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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Hausdorff dimension of $ \lambda$-expansions with deleted digits


Authors: Mark Pollicott and Károly Simon
Journal: Trans. Amer. Math. Soc. 347 (1995), 967-983
MSC: Primary 11K55; Secondary 28A78
MathSciNet review: 1290729
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Abstract: In this article we examine the continuity of the Hausdorff dimension of the one parameter family of Cantor sets $ \Lambda (\lambda ) = \{ \sum\nolimits_{k = 1}^\infty {{i_k}{\lambda ^k}:{i_k} \in S\} } $, where $ S \subset \{ 0,1, \ldots ,(n - 1)\} $. In particular, we show that for almost all (Lebesgue) $ \lambda \in [\tfrac{1} {n},\tfrac{1} {l}]$ we have that $ {\dim _H}(\Lambda (\lambda )) = \frac{{\log l}} {{ - \log \lambda }}$ where $ l = \operatorname{Card} (S)$. In contrast, we show that under appropriate conditions on $ S$ we have that for a dense set of $ \lambda \in [\tfrac{1} {n},\tfrac{1} {l}]$ we have $ {\dim _H}(\Lambda (\lambda )) < \frac{{\log l}} {{ - \log \lambda }}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1290729-0
PII: S 0002-9947(1995)1290729-0
Keywords: Hausdorff dimension, box dimension, potential method, $ \lambda $-expansions, Newhouse thickness
Article copyright: © Copyright 1995 American Mathematical Society