Summary: We consider the behavior of piecewise isometries in Euclidean spaces. We show that if is odd and the system contains no orientation-reversing isometries, then recurrent orbits with rational coding are not expected. More precisely, a prevalent set of piecewise isometrics do not have recurrent points having rational coding. This implies that when all atoms are convex, no periodic points exist for almost every piecewise isometry.
By contrast, if is even then periodic points are stable for almost every piecewise isometry whose set of defining isometries are not orientation reversing. If, in addition, the defining isometries satisfy an incommensurability condition then all unbounded orbits must be irrationally coded.