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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The first return time properties of an irrational rotation


Authors: Dong Han Kim and Kyewon Koh Park
Journal: Proc. Amer. Math. Soc. 136 (2008), 3941-3951
MSC (2000): Primary 37E10, 11K50
Published electronically: June 2, 2008
MathSciNet review: 2425734
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Abstract: If an ergodic system has positive entropy, then the Shannon-McMillan-Breiman theorem provides a relationship between the entropy and the size of an atom of the iterated partition. The system also has Ornstein-Weiss' first return time property, which offers a method of computing the entropy via an orbit. We consider irrational rotations which are the simplest model of zero entropy. We prove that almost every irrational rotation has the analogous properties if properly normalized. However there are some irrational rotations that exhibit different behavior.


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Additional Information

Dong Han Kim
Affiliation: Department of Mathematics, The University of Suwon, Hwaseong 445-743, Korea
Email: kimdh@suwon.ac.kr

Kyewon Koh Park
Affiliation: Department of Mathematics, Ajou University, Suwon 443-749, Korea
Email: kkpark@ajou.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09388-X
PII: S 0002-9939(08)09388-X
Keywords: Recurrence time, the first return time, irrational rotations
Received by editor(s): June 1, 2007
Received by editor(s) in revised form: October 2, 2007
Published electronically: June 2, 2008
Additional Notes: The first author was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-331-C00016).
The second author was supported in part by KRF 2007-313-C00044
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2008 American Mathematical Society