Abstract

This paper attempts to study the irreducibility on complete prefix code (CPC-irreducibility) of a Markov shift over a free group, a topological mixing property first considered for shift spaces over free semigroups that induces chaotic behavior such as the existence of a dense set of periodic points. An example shows that the (d,c)-reduction, an effective algorithm of determination of CPC-irreducibility of Markov shifts over free semigroups (Ban et al. in J Stat Phys 177:1043–1062, 2019), fails for general Markov shifts over free groups. This paper reveals an algorithm for determining the CPC-irreducibility of Markov shifts over both free semigroups and groups. Furthermore, such an examination is finitely checkable, and an upper bound for the complexity of the algorithm is provided.