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Cone exchange transformations and boundedness of orbits

Published online by Cambridge University Press:  07 September 2009

PETER ASHWIN  and
AREK GOETZ
PETER ASHWIN
Affiliation:
Mathematics Research Institute, University of Exeter, Exeter EX4 4QF, UK (email: P.Ashwin@ex.ac.uk)
AREK GOETZ
Affiliation:
San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, USA (email: goetz@sfsu.edu)
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Abstract

We introduce a class of two-dimensional piecewise isometries on the plane that we refer to as cone exchange transformations (CETs). These are generalizations of interval exchange transformations (IETs) to 2D unbounded domains. We show for a typical CET that boundedness of orbits is determined by ergodic properties of an associated IET and a quantity we refer to as the ‘flux at infinity’. In particular we show, under an assumption of unique ergodicity of the associated IET, that a positive flux at infinity implies unboundedness of almost all orbits outside some bounded region, while a negative flux at infinity implies boundedness of all orbits. We also discuss some examples of CETs for which the flux is zero and/or we do not have unique ergodicity of the associated IET; in these cases (which are of great interest from the point of view of applications such as dual billiards) it remains an outstanding problem to find computable necessary and sufficient conditions for boundedness of orbits.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 5 , October 2010 , pp. 1311 - 1330
Copyright
Copyright © Cambridge University Press 2009

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