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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The first return time properties of an irrational rotation
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by Dong Han Kim and Kyewon Koh Park PDF
Proc. Amer. Math. Soc. 136 (2008), 3941-3951 Request permission


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
  • MR Author ID: 630927
  • Email:
  • Kyewon Koh Park
  • Affiliation: Department of Mathematics, Ajou University, Suwon 443-749, Korea
  • MR Author ID: 136240
  • Email:
  • 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
  • © Copyright 2008 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 3941-3951
  • MSC (2000): Primary 37E10, 11K50
  • DOI:
  • MathSciNet review: 2425734