  Berend gives necessary and sufficient conditions on a $Z^r$-action $\alpha$
on a torus $T^d$ by toral automorphisms in order for every orbit be either
finite or dense. One of these conditions is that on every common eigendirection
of the $Z^r$-action there is an element $\mathbf n \in Z^r$ so that
$\alpha^\mathbf n$ expands this direction. In this paper, we investigate what
happens when this condition is removed; more generally, we consider a partial
orbit $\{\alpha^\mathbf n.x : \mathbf n \in \Omega\}$ where $\Omega$ is a set
of elements which acts approximately as the identity on a given set of
eigendirections. This analysis is used in an essential way in the work of the
author with E. Lindenstrauss classifying topological self-joinings of maximal
$Z^r$-actions on tori for $r\geq 3$.
