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Irregular sets, the $ \beta$-transformation and the almost specification property


Author: Daniel J. Thompson
Journal: Trans. Amer. Math. Soc. 364 (2012), 5395-5414
MSC (2010): Primary 37C45
DOI: https://doi.org/10.1090/S0002-9947-2012-05540-1
Published electronically: May 8, 2012
MathSciNet review: 2931333
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Abstract: Let $ (X,d)$ be a compact metric space, $ f:X \mapsto X$ be a continuous map satisfying a property we call almost specification (which is slightly weaker than the $ g$-almost product property of Pfister and Sullivan), and $ \varphi : X \mapsto \mathbb{R}$ be a continuous function. We show that the set of points for which the Birkhoff average of $ \varphi $ does not exist (which we call the irregular set) is either empty or has full topological entropy. Every $ \beta $-shift satisfies almost specification and we show that the irregular set for any $ \beta $-shift or $ \beta $-transformation is either empty or has full topological entropy and Hausdorff dimension.


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Additional Information

Daniel J. Thompson
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Address at time of publication: Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, Ohio 43210
Email: thompson@math.psu.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05540-1
Received by editor(s): May 6, 2009
Received by editor(s) in revised form: November 24, 2010
Published electronically: May 8, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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