Generalized dimensions of measures on self-affine sets
نویسندگان
چکیده
منابع مشابه
Overlapping Self-affine Sets
We study families of possibly overlapping self-affine sets. Our main example is a family that can be considered the self-affine version of Bernoulli convolutions and was studied, in the non-overlapping case, by F. Przytycki and M. Urbański [23]. We extend their results to the overlapping region and also consider some extensions and generalizations.
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We compute the Minkowski dimension for a family of self-affine sets on R. Our result holds for every (rather than generic) set in the class. Moreover, we exhibit explicit open subsets of this class where we allow overlapping, and do not impose any conditions on the norms of the linear maps. The family under consideration was inspired by the theory of Kakeya sets.
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Let fi(x) = Aix + bi (1 ≤ i ≤ n) be affine maps of Euclidean space RN with each Ai nonsingular and each fi contractive. We prove that the self-affine set K of {f1, . . . , fn} is uniformly perfect if it is not a singleton.
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ژورنال
عنوان ژورنال: Nonlinearity
سال: 1999
ISSN: 0951-7715,1361-6544
DOI: 10.1088/0951-7715/12/4/308