An equivalent definition of packing dimension and its application
type: Item
instance of: scholarly article
title: An equivalent definition of packing dimension and its application
zbMATH Open document ID: 1175.28005
author: Wenxia Li
published in: Nonlinear Analysis. Real World Applications
publication date: 2009-05-07T00:00:00Z
review text: For a set \(E\subseteq \mathbb{R}^d\) and \(\delta>0\), a \(\delta\)-packing of \(E\) is usually defined as a collection of at most countable number of disjoint balls of radii at most \(\delta\) with centers in \(E\); fixing \(s\geq 0\), the \(s\)-dimensional packing measure of \(E\) is defined by \(P^s(E)=\inf \{\sum_iP^s_0(E_i):E\subseteq \bigcup_i^\infty E_i\}\), where \(P_0^s(E_i) =\lim_{\delta\to 0^+}\sup\sum_j|B_j|^s\) and the sup is taken over all \(\delta\)-packings \(\{B_j\}\) of \(E_i\). The packing dimension of \(E\) is then given by \(\dim_pE=\sup\{s:P^s(E)=\infty\}=\inf\{s:P^s (E)=0\}\). The author wants to avoid the inconvenience for evaluating the packing dimension of \(E\), adjacent with the above definition -- in fact the packing used for \(P_0^s(E_i)\) are restricted to the disjoint balls centered at \(E_i\). The author takes the help of Moran sets in his definition only to find out an alternative way to relax the restriction on a packing and at the same time to induce the same value for the packing dimension as introduced earlier. As a direct application, the author has the packing dimension of a class of subsets with prescribed relative group frequencies.
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 51E23
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 5551926
zbMATH Keywords: Moran set
zbMATH Keywords: component set
zbMATH Keywords: packing dimension
zbMATH Keywords: \(J\)-type packing
reviewed by: Sobhakar Ganguly
MaRDI profile type: Publication
full work available at URL: j.nonrwa.2008.02.004
OpenAlex ID: W2047966392
cites work: Multifractal decompositions of Moran fractals
cites work: Techniques in fractal geometry
cites work: Q3995301
cites work: Hausdorff dimension of subsets of Moran fractals with prescribed group frequency of their codings
cites work: Q5603915
cites work: Measures and self similarity
cites work: Packing Measure, and its Evaluation for a Brownian Path
cites work: Q4520253
DOI: 10.1016/J.NONRWA.2008.02.004
description: scientific article; zbMATH DE number 5551926