Is Every Irreducible Shift of Finite Type Flow Equivalent to a Renewal System?

TL;DR

This paper investigates whether all irreducible shifts of finite type are flow equivalent to renewal systems, exploring the Bowen--Franks invariant and constructing renewal systems with various invariants.

Contribution

It provides the first partial results on the Bowen--Franks invariant range for renewal systems and shows cyclicity of the invariant in certain classes.

Findings

Bowen--Franks group is cyclic for specific renewal systems

Constructed renewal systems with non-trivial Bowen--Franks invariants

Partial results on the invariant's range over renewal systems

Abstract

Is every irreducible shift of finite type flow equivalent to a renewal system? For the first time, this variation of a classic problem formulated by Adler is investigated, and several partial results are obtained in an attempt to find the range of the Bowen--Franks invariant over the set of renewal systems of finite type. In particular, it is shown that the Bowen--Franks group is cyclic for every member of a class of renewal systems known to attain all entropies realised by shifts of finite type, and several classes of renewal systems with non--trivial values of the invariant are constructed.