Classify group shifts

Definitions and mathematical statement

Fix $d\ge1$. A group shift is a closed shift-invariant subgroup $X\le G^{\mathbb Z^d}$ for some finite group $G$, with the product topology and coordinate shifts $\sigma^v$. Different finite alphabet groups and subgroup presentations are allowed. Two group shifts $X,Y$ are topologically conjugate when there is a homeomorphism \[ \phi:X\to Y,\qquad \phi\sigma^v=\sigma^v\phi \quad(v\in\mathbb Z^d). \] The homeomorphism need not preserve multiplication. By compactness, such conjugacies and their inverses are described by finite local coding rules on the configurations.

The problem is to classify group shifts up to this equivalence: find invariants or structural data that agree exactly when such a conjugacy exists, with a corresponding realization theory. The particularly emphasized case consists of abelian group shifts with completely positive entropy. For this condition use normalized Haar measure: every nontrivial measurable factor must have positive measure entropy. A factor is trivial when its probability space is a single point modulo null sets.

The desired equivalence is topological, not merely measure-theoretic or algebraic. In particular, having equal entropy or mutually factoring through continuous homomorphisms is not stipulated to imply conjugacy. The dimension $d$ is part of the action and is held fixed in each classification problem. Known one-dimensional descriptions motivate the question, but do not themselves classify higher-rank lattice actions. The complete problem includes groups without the completely-positive-entropy restriction as well as the highlighted subclass.

Short English statement

Classify topological conjugacy of group shifts, especially abelian shifts with completely positive entropy.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600027 (AMR-045-0027), 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 17.3. Original source indexed by AMR-045-0027. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.