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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 $G$.