Fractal boundaries are not typical
type: Item
instance of: scholarly article
title: Fractal boundaries are not typical
zbMATH Open document ID: 1127.54005
author: W. L. Bloch
published in: Topology and its Applications
publication date: 2007-02-02T00:00:00Z
review text: Let \(M\subset \mathbb{R}^{n}\) be a \(C^{1}\) compact \(n\)-dimensional manifold with boundary. For a \(C^{1}\) embedding \(f\in \partial M\times\left[0,\infty\right)\rightarrow M \) such that \(f((x,0))=x\) for \(x\in\partial M\), the author defines the space \({\mathcal{F}}_{f}(M)\), the frame-approximation space of \(M\), the elements of which are obtained by deforming \(\partial M .\) The elements of \({\mathcal{F}}_{f}(M)\) are defined to be sets \(\{f(x,g(x))\in\mathbb{R}^{n}, x\in\partial M\}\), where \(g\in C^{0}(\partial M,\left[0,\infty\right)).\) The space \({\mathcal{F}}_{f}(M)\) endowed with the Hausdorff metric is a complete metric space. It is proved that a typical element of \({\mathcal{F}}_{f}(M)\) is nowhere-differentiable and has Hausdorff dimension equal to \(n-1\).
reviewed by: Irmina Herburt
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54E50
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 57N40
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 5121445
zbMATH Keywords: manifold
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: hyperspace
zbMATH Keywords: Baire category theorem
MaRDI profile type: Publication
full work available at URL: j.topol.2006.07.006
OpenAlex ID: W2082622057
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DOI: 10.1016/J.TOPOL.2006.07.006
description: scientific article; zbMATH DE number 5121445