Finitary images of IID processes

Definitions and mathematical statement

Let $X\subseteq A^{\mathbb Z^d}$, with finite alphabet $A$, be a shift of finite type: it consists of the configurations avoiding a fixed finite list of forbidden patterns and their translates. Suppose it has a unique shift-invariant probability measure $\mu$ of maximal entropy, meaning $h_\mu(\sigma)=h_{\rm top}(X)$. Assume also that this measure-preserving lattice action is Bernoulli, that is, measurably isomorphic to an independent product process.

Does there exist an independent identically distributed process $Y\in B^{\mathbb Z^d}$ and a measurable shift-equivariant map $\Phi:B^{\mathbb Z^d}\to X$ such that $\Phi_*\mathbb P=\mu$ and $\Phi$ is finitary? Here finitary means that almost surely there is a finite random radius $R(Y)$ such that the output symbol at the origin is determined by the input coordinates in $[-R,R]^d\cap\mathbb Z^d$. The same local rule, translated, determines every other output coordinate.

The input alphabet and its probability law may be chosen to suit $X$. No tail estimate or finite expectation for $R$ is required. The map is only required to be a factor onto the probability law $\mu$, not a finitary isomorphism with a finitary inverse. Bernoullicity already gives a measurable coding; the question is whether one can arrange that every individual output coordinate is decided by a finite, configuration-dependent window. A single deterministic uniform coding radius is not requested.

Short English statement

Must a Bernoulli measure of maximal entropy on a shift of finite type be a finitary factor of an IID process?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600029 (AMR-045-0029), 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 18.2. Original source indexed by AMR-045-0029. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.