelations.

Findings

Positively expansive automata do not exist on k-ary trees for k>=2.

Topological properties such as permutivity, surjectivity, and openness are studied.

Relationships among these properties are characterized.

Abstract

We prove that there do not exist positively expansive cellular automata defined on the full k-ary tree shift (for k>=2). Moreover, we investigate some topological properties of these automata and their relationships, namely permutivity, surjectivity, preinjectivity, right-closingness and openness.