Equal-entropy SFT covers

Definitions and mathematical statement

Fix $d\ge2$. A $\mathbb Z^d$ subshift is a nonempty closed, translation-invariant set $Y\subset B^{\mathbb Z^d}$ over a finite alphabet. A shift of finite type (SFT) is defined by excluding finitely many finite patterns and all their translates. A factor map is a continuous surjection commuting with every translation; equivalently, it applies one finite-window local rule at every site. A subshift is sofic if it is a factor of some $\mathbb Z^d$ SFT.

For a subshift $Z$, let $\mathcal L_{Q_n}(Z)$ be the patterns it realizes on $Q_n=\{0,\ldots,n-1\}^d$. Its entropy per site is \[ h(Z)=\lim_{n\to\infty}\frac{1}{n^d} \log|\mathcal L_{Q_n}(Z)|. \] Use natural logarithms and the same normalization for all spaces. The question is whether every such sofic $Y$ admits an SFT $X$ and a factor map $\pi:X\to Y$ with $h(X)=h(Y)$.

The alphabet and the local constraints of $X$ may be chosen freely, but its lattice dimension must remain $d$. Equality is exact; an SFT cover whose entropy is merely arbitrarily close to $h(Y)$ does not answer the question. No injectivity, finite-to-one condition, or mixing hypothesis is imposed on the cover. The problem includes zero-entropy sofic shifts and asks for an equal-entropy finite-type realization in every dimension greater than one.

Short English statement

Can every multidimensional sofic shift be covered by a finite-type shift without adding entropy?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600020 (AMR-045-0020), 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 15.1. Original source indexed by AMR-045-0020. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.