On the existence of open and bi-continuing codes

TL;DR

This paper characterizes when a lower-entropy irreducible SFT is a factor of an irreducible sofic shift via open bi-continuing codes, extending results to almost specified shifts and discussing code extensions.

Contribution

It establishes necessary and sufficient conditions for factorization through open bi-continuing codes and extends these results to almost specified shifts.

Findings

Characterization of factors via open bi-continuing codes

Extension of code from subshifts to the entire shift under certain conditions

Results valid for almost specified shifts, broadening applicability

Abstract

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