Expansive directions of two-dimensional SFTs

Definitions and mathematical statement

Let $X\subset A^{\mathbb Z^2}$ be an infinite nonempty subshift over a finite alphabet, defined by finitely many forbidden finite patterns. Its shift action is $(\alpha^v x)(u)=x(u+v)$ for $u,v\in\mathbb Z^2$. Fix any metric $d_X$ compatible with the product topology. A line $L\subset\mathbb R^2$ through the origin is expansive if there exist $R,\delta>0$ such that \[ d_X(\alpha^v x,\alpha^v y)<\delta\quad \text{for every }v\in\mathbb Z^2\text{ with }\operatorname{dist}(v,L)<R \quad\Longrightarrow\quad x=y. \] Define $E_1(\alpha)=\{w\in\mathbb R^2\setminus\{0\}:\mathbb Rw \text{ is expansive}\}$. This definition is independent of the chosen compatible metric.

Impose the additional hypothesis that, for at least one nonzero $v_0\in\mathbb Z^2$, the single homeomorphism $\alpha^{v_0}:X\to X$ is topologically conjugate to a two-sided shift of finite type. Characterize precisely the sets $E\subset\mathbb R^2\setminus\{0\}$ that can arise as $E_1(\alpha)$ under this hypothesis. A classification must give both necessary restrictions and realizations of every permitted set.

Directions are unoriented lines, so $w$ and $-w$ determine the same test; the set records all real directions, including irrational slopes. This is the constrained part of Boyle's Problem 11.1. The unrestricted realization question, without the finite-type element hypothesis, is recorded as solved on the author's update page.

Short English statement

Which expansive directions can a two-dimensional finite-type shift have when one lattice direction itself acts as a finite-type shift?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600012 (AMR-045-0012), 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 11.1. Original source indexed by AMR-045-0012. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.
• S3 Boyle, solution updates for Problem 11.1(i). https://math.umd.edu/~mboyle/open/.