Orientation of piecewise powers of a minimal homeomorphism

We show that given a compact minimal system $(X,g)$ and an element $h$ of the topological full group $\tau[g]$ of $g$, then the infinite orbits of $h$ admit a locally constant orientation with respect to the orbits of $g$. We use this to obtain a clopen partition of $(X,G)$ into minimal and periodic parts, where $G$ is any virtually polycyclic subgroup of $\tau[g]$. We also use the orientation of orbits to give a refinement of the index map and to describe the role in $\tau[g]$ of the submonoid generated by the induced transformations of $g$. Finally, we consider the problem, given a homeomorphism $h$ of the Cantor space $X$, of determining whether or not there exists a minimal homeomorphism $g$ of $X$ such that $h \in \tau[g]$.