Tangent-Free Property for Periodic Cells Generated by Some General Piecewise Isometries

Iterating an orientation-preserving piecewise isometry of -dimensional Euclidean space, the phase space can be partitioned with full measure into the union of the rational set consisting of periodically coded points, and the complement of the rational set is usually called the exceptional set. The tangencies between the periodic cells have been studied in some previous papers, and the results showed that almost all disk packings for certain families of planar piecewise isometries have no tangencies. In this paper, the authors further investigate the structure of any periodic cells for a general piecewise isometry of even dimensional Euclidean space and the tangencies between the periodic cells. First, we show that each periodic cell is a symmetrical body to a center if the piecewise isometry is irrational; this result is a generalization of the results in some previously published papers. Second, we show that the periodic cell packing induced by an invertible irrational planar piecewise rotation, such as the Sigma-Delta map and the overflow map, has no tangencies. And furthermore, we generalize the result to general even dimensional Euclidean spaces. Our results generalize and strengthen former research results on this topic.