Pseudo-cyclic renewal systems

Abstract

A finite set W of words over an alphabet A is cyclic if, whenever u , v ∈ A ∗ and u v , v u ∈ 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 u v , v u ∈ 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.