On the geometry of orientation-preserving planar piecewise isometries

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On the geometry of orientation-preserving planar piecewise isometries

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dc.contributor.author Ashwin, Peter en_GB
dc.contributor.author Fu, Xin-Chu en_GB
dc.contributor.department University of Exeter en_GB
dc.date.accessioned 2009-04-08T13:48:40Z en_GB
dc.date.accessioned 2011-01-25T10:33:54Z en_US
dc.date.accessioned 2013-03-20T12:30:48Z
dc.date.issued 2002 en_GB
dc.description.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. en_GB
dc.identifier.citation 12 (3), pp. 207-240 en_GB
dc.identifier.doi 10.1007/s00332-002-0477-1 en_GB
dc.identifier.uri http://hdl.handle.net/10036/64713 en_GB
dc.language.iso en en_GB
dc.publisher Springer en_GB
dc.relation.url http://www.springerlink.com/content/jr1d7qjn9y00vmr0/ en_GB
dc.relation.url http://dx.doi.org/10.1007/s00332-002-0477-1 en_GB
dc.subject piecewise isometry en_GB
dc.subject discontinuous dynamics en_GB
dc.subject nonhyperbolic dynamics en_GB
dc.title On the geometry of orientation-preserving planar piecewise isometries en_GB
dc.type Article en_GB
dc.type Preprint en_GB
dc.date.available 2009-04-08T13:48:40Z en_GB
dc.date.available 2011-01-25T10:33:54Z en_US
dc.date.available 2013-03-20T12:30:48Z
dc.identifier.issn 0938-8974 en_GB
dc.identifier.issn 1432-1467 en_GB
dc.description 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 en_GB
dc.identifier.journal Journal of Nonlinear Science en_GB


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