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Existence and Properties of ℎ-Sets

Existence and Properties of ℎ-Sets In this note we shall consider the following problem: which conditions should satisfy a function ℎ : (0, 1) → ℝ in order to guarantee the existence of a (regular) measure μ in with compact support and for some positive constants 𝑐 2 , and 𝑐 2 independent of γ ∈ Γ and 𝑟 ∈ (0,1)? The theory of self-similar fractals provides outstanding examples of sets fulfilling (♡) with ℎ(𝑟) = 𝑟 𝑑 , 0 ≤ 𝑑 ≤ 𝑛, and a suitable measure μ . Analogously, we shall rely on some recent techniques for the construction of pseudo self-similar fractals in order to deal with our more general task. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Georgian Mathematical Journal de Gruyter

Existence and Properties of ℎ-Sets

Bricchi, Michele
Georgian Mathematical Journal , Volume 9 (1) – Mar 1, 2002
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Publisher
de Gruyter
Copyright
© Heldermann Verlag
ISSN
1072-947X
eISSN
1072-9176
DOI
10.1515/GMJ.2002.13
Publisher site
See Article on Publisher Site

Abstract

In this note we shall consider the following problem: which conditions should satisfy a function ℎ : (0, 1) → ℝ in order to guarantee the existence of a (regular) measure μ in with compact support and for some positive constants 𝑐 2 , and 𝑐 2 independent of γ ∈ Γ and 𝑟 ∈ (0,1)? The theory of self-similar fractals provides outstanding examples of sets fulfilling (♡) with ℎ(𝑟) = 𝑟 𝑑 , 0 ≤ 𝑑 ≤ 𝑛, and a suitable measure μ . Analogously, we shall rely on some recent techniques for the construction of pseudo self-similar fractals in order to deal with our more general task.

Journal

Georgian Mathematical Journal – de Gruyter

Published: Mar 1, 2002

Keywords: 𝑑-set; Ahlfors-type measure; pseudo self-similar fractal; Hausdorff ℎ-measure

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