Dynamics of flat actions on totally disconnected, locally compact groups

Let G be a totally disconnected, locally compact group and let H be a virtually flat (for example, polycyclic) group of automorphisms of G. We study the structure of, and relationships between, various subgroups of G defined by the dynamics of H. In particular, we consider the following four subgroups: the intersection of all tidy subgroups for H on G (in the case that H is flat); the intersection of all H-invariant open subgroups of G; the smallest closed H-invariant subgroup D such that no H-orbit on G/D accumulates at the trivial coset; and the group generated by the closures of contraction groups of elements of H on G.