Some typical properties of dimensions of sets and measures.

Myjak, Józef

Displaying similar documents to “Some typical properties of dimensions of sets and measures.”

Dimension of a measure

Pertti Mattila, Manuel Morán, José-Manuel Rey (2000)

Studia Mathematica

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We propose a framework to define dimensions of Borel measures in a metric space by formulating a set of natural properties for a measure-dimension mapping, namely monotonicity, bi-Lipschitz invariance, (σ-)stability, etc. We study the behaviour of most popular definitions of measure dimensions in regard to our list, with special attention to the standard correlation dimensions and their modified versions.

Thickness measures for Cantor sets.

Astels, S. (1999)

Electronic Research Announcements of the American Mathematical Society [electronic only]

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The multifractal spectrum of discrete measures

Vincenzo Aversa, Christoph Bandt (1990)

Acta Universitatis Carolinae. Mathematica et Physica

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On the connection between Hausdorff measures and generalized capacity

R. Darst (1973)

Fundamenta Mathematicae

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Dimension of measures

Herrmann Haase (1990)

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Some generic properties of concentration dimension of measure

Józef Myjak, Tomasz Szarek (2003)

Bollettino dell'Unione Matematica Italiana

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Let $K$ be a compact quasi self-similar set in a complete metric space $X$ and let $M_{1} (K)$ denote the space of all probability measures on $K$ , endowed with the Fortet-Mourier metric. We will show that for a typical (in the sense of Baire category) measure in $M_{1} (K)$ the lower concentration dimension is equal to $0$ , while the upper concentration dimension is equal to the Hausdorff dimension of $K$ .