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Classifying Cantor sets by their fractal dimensions
Author(s):
Carlos
A.
Cabrelli;
Kathryn
E.
Hare;
Ursula
M.
Molter
Journal:
Proc. Amer. Math. Soc.
138
(2010),
3965-3974.
MSC (2010):
Primary 28A78, 28A80
Posted:
May 14, 2010
MathSciNet review:
2679618
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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  -Hausdorff and  -packing measures, for the family of dimension functions  , and characterize this classification in terms of the underlying sequences.
References:
-
- 1.
- 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 0064849 (16:344d)
- 2.
- C. Cabrelli, F. Mendivil, U. Molter and R. Shonkwiler, On the
 -Hausdorff measure of Cantor sets, Pac. J. of Math. 217(2004), 29-43. MR 2105765 (2005h:28013) - 3.
- C. Cabrelli, U. Molter, V. Paulauskas and R. Shonkwiler, The Hausdorff dimension of
 -Cantor sets, Real Anal. Exchange 30(2004/05), no. 2, 413-433. MR 2177411 (2006g:28012) - 4.
- K. Falconer, Techniques in fractal geometry, Wiley and Sons, Chichester, 1997. MR 1449135 (99f:28013)
- 5.
- I. Garcia, U. Molter and R. Scotto, Dimension functions of Cantor sets, Proc. Amer. Math. Soc. 135(2007), 3151-3161. MR 2322745 (2008i:28004)
- 6.
- C. A. Rogers, Hausdorff measures, Cambridge Math Library, Cambridge University Press, Cambridge, 1998. MR 1692618 (2000b:28009)
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Additional 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
Email:
cabrelli@dm.uba.ar
Kathryn
E.
Hare
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada
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
Email:
umolter@dm.uba.ar
DOI:
10.1090/S0002-9939-2010-10396-9
PII:
S 0002-9939(2010)10396-9
Keywords:
Hausdorff dimension,
packing dimension,
Cantor set,
cut-out set
Received by editor(s):
May 11, 2009
Received by editor(s) in revised form:
January 15, 2010
Posted:
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 of article:
Copyright
2010,
American Mathematical Society
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