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Piecewise isometries, uniform distribution and 3log 2−π2/8

Published online by Cambridge University Press:  23 November 2011

YITWAH CHEUNG ,
AREK GOETZ  and
ANTHONY QUAS
YITWAH CHEUNG
Affiliation:
Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, USA (email: cheung@math.sfsu.edu, goetz@sfsu.edu)
AREK GOETZ
Affiliation:
Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, USA (email: cheung@math.sfsu.edu, goetz@sfsu.edu)
ANTHONY QUAS
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria BC, V8W 3R4, Canada (email: aquas@uvic.ca)
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Abstract

We use analytic tools to study a simple family of piecewise isometries of the plane parameterized by an angle. In previous work, we showed the existence of large numbers of periodic points, each surrounded by a ‘periodic island’. We also proved conservativity of the systems as infinite measure-preserving transformations. In experiments it is observed that the periodic islands fill up a large part of the phase space and it has been asked whether the periodic islands form a set of full measure. In this paper we study the periodic islands around an important family of periodic orbits and demonstrate that for all angle parameters that are irrational multiples of π, the islands have asymptotic density in the plane of 3log 2−π2/8≈0.846.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 6 , December 2012 , pp. 1862 - 1888
Copyright
Copyright © Cambridge University Press 2011

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