Pseudo-cyclic renewal systems

A finite set W of words over an alphabet A is cyclic if, whenever u,v∈A∗ and uv,vu∈W∗, we have u,v∈W∗. If it is only assumed that the property holds for all u,v∈A∗ with a large length, then W is called pseudo-cyclic, that is, there is N∈N such that, whenever u,v∈A∗ with |u|, |v|≥N and uv,vu∈W∗, we have u,v∈W∗. We analyze the class of pseudo-cyclic sets and describe how it is related to the open question which asks whether every irreducible shift of finite type is conjugate to a renewal system.