Sum of Cantor sets: Self-similarity and measure

Author:
Pedro Mendes

Journal:
Proc. Amer. Math. Soc. **127** (1999), 3305-3308

MSC (1991):
Primary 28A78, 58F14

DOI:
https://doi.org/10.1090/S0002-9939-99-05107-2

Published electronically:
May 13, 1999

MathSciNet review:
1637408

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Abstract | References | Similar Articles | Additional Information

Abstract: In this note it is shown that the sum of two homogeneous Cantor sets is often a uniformly contracting self-similar set and it is given a sufficient condition for such a set to be of Lebesgue measure zero (in fact, of Hausdorff dimension less than one and positive Hausdorff measure at this dimension).

**1.**C. Brant, S. Graf,*Self-similar sets 7. A characterization of self-similar fractals with positive Hausdorff measure*, Proc. Amer. Math. Soc.**114**(1992), 995 - 1001. MR**93d:28014****2.**J. E. Hutchinson,*Fractals and self-similarity*, Indiana Univ. Math. J.**30**(1981), 713 - 747. MR**82h:49026****3.**B. Mandelbrot,,*Fractals, Form, Chance and Dimension*, Freeman, San Francisco, 1977. MR**57:11224****4.**P. Mendes, F. Oliveira,*On the topological structure of the arithmetic sum of two Cantor sets.*, Nonlinearity**7**(1994), 329 - 343. MR**95j:58123****5.**C. G. T. de A. Moreira,*Stable intersections of Cantor sets and homoclinic bifurcations*, Ann. Inst. Henri Poincaré: Analyse non linéaire**13**(1996), 741 - 781.**6.**C. G. T. de A. Moreira, C. Yoccoz,*Stable intersections of Cantor sets with large Hausdorff dimension*(to appear).**7.**J. Palis, F. Takens,*Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations: fractal dimensions and infinitely many attractors.*, Cambridge Univ. Press, 1993. MR**94h:58129****8.**B. Solomyak,*On the measure of arithmetic sums of Cantor sets.*, Indag. Mathem., N.S.**8**(1997), 133 - 141. CMP**98:12****9.**M. P, W. Zerner,*Weak separation properties for self-similar sets*, Proc. Amer. Math. Soc.**124**(1996), 3529 - 3539. MR**97c:54035**

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Additional Information

**Pedro Mendes**

Affiliation:
Departamento de Matemática, ICEx, UFMG Av. Antonio Carlos 6627 31270.901 Belo Horizonte, MG, Brazil

Email:
pmendes@mat.ufmg.br

DOI:
https://doi.org/10.1090/S0002-9939-99-05107-2

Keywords:
Self-similar set,
homogeneous Cantor set,
Hausdorff dimension,
Hausdorff measure

Received by editor(s):
February 6, 1998

Published electronically:
May 13, 1999

Communicated by:
Michael Handel

Article copyright:
© Copyright 1999
American Mathematical Society