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Polygonal invariant curves for a planar piecewise isometry
Author(s):
Peter
Ashwin;
Arek
Goetz
Journal:
Trans. Amer. Math. Soc.
358
(2006),
373-390.
MSC (2000):
Primary 37B10, 37E15;
Secondary 11R11, 20C20, 68W30
Posted:
March 31, 2005
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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
DOI:
10.1090/S0002-9947-05-03670-6
PII:
S 0002-9947(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
Posted:
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 {\em Mathematica}
routines, some of which were developed in connection
with a project
by Goetz and Poggiaspalla.
Copyright of article:
Copyright
2005,
American Mathematical Society
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