Measure of Self-Affine Sets and Associated Densities
نویسندگان
چکیده
منابع مشابه
Self-Affine Sets with Positive Lebesgue Measure
Using techniques introduced by C. Güntürk, we prove that the attractors of a family of overlapping self-affine iterated function systems contain a neighbourhood of zero for all parameters in a certain range. This corresponds to giving conditions under which a single sequence may serve as a ‘simultaneous β-expansion’ of different numbers in different bases.
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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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Let K ⊂ R be a self-similar or self-affine set, let μ be a self-similar or self-affine measure on it, and let G be the group of affine maps, similitudes, isometries or translations of R. Under various assumptions (such as separation conditions or we assume that the transformations are small perturbations or that K is a so called Sierpiński sponge) we prove theorems of the following types, which...
متن کاملOverlapping Self-affine Sets of Kakeya Type
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.
متن کاملUniform Perfectness of Self-affine Sets
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: Constructive Approximation
سال: 2014
ISSN: 0176-4276,1432-0940
DOI: 10.1007/s00365-014-9258-y