Let $X$ be the Cantor set, and let $g$ be a minimal homeomorphism of $X$. Let $h$ be a homeomorphism in the topological full group of $g$, that is, for every $x \in X$, there is a neighbourhood of $x$ such that $h$ restricts to a power of $g$ on that neighbourhood. Write $\langle h \rangle$ for the group generated by $h$.

Certainly, the action of $\langle h \rangle$ is not minimal in general, because it can preserve proper nonempty clopen subsets of $X$. But suppose we take the closure $Y$ of an $\langle h \rangle$-orbit. Does $\langle h \rangle$ act minimally on $Y$?

Equivalently, let $x \in X$, let $U$ be a neighbourhood of $x$ and let $T$ be the set of elements $h^n$ of $\langle h \rangle$ such that $h^nx \in U$. Is $T$ necessarily syndetic in $\langle h \rangle$, i.e. $\langle h \rangle = FT$ for some finite set $F$?

There is a lot known about topological full groups of minimal homeomorphisms on the Cantor set, but I couldn't find any reference to this property.

Keane, Michael Interval exchange transformations. Math. Z. 141 (1975), 25-31): essentially an interval exchange $f$ yields a decomposition in multitintervals $I_0\sqcup \dots \sqcup I_n$ with each $I_j$ $f$-invariant, f has finite order on $I_0$ and is minimal on $I_j$ for all $j\ge 1$. It would be interesting to find a similar result in the context of topological-full groups. $\endgroup$