Polygonal invariant curves for a planar piecewise isometry

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Polygonal invariant curves for a planar piecewise isometry

Please use this identifier to cite or link to this item: http://hdl.handle.net/10036/19239

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Title: Polygonal invariant curves for a planar piecewise isometry
Author: Ashwin, Peter
Goetz, Arek
Citation: 358 (1), pp. 373-390
Publisher: American Mathematical Society
Journal: Transactions of the American Mathematical Society
Date Issued: 2005
URI: http://hdl.handle.net/10036/19239
Links: http://www.ams.org/tran/2006-358-01/S0002-9947-05-03670-6/home.html
Abstract: We investigate a remarkable new planar piecewise isometry whose generating map is a permutation of four cones. For this system we prove the coexistence of an infinite number of periodic components and an uncountable number of transitive components. The union of all periodic components is an invariant pentagon with unequal sides. Transitive components are invariant curves on which the dynamics are conjugate to a transitive interval exchange. The restriction of the map to the invariant pentagonal region is the first known piecewise isometric system for which there exist an infinite number of periodic components but the only aperiodic points are on the boundary of the region. The proofs are based on exact calculations in a rational cyclotomic field. We use the system to shed some light on a conjecture that PWIs can possess transitive invariant curves that are not smooth.
Type: Article
Description: Copyright © 2005 American Mathematical Society. First published in Transactions of the American Mathematical Society in Vol. 358 no. 1 (2005), published by the American Mathematical Society.
Keywords: piecewise isometryinterval exchange transformationcells
ISSN: 0002-9947


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