Pingree open problems — Boyle problem 1

Definitions and mathematical statement

A two-sided subshift is a closed subset $X\subset A^{\mathbb Z}$, for a finite discrete alphabet $A$, invariant under the homeomorphism $(\sigma x)_j=x_{j+1}$ and its inverse. The topology is the product topology. It is a shift of finite type (SFT) if a finite list of forbidden finite words defines $X$. It is mixing if for any nonempty relatively open $U,V\subset X$ there is $N$ such that $\sigma^n(U)\cap V\ne\varnothing$ for every $n\ge N$.

Two such systems $(X,\sigma)$ and $(Y,\tau)$ are topologically orbit equivalent if there is a homeomorphism $H:X\to Y$ satisfying \[ H(\{\sigma^n x:n\in\mathbb Z\}) =\{\tau^n H(x):n\in\mathbb Z\}\qquad(x\in X). \] No requirement that $H\sigma=\tau H$ is imposed, and the integers relating successive images need not form a continuous cocycle. The target is a classification of mixing SFTs under this equivalence: give a complete set of invariants, or necessary and sufficient conditions on their finite graph presentations, for such an $H$ to exist.

An orbit homeomorphism preserves the cardinality of every finite orbit; therefore equal numbers of periodic orbits of each length are necessary. The classification must account for infinite orbits and their accumulation as well. In particular, merely matching finite orbit counts does not by itself provide the required homeomorphism of the compact spaces.

Short English statement

Classify mixing shifts defined by finitely many local rules when a homeomorphism is allowed to rearrange time along each orbit.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4400007 (AMR-043-0007), 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 1. Original source indexed by AMR-043-0007. https://math.huji.ac.il/~mhochman/open-problems/pingree-open-problems.pdf.