Pingree open problems — Boyle problem 2

Definitions and mathematical statement

Let $X\subset A^{\mathbb Z}$ and $Y\subset B^{\mathbb Z}$ be nonempty two-sided subshifts over finite alphabets, with shifts $\sigma$ and $\tau$. Both carry their product topologies. Assume that $X$ is defined by finitely many forbidden words and is topologically mixing: for every pair of nonempty open sets $U,V\subset X$, all sufficiently large $n$ satisfy $\sigma^n(U)\cap V\ne\varnothing$.

Suppose a homeomorphism $H:X\to Y$ carries each entire shift orbit onto an entire shift orbit, meaning \[ H(\{\sigma^n x:n\in\mathbb Z\}) =\{\tau^n H(x):n\in\mathbb Z\}\quad\text{for every }x\in X. \] Must $Y$ also be a mixing shift of finite type? Both conclusions are part of the question: a finite forbidden-word presentation must exist for $Y$, and its shift must be mixing.

The map $H$ is not assumed to commute with the shifts, preserve the direction of time, or have a continuous integer-valued orbit cocycle. Thus one cannot transfer a defining local rule through $H$ as through a sliding block conjugacy. The assumption that the target is a subshift is substantive: it requires a finite symbolic alphabet and an expansive shift homeomorphism. The question concerns orbit equivalence of the whole compact systems, including their periodic points, rather than equivalence after discarding exceptional or measure-zero subsets.

Short English statement

Does topological orbit equivalence to a mixing finite-type shift force another subshift to be mixing and of finite type?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4400008 (AMR-043-0008), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open problems from the 3rd Pingree Workshop on Dynamical Systems. (2010), Mike Boyle, Problem 2. Original source indexed by AMR-043-0008. https://math.huji.ac.il/~mhochman/open-problems/pingree-open-problems.pdf.