Commuting powers conjecture

Definitions and mathematical statement

Let $(X,S)$ and $(Y,T)$ be nontrivial mixing two-sided shifts of finite type over finite alphabets. Nontrivial here excludes the one-point system; in this setting the spaces are Cantor spaces and have positive entropy. Mixing means that for any nonempty open sets $U,V$, every sufficiently large iterate of $U$ meets $V$.

Say that two homeomorphisms can commute if they have topologically conjugate representatives on a common compact space which commute. The commuting powers conjecture asserts that there is $N=N(S,T)$ such that for every pair of integers $i,j\ge N$ there exists a homeomorphism $H_{i,j}:Y\to X$ for which \[ S^i\left(H_{i,j}T^jH_{i,j}^{-1}\right) =\left(H_{i,j}T^jH_{i,j}^{-1}\right)S^i. \] Equivalently, $S^i$ and $T^j$ can be realized as two commuting generators of a $\mathbb Z^2$ action, each retaining its specified conjugacy class.

The same threshold must work for every sufficiently large pair, but the identification $H_{i,j}$ may depend on that pair. No common identification for all powers is requested. The nontrivial convention is necessary: a one-point mixing shift cannot be placed on the same underlying space as an infinite one. The conjecture concerns two-sided shifts, with no requirement that their defining adjacency matrices commute or have equal entropy.

Short English statement

Can every pair of nontrivial mixing finite-type shifts be made to commute after taking any sufficiently large powers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600016 (AMR-045-0016), 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 13.1. Original source indexed by AMR-045-0016. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.