Relationships between different dimensions of a measure
type: Item
instance of: scholarly article
title: Relationships between different dimensions of a measure
zbMATH Open document ID: 0996.28001
author: Ai-Hua Fan
author: Ka-Sing Lau
author: Hui Rao
published in: Monatshefte für Mathematik
publication date: 2002-06-27T00:00:00Z
review text: The authors studied various relationships of the Hausdorff dimension, entropy dimension and \(L^p\)-dimension of a measure. The main results of the paper are: Let \(0<p<1<q\), then \[ {\underline{\dim}}_q\mu\leq\dim_* \mu \leq {\underline{\dim}}_e\mu\leq {\overline{\dim}}_e\mu\leq\dim^*\mu\leq {\overline{\dim}}_p, \] where \({\underline{\dim}}_q\), \(\dim_*\mu\), \({\underline{\dim}}_e\mu\), \({\overline{\dim}}_e\mu\), \(\dim^*\mu\), \({\overline{\dim}}_p\) are, respectively, the lower \(L^q\)-dimension, lower Hausdorff dimension, lower entropy dimension, upper entropy dimension, upper packing dimension and upper \(L^p\)-dimension of the measure \(\mu\). The result extends a well-known result of Young: if the local dimension of \(\mu\) exists and is equal to \(\alpha\), \(\mu\)-a.e., then \(\dim_e\mu=\dim_H\mu=\alpha\), without assuming that the local dimension of \(\mu\) exists \(\mu\)-a.e.
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 1761122
zbMATH Keywords: Bernoulli product
zbMATH Keywords: entropy
zbMATH Keywords: fractals
zbMATH Keywords: Hausdorff dimension
reviewed by: Hua Su
MaRDI profile type: Publication
full work available at URL: s006050200016
OpenAlex ID: W2087049094
DOI: 10.1007/S006050200016
description: scientific article; zbMATH DE number 1761122