Irregular sets, the $\beta$-transformation and the almost specification property
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- by Daniel J. Thompson PDF
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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.References
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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
- MR Author ID: 886080
- Email: thompson@math.psu.edu
- Received by editor(s): May 6, 2009
- Received by editor(s) in revised form: November 24, 2010
- Published electronically: May 8, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 5395-5414
- MSC (2010): Primary 37C45
- DOI: https://doi.org/10.1090/S0002-9947-2012-05540-1
- MathSciNet review: 2931333