Stem and topological entropy on Cayley trees

TL;DR

This paper investigates the existence and equality of topological and stem entropy for shift spaces on Cayley trees, including free semigroups and groups, providing conditions for their equivalence.

Contribution

It introduces the concept of stem entropy, establishes its relation to topological entropy, and provides conditions ensuring their existence and equality on Cayley trees.

Findings

Stem entropy coincides with topological entropy for free semigroups.

Sufficient conditions are identified for the existence of stem entropy.

Topological entropy exists and equals stem entropy in many cases.

Abstract

We consider the existence of the topological entropy of shift spaces on a finitely generated semigroup whose Cayley graph is a tree. The considered semigroups include free groups. On the other hand, the notion of stem entropy is introduced. For shift spaces on a strict free semigroup, the stem entropy coincides with the topological entropy. We reveal a sufficient condition for the existence of the stem entropy of shift spaces on a semigroup. Furthermore, we demonstrate that the topological entropy exists in many cases and is identical to the stem entropy.