Commuting expansive automorphisms

Definitions and mathematical statement

Let $X$ be a two-sided shift of finite type on a finite alphabet, with shift $\sigma$. Assume it is irreducible: for any two words occurring in $X$, some finite connecting word makes their concatenation occur in $X$. Equivalently, a nonempty edge presentation may be chosen with strongly connected directed graph. Let $F:X\to X$ be an automorphism, meaning a homeomorphism satisfying $F\sigma=\sigma F$.

Suppose that $F$, considered as a dynamical system in its own right, is expansive. Thus for a compatible metric $d$ on $X$ there exists $\delta>0$ such that \[ \left[d(F^n x,F^n y)<\delta\text{ for every }n\in\mathbb Z\right] \quad\Longrightarrow\quad x=y. \] Must there exist a finite alphabet $B$, a two-sided finite-type shift $Y\subset B^{\mathbb Z}$, and a homeomorphism $H:X\to Y$ such that $HF=\sigma_YH$?

The finite-type conclusion concerns the dynamics of $F$, which need not be the original shift $\sigma$. It asks for some finite forbidden-word presentation after a conjugacy, rather than requiring the coordinate description already used for $X$ to serve that purpose. Expansiveness is two-sided in time, and no positive expansiveness assumption is made. Irreducibility is retained throughout: allowing arbitrary reducible finite-type shifts changes the question and admits the counterexamples discussed in the source.

Short English statement

Must an expansive automorphism of an irreducible finite-type shift itself admit a finite-type symbolic presentation?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600014 (AMR-045-0014), 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 12.1. Original source indexed by AMR-045-0014. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.