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.

Dimension of measures

Herrmann Haase (1990)

Acta Universitatis Carolinae. Mathematica et Physica

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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 .

Inhomogeneous self-similar sets and box dimensions

Jonathan M. Fraser (2012)

Studia Mathematica

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We investigate the box dimensions of inhomogeneous self-similar sets. Firstly, we extend some results of Olsen and Snigireva by computing the upper box dimensions assuming some mild separation conditions. Secondly, we investigate the more difficult problem of computing the lower box dimension. We give some non-trivial bounds and provide examples to show that lower box dimension behaves much more strangely than upper box dimension, Hausdorff dimension and packing dimension.

On the Information Dimensions

Józef Myjak, Ryszard Rudnicki (2007)

Bollettino dell'Unione Matematica Italiana

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A relationship between the information dimension and the average dimension of a measure is given. Properties of the average dimension are studied.