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Polygonal Invariant Curves for a Planar Piecewise Isometry

Peter Ashwin and Arek Goetz
Transactions of the American Mathematical Society
Vol. 358, No. 1 (Jan., 2006), pp. 373-390
Stable URL: http://www.jstor.org/stable/3845462
Page Count: 18
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Polygonal Invariant Curves for a Planar Piecewise Isometry
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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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