Orientation of piecewise powers of a minimal homeomorphism

We show that, given a compact minimal system and an element h of the topological full group of g, 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 into minimal and periodic parts, where G is any virtually polycyclic subgroup of. We also use the orientation of orbits to give a refinement of the index map and to describe the role in 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.