Extension of a block code I

Definitions and mathematical statement

In the usual two-sided convention, a shift space is a closed shift-invariant subset of $A^{\mathbb Z}$ for a finite alphabet $A$. A shift of finite type (SFT) is specified by finitely many forbidden finite words; a sofic shift is the image of an SFT under a continuous shift-commuting map. A block code is such a map, given by a local rule on a fixed finite window. A shift is mixing if any two allowed words can be connected by allowed words of every sufficiently large length. A fixed point is a constant configuration $a^{\mathbb Z}$.

The source assumes a mixing sofic shift $T$ with a "receptive fixed point" and asks when there are a block code $f:T\to T$ and an SFT $T'\supset T$ for which $f(T')\subset T$. As written, $f(T')$ is not defined when $f$ has domain only $T$. An extension interpretation would require an additional block code $\widetilde f:T'\to T$ satisfying \[ \widetilde f|_T=f,\qquad T\subseteq T',\qquad T'\text{ is an SFT}. \] The task would then be to characterize the admissible $T$ and/or $f$, but the quantification of $f$ is unresolved: the statement asks for the existence of $f$, whereas the summary discusses extension of a block code already on the sofic shift.

This distinction matters. With no surjectivity, identity-on-$T$, or other requirement on an existentially chosen code, a fixed point allows the constant local rule taking every input to that fixed point, including on an ambient full shift. Thus an intended nontrivial extension problem needs the missing restriction or the intended given-code quantifier. No such restriction is inserted here. Likewise the technical adjective "receptive" is not defined in the supplied excerpt and is retained without a guessed replacement. The source does not specify one- versus two-sided shifts, whether containment is proper, or additional mixing conditions on $T'$.

The research summary relates this question to item 16.3 and calls the general code-extension characterization open, mentioning mixing/closed SFT models. Those descriptions do not supply the missing definition or code conditions. The 2008 source's open label has no recorded later authoritative resolution check.

Short English statement

When can a block code on a mixing sofic shift with a receptive fixed point extend to an SFT containing that shift, with image still inside it? The excerpt does not define receptive, uses the code outside its stated domain, and leaves unclear whether the code is given or freely chosen; these omissions affect the intended problem.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $4600023$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Boyle, Open Problems in Symbolic Dynamics (2008), Problem/Question/Conjecture 16.2: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf. Supplied PDF-text extraction accessed 2026-07-29; no later authoritative resolution check is recorded.
• The research summary mentions related item 16.3 and Boyle's code-extension/SFT-model work but supplies no further identifiable publication or precise characterization theorem.