Dynamical Parallelepipeds in Minimal Systems

For a topological dynamical system ( X , T ) and d ∈ N , the associated dynamical parallelepiped Q [ d ] was defined by Host–Kra–Maass. For a minimal distal system it was shown by them that the relation ∼ d − 1 defined on Q [ d − 1 ] is an equivalence relation; the closing parallelepiped property holds, and for each x ∈ X the collection of points in Q [ d ] with first coordinate x is a minimal subset under the face transformations. We give examples showing that the results do not extend to general minimal systems.