Classifying Cantor sets by their fractal dimensions
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- by Carlos A. Cabrelli, Kathryn E. Hare and Ursula M. Molter
- Proc. Amer. Math. Soc. 138 (2010), 3965-3974
- DOI: https://doi.org/10.1090/S0002-9939-2010-10396-9
- Published electronically: May 14, 2010
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Abstract:
In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovich and Taylor. We classify these Cantor sets in terms of their $h$-Hausdorff and $h$-packing measures, for the family of dimension functions $h$, and characterize this classification in terms of the underlying sequences.References
- A. S. Besicovitch and S. J. Taylor, On the complementary intervals of a linear closed set of zero Lebesgue measure, J. London Math. Soc. 29 (1954), 449–459. MR 64849, DOI 10.1112/jlms/s1-29.4.449
- Carlos Cabrelli, Franklin Mendivil, Ursula M. Molter, and Ronald Shonkwiler, On the Hausdorff $h$-measure of Cantor sets, Pacific J. Math. 217 (2004), no. 1, 45–59. MR 2105765, DOI 10.2140/pjm.2004.217.45
- C. Cabrelli, U. Molter, V. Paulauskas, and R. Shonkwiler, Hausdorff measure of $p$-Cantor sets, Real Anal. Exchange 30 (2004/05), no. 2, 413–433. MR 2177411, DOI 10.14321/realanalexch.30.2.0413
- Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
- Ignacio Garcia, Ursula Molter, and Roberto Scotto, Dimension functions of Cantor sets, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3151–3161. MR 2322745, DOI 10.1090/S0002-9939-07-09019-3
- C. A. Rogers, Hausdorff measures, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1970 original; With a foreword by K. J. Falconer. MR 1692618
- Claude Tricot Jr., Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 1, 57–74. MR 633256, DOI 10.1017/S0305004100059119
- S. James Taylor and Claude Tricot, Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), no. 2, 679–699. MR 776398, DOI 10.1090/S0002-9947-1985-0776398-8
Bibliographic Information
- Carlos A. Cabrelli
- Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, C1428EGA C.A.B.A., Argentina – and – CONICET, Argentina
- MR Author ID: 308374
- ORCID: 0000-0002-6473-2636
- Email: cabrelli@dm.uba.ar
- Kathryn E. Hare
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada
- MR Author ID: 246969
- Email: kehare@uwaterloo.edu
- Ursula M. Molter
- Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, C1428EGA C.A.B.A., Argentina – and – CONICET, Argentina
- MR Author ID: 126270
- Email: umolter@dm.uba.ar
- Received by editor(s): May 11, 2009
- Received by editor(s) in revised form: January 15, 2010
- Published electronically: May 14, 2010
- Additional Notes: The first and third authors were partially supported by Grants UBACyT X149 and X028 (UBA), PICT 2006-00177 (ANPCyT), and PIP 112-200801-00398 (CONICET)
The second author was partially supported by NSERC - Communicated by: Michael T. Lacey
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 3965-3974
- MSC (2010): Primary 28A78, 28A80
- DOI: https://doi.org/10.1090/S0002-9939-2010-10396-9
- MathSciNet review: 2679618