On intrinsic invariance of fractal dimensions
type: Item
instance of: scholarly article
title: On intrinsic invariance of fractal dimensions
zbMATH Open document ID: 0834.28006
DOI: 10.1007/BF01264058
author: Irmina Herburt
author: Maria Moszyńska
author: Krzysztof Rudnik
published in: Geometriae Dedicata
publication date: 1996-03-31T00:00:00Z
review text: In this paper a fractal dimension function \(\dim_F\) is defined as a function which assigns to a metric space \((X, \rho)\) a real number at least \(- 1\) and satisfies the axioms of Lipschitz subinvariance, of stability and of equality to the topological dimension for smooth submanifolds \(X\) of \(\mathbb{R}^n\). The authors consider geometrically acceptable metric spaces \((X, \rho)\) introduced by Borsuk, i.e., they are strongly arcwise connected and the intrinsic metric \(\rho^*\) is topologically equivalent to \(\rho\). Metrics will be compared as follows. If \(\rho_1, \rho_2\) are metrics for \(X\) then \(\rho_1 \prec \rho_2\) iff \(\exists \alpha > 0 : \rho_1 \leq \alpha \rho_2\). An intrinsic isometry on the spaces \(X\) and \(Y\) is an isometry with respect to \(\rho^*_X, \rho^*_Y\). The authors show that in the general setting intrinsic isometries do not preserve Hausdorff as well as box dimension, both are examples for \(\dim_F\). They give also a sufficient condition for dimension invariance with respect to such isometries. The key is the above order for the metrics.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 54E40
zbMATH DE Number: 806270
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: fractal dimension
zbMATH Keywords: metric spaces
zbMATH Keywords: intrinsic isometry
zbMATH Keywords: box dimension
zbMATH Keywords: dimension invariance
reviewed by: Hermann Haase
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 806270