Range of the dimension representation

Definitions and mathematical statement

Let $A$ be a primitive square matrix with nonnegative integer entries. The edge shift $X_A$ consists of bi-infinite paths in its directed multigraph, with shift $\sigma$ deleting the initial coordinate. Primitivity means that some power of $A$ is strictly positive and ensures mixing. Use row vectors and define the dimension group \[ G_A=\varinjlim(\mathbb Z^N,\ v\mapsto vA), \qquad [v,j]=[vA,j+1]. \] Its positive cone consists of classes represented by a nonnegative integer vector at some stage. The dimension automorphism is $\delta_A[v,j]=[vA,j]$; its inverse is $[v,j]\mapsto[v,j+1]$.

Every topological automorphism $\phi$ of $(X_A,\sigma)$ induces an order-preserving automorphism $\rho_A(\phi)$ of $G_A$ commuting with $\delta_A$. To specify this representation concretely, a strong shift equivalence step $A=UV$, $B=VU$ induces the map $[v,j]\mapsto[vU,j]$. A chain of such nonnegative integral steps, followed by a shift power, represents any conjugacy and defines its induced map; the result is independent of that representation.

The problem is to determine exactly the subgroup \[ \rho_A(\operatorname{Aut}(X_A,\sigma)) \subseteq\operatorname{Aut}(G_A,G_A^+,\delta_A). \] Thus one seeks necessary and sufficient conditions for a dimension-module automorphism to lift to a sliding-block automorphism of the shift. Merely describing all algebraic automorphisms of the ordered group does not settle liftability. The matrix $A$ is arbitrary, and both the positive cone and the commuting condition are part of the target structure.

Short English statement

Determine which automorphisms of a mixing shift’s ordered dimension module come from actual shift automorphisms.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600003 (AMR-045-0003), 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 4.1. Original source indexed by AMR-045-0003. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.