Extension of a block code II

Definitions and mathematical statement

Let $T\subset A^{\mathbb Z}$ be a mixing sofic shift over a finite alphabet. Sofic means that $T$ is presented by the labels of paths in a finite directed graph. Mixing means that any two allowed words can be connected by allowed intervening words of every sufficiently large length. Let $f:T\to T$ be a surjective sliding block code: it is continuous and commutes with the shift.

Determine necessary and sufficient conditions for the existence of a shift of finite type $T'\subset A^{\mathbb Z}$ containing $T$ and a sliding block code $\widetilde f:T'\to T$ such that \[ \widetilde f|_T=f. \] In the source this is abbreviated as $f(T')\subset T$: the local rule of the given map is extended to the larger domain. A radius may be increased when presenting the same map on $T$, and values on newly allowed input blocks must be specified consistently.

No proper containment is required; if $T$ already is of finite type, $T'=T$ is permitted. Since $f$ is onto, any such extension is automatically onto $T$ as well. The significant constraint is that $T'$ have finitely many forbidden words while its image lies entirely inside $T$. The question asks for an extension of the specified map, not merely the existence of some unrelated self-map of the sofic shift.

Short English statement

When does a specified onto code of a mixing sofic shift extend to a finite-type neighborhood while still mapping into that shift?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600024 (AMR-045-0024), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Boyle - Open Problems in Symbolic Dynamics (2008), Problem/Question/Conjecture 16.3. Original source indexed by AMR-045-0024. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.