# On the Geometry of Orientation-Preserving Planar Piecewise Isometries

DOI: 10.1007/s00332-002-0477-1

- Cite this article as:
- Ashwin & Fu J. Nonlinear Sci. (2002) 12: 207. doi:10.1007/s00332-002-0477-1

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## Summary.

{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 orientation-preserving 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 PWI s 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.}