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Polygonal invariant curves for a planar piecewise isometry

Authors: Peter Ashwin and Arek Goetz
Journal: Trans. Amer. Math. Soc. 358 (2006), 373-390
MSC (2000): Primary 37B10, 37E15; Secondary 11R11, 20C20, 68W30
Published electronically: March 31, 2005
MathSciNet review: 2171238
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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.

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Additional Information

Peter Ashwin
Affiliation: Department of Mathematical Sciences, University of Exeter, Exeter EX4 4QE, United Kingdom

Arek Goetz
Affiliation: Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, California 94132

Keywords: Piecewise isometry, interval exchange transformation, cells
Received by editor(s): November 22, 2002
Received by editor(s) in revised form: March 22, 2004
Published electronically: March 31, 2005
Additional Notes: The work on this article commenced during the second author’s visit to Exeter sponsored by the LMS. The second author was partially supported by NSF research grant DMS 0103882, and the San Francisco State Presidential Research Leave. We thank Michael Boshernitzan for interesting conversations and helpful suggestions. Symbolic computations were aided by Mathematica routines, some of which were developed in connection with a project by Goetz and Poggiaspalla.
Article copyright: © Copyright 2005 American Mathematical Society

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