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The first return time properties of an irrational rotation
Author(s):
Dong Han
Kim;
Kyewon
Koh
Park
Journal:
Proc. Amer. Math. Soc.
136
(2008),
3941-3951.
MSC (2000):
Primary 37E10, 11K50
Posted:
June 2, 2008
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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:
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
Posted:
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 of article:
Copyright
2008,
American Mathematical Society
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