Amenable Cantor actions

Definitions and mathematical statement

Let $G$ be a countable discrete amenable group. Here amenability can be expressed by a Følner sequence of finite nonempty sets $F_n\subset G$ with $|gF_n\mathbin{\triangle} F_n|/|F_n|\to0$ for every $g\in G$. Let $G$ act by homeomorphisms on a Cantor space $X$, a nonempty compact metrizable perfect totally disconnected space. Assume the action is minimal: every orbit $Gx$ is dense in $X$.

The question is whether there exists a homeomorphism $T:X\to X$ whose integer iterates have exactly the same orbit sets: \[ \{T^n x:n\in\mathbb Z\}=\{gx:g\in G\} \qquad(x\in X). \] Equivalently, is the given action topologically orbit equivalent to a minimal $\mathbb Z$-action on a Cantor space, through a homeomorphism carrying each orbit onto an orbit? Using $X$ itself loses no generality, since such a homeomorphism can transport the proposed $\mathbb Z$-action.

The original group action is not assumed free. Orbit equivalence compares orbit sets and does not require a group isomorphism or continuous orbit cocycles. Conversely a merely measurable or Borel orbit equivalence does not answer the topological question. Minimality would automatically hold for $T$ once its orbits agree with the given dense orbits. The problem concerns all countable amenable groups, beyond the special lattice actions for which stronger classification results are already available.

Short English statement

Can every minimal amenable-group action on a Cantor space be generated, at the level of orbit sets, by one homeomorphism?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600037 (AMR-045-0037), 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 23.3. Original source indexed by AMR-045-0037. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.