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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Exact Hausdorff measure and intervals of maximum density for Cantor sets


Authors: Elizabeth Ayer and Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 351 (1999), 3725-3741
MSC (1991): Primary 28A80, 28A78
DOI: https://doi.org/10.1090/S0002-9947-99-01982-0
Published electronically: January 26, 1999
MathSciNet review: 1433110
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Abstract: Consider a linear Cantor set $K$, which is the attractor of a linear iterated function system (i.f.s.) $S_{j}x = \rho _{j}x+b_{j}$, $j = 1,\ldots ,m$, on the line satisfying the open set condition (where the open set is an interval). It is known that $K$ has Hausdorff dimension $\alpha $ given by the equation $\sum ^{m}_{j=1} \rho ^{\alpha }_{j} = 1$, and that $\mathcal{H}_{\alpha }(K)$ is finite and positive, where $\mathcal{H}_{\alpha }$ denotes Hausdorff measure of dimension $\alpha $. We give an algorithm for computing $\mathcal{H}_{\alpha }(K)$ exactly as the maximum of a finite set of elementary functions of the parameters of the i.f.s. When $\rho _{1} = \rho _{m}$ (or more generally, if $\log \rho _{1}$ and $\log \rho _{m}$ are commensurable), the algorithm also gives an interval $I$ that maximizes the density $d(I) = \mathcal{H}_{\alpha }(K \cap I)/|I|^{\alpha }$. The Hausdorff measure $\mathcal{H}_{\alpha }(K)$ is not a continuous function of the i.f.s. parameters. We also show that given the contraction parameters $\rho _{j}$, it is possible to choose the translation parameters $b_{j}$ in such a way that $\mathcal{H}_{\alpha }(K) = |K|^{\alpha }$, so the maximum density is one. Most of the results presented here were discovered through computer experiments, but we give traditional mathematical proofs.


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Additional Information

Elizabeth Ayer
Affiliation: Department of Mathematics, Duke University, Durham, North Carolina 27708
Address at time of publication: Churchill College, Cambridge, CB3 ODS, U.K.
Email: eca23@cus.cam.ac.uk

Robert S. Strichartz
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
Email: str@math.cornell.edu

DOI: https://doi.org/10.1090/S0002-9947-99-01982-0
Received by editor(s): July 28, 1995
Received by editor(s) in revised form: November 13, 1996
Published electronically: January 26, 1999
Additional Notes: Research supported by the National Science Foundation through the REU program (Ayer) and through Grant DMS–9303718 (Strichartz)
Article copyright: © Copyright 1999 American Mathematical Society