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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 }}$.

References [Enhancements On Off] (What's this?)

  • [F1] K. J. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 2, 339–350. MR 923687, 10.1017/S0305004100064926
  • [F] -, Fractal geometry, Cambridge Univ. Press, Cambridge, 1990.
  • [KSS] M. Keane, M. Smorodinsky and B. Solomyak, Cantor criticality, Talk delivered by M. Keane (Warwick conference on $ {\mathbb{Z}^n}$-actions), September 1993.
  • [PT] Jacob Palis and Floris Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge University Press, Cambridge, 1993. Fractal dimensions and infinitely many attractors. MR 1237641

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Keywords: Hausdorff dimension, box dimension, potential method, $ \lambda $-expansions, Newhouse thickness
Article copyright: © Copyright 1995 American Mathematical Society