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 | References | Similar Articles | Additional Information

Abstract: Consider a linear Cantor set , which is the attractor of a linear iterated function system (i.f.s.) , , on the line satisfying the open set condition (where the open set is an interval). It is known that has Hausdorff dimension given by the equation , and that is finite and positive, where denotes Hausdorff measure of dimension . We give an algorithm for computing exactly as the maximum of a finite set of elementary functions of the parameters of the i.f.s. When (or more generally, if and are commensurable), the algorithm also gives an interval that maximizes the density . The Hausdorff measure is not a continuous function of the i.f.s. parameters. We also show that given the contraction parameters , it is possible to choose the translation parameters in such a way that , so the maximum density is one. Most of the results presented here were discovered through computer experiments, but we give traditional mathematical proofs.

**[CV]**P. M. Centore and E. R. Vrscay,*Continuity of attractors and invariant measures for iterated function systems*, Canad. Math. Bull.**37 (3)**(1994), 315-329. MR**95g:58126****[F]**K. J. Falconer,*Fractal geometry: mathematical foundations and applications*, John Wiley, 1990. MR**92j:28008****[M1]**J. Marion,*Mesure de Hausdorff d'un fractal à similitude interne*, Ann. Sc. Math. Québec**10**(1986), 51-84. MR**87h:28009****[M2]**J. Marion,*Mesures de Hausdorff d'ensembles fractals*, Ann. Sc. Math. Québec**11**(1987), 111-132. MR**88k:28011****[STZ]**R. S. Strichartz, A. Taylor and T. Zhang,*Densities of self-similar measures on the line*, Experimental Math.**4**(1995), 101-128. MR**97c:28014**.

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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