Equal-entropy subcovers

Definitions and mathematical statement

Fix $d\ge2$, a nonempty $\mathbb Z^d$ shift of finite type $X\subset A^{\mathbb Z^d}$, and a factor map $\pi:X\to Y\subset B^{\mathbb Z^d}$. Here the alphabets are finite, $X$ is defined by finitely many forbidden patterns, and $\pi$ is a continuous surjection commuting with every lattice shift. Its image $Y$ is consequently a sofic shift. A subshift is sofic when it is the image of some finite-type shift under such a map.

For any subshift $Z$, define entropy by \[ h(Z)=\lim_{n\to\infty}n^{-d} \log\#\{z|_{\{0,\ldots,n-1\}^d}:z\in Z\}. \] Must there be a nonempty sofic subshift $W\subset X$ such that \[ \pi(W)=Y\quad\hbox{and}\quad h(W)=h(Y)? \] The covering map is fixed as part of the input. Thus the question asks to remove surplus configurations inside this particular cover, rather than to choose an unrelated cover of $Y$. The restriction $\pi|_W$ must remain onto every point of $Y$, including exceptional configurations.

The desired $W$ must be closed and invariant under the full lattice action, and must admit a finite-state sofic presentation. It need not itself be of finite type. Exact entropy equality is required, rather than a family of subcovers approaching the target entropy. Neither the input spaces nor the chosen subcover are assumed mixing or irreducible.

Short English statement

Inside a given multidimensional finite-type cover, can one retain an onto sofic subcover with exactly the target entropy?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600021 (AMR-045-0021), 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 15.2. Original source indexed by AMR-045-0021. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.