On the geometry of orientation-preserving planar piecewise isometries

Abstract

Planar piecewise isometries (PWIs) are iterated mappings of subsets of the plane that preserve length (and hence angle and area) on each of a number of disjoint regions. They arise naturally in several applications and are a natural generalization of the well-studied interval exchange transformations. The aim of this paper is to propose and investigate basic properties of orientationpreserving PWIs. We develop a framework with which one can classify PWIs of a polygonal region of the plane with polygonal partition. Basic dynamical properties of such maps are discussed and a number of results are proved that relate dynamical properties of the maps to the geometry of the partition. It is shown that the set of such mappings on a given number of polygons splits into a finite number of families; we call these classes. These classes may be of varying dimension and may or may not be connected. The classification of PWIs on n triangles for n up to 3 is discussed in some detail, and several specific cases where n is larger than three are examined. To perform this classification, equivalence under similarity is considered, and an associated perturbation dimension is defined as the dimension of a class of maps modulo this equivalence. A class of PWIs is said to be rigid if this perturbation dimension is zero. A variety of rigid and nonrigid classes and several of these rigid classes of PWIs are found. In particular, those with angles that are multiples of π/n for n = 3, 4, and 5 give rise to self-similar structures in their dynamical refinements that are considerably simpler than those observed for other angles.