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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Polygonal invariant curves for a planar piecewise isometry
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by Peter Ashwin and Arek Goetz PDF
Trans. Amer. Math. Soc. 358 (2006), 373-390 Request permission


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
  • 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.
  • © Copyright 2005 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 373-390
  • MSC (2000): Primary 37B10, 37E15; Secondary 11R11, 20C20, 68W30
  • DOI:
  • MathSciNet review: 2171238