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Global properties of a family of piecewise isometries

Published online by Cambridge University Press:  01 April 2009

AREK GOETZ  and
ANTHONY QUAS
AREK GOETZ
Affiliation:
Department of Mathematics, San Francisco State University, San Francisco, CA 94132, USA (email: goetz@sfsu.edu)
ANTHONY QUAS
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, Canada V8W 3P4 (email: aquas@uvic.ca)
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Abstract

We investigate a basic system of a piecewise rotations acting on two half-planes. We prove that for invertible systems, an arbitrary neighbourhood of infinity contains infinitely many periodic points surrounded by periodic cells. In the case where the underlying rotation is rational, we show that all orbits remain bounded, whereas in the case where the underlying rotation is irrational, we show that the map is conservative (satisfies the Poincaré recurrence property). A key part of the proof is the construction of periodic orbits that shadow orbits for certain rational rotations of the plane.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 2 , April 2009 , pp. 545 - 568
Copyright
Copyright © Cambridge University Press 2008

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