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 | References | Similar Articles | Additional Information

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

DOI:
http://dx.doi.org/10.1090/S0002-9947-05-03670-6

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.

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© Copyright 2005
American Mathematical Society