Automorphism groups of full shifts

Definitions and mathematical statement

For $N\ge2$, put $X_N=\{0,\ldots,N-1\}^{\mathbb Z}$ with the product topology, and let $(\sigma_Nx)_j=x_{j+1}$. Define \[ \operatorname{Aut}(\sigma_N)= \{F:X_N\to X_N:F\text{ is a homeomorphism and } F\sigma_N=\sigma_NF\}, \] with group operation composition. By the finite-window description of continuous shift-commuting maps, each element and its inverse can be expressed by local rules on finitely many coordinates.

Are $\operatorname{Aut}(\sigma_2)$ and $\operatorname{Aut}(\sigma_3)$ isomorphic as abstract groups? An affirmative answer requires a bijection $\Psi$ between the groups satisfying $\Psi(FG)=\Psi(F)\Psi(G)$ for every pair of automorphisms; a negative answer requires an abstract group property distinguishing them.

No topology on the automorphism groups is part of the question. Nor is an isomorphism required to respect local rule radii, to send the distinguished shift to the other shift, or to be induced by a homeomorphism between the underlying configuration spaces. These extra requirements would change the target. The alphabets are fixed at two and three symbols, and both shifts are two-sided. Thus the problem compares the complete algebraic symmetry groups of the two systems, rather than asking whether the shifts themselves are dynamically conjugate or whether either symmetry group embeds into the other.

Short English statement

Are the abstract automorphism groups of the full binary and ternary shifts isomorphic?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600035 (AMR-045-0035), 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 22.1. Original source indexed by AMR-045-0035. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.
• S3 Hochman, Groups of automorphisms of SFTs; precise abstract-group formulation. https://math.huji.ac.il/~mhochman/problems/automorphisms.pdf.