On the existence of open and bi-continuing codes

Abstract

Given an irreducible sofic shift X X , we show that an irreducible shift of finite type Y Y of lower entropy is a factor of X X if and only if it is a factor of X X by an open bi-continuing code. If these equivalent conditions hold and Y Y is mixing, then any code from a proper subshift of X X to Y Y can be extended to an open bi-continuing code on X X . These results are still valid when X X is assumed to be only an almost specified shift, i.e., a subshift satisfying an irreducible version of the specification property.