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.



Topological full groups and minimal orbit closures
Having worked on this problem, I can now say the answer is yes, $h$ has minimal orbit closures. Actually YCor's comment is spot on: analogous to Keane's interval exchange transformations, the space decomposes into finitely many clopen $\langle h \rangle$-invariant pieces such that on each piece, the restriction of $h$ is either finite order or minimal. The key idea is something François Le Maître suggested to me, which is to think about infinite orbits of $h$ as moving either 'in the same direction' as the $g$-orbit or 'in the opposite direction'. The main arguments are in section 3 of the following article:


<https://arxiv.org/abs/1812.00480>


