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On the geometry of orientation-preserving planar piecewise isometries
University of Exeter
Journal of Nonlinear Science
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.
Copyright © Springer 2002. NOTICE: This is the author’s pre-print version of a work accepted for publication by Springer. The original publication is available at www.springerlink.com
12 (3), pp. 207-240