Topological entropy for countable Markov shifts and Exel--Laca algebras

TL;DR

This paper establishes a connection between topological entropy in countable Markov shifts and non-commutative topological entropy in Exel--Laca algebras, highlighting cases like the renewal shift.

Contribution

It proves the equality of these entropies under certain conditions and explores examples beyond locally finite matrices.

Findings

Topological entropy for countable Markov shifts matches non-commutative entropy for associated algebras.

The renewal shift is a key example satisfying the conditions.

Open questions are posed for non-locally finite transition matrices.

Abstract

We show that the (Gurevich) topological entropy for the countable Markov shift associated with an infinite transition matrix $A$ coincides with the non-commutative topological entropy for the Exel--Laca algebra associated with $A$, under certain conditions on $A$. An important example satisfying the conditions is the renewal shift, which is not locally finite. We also pose interesting questions for future research on non-commutative topological entropy for non-locally finite transition matrices.