On the Hausdorff dimension of fibres
type: Item
instance of: scholarly article
title: On the Hausdorff dimension of fibres
zbMATH Open document ID: 0748.28004
DOI: 10.1007/BF02775791
author: Itai Benjamini
author: Yuval Peres
published in: Israel Journal of Mathematics
publication date: 1992-06-28T00:00:00Z
review text: The unit square is partitioned into four congruent subsquares, and an arbitrary one of them is deleted. This operation will be applied to all remaining squares ad infinitum. The authors prove that for the resulting limit set \(F\) its fibre sets \(F_ x=\{y\in[0,1]\mid(x,y)\in F\}\) satisfy \(\dim(F_ x)\geq 1/2\) for almost all \(x\in[0,1]\) with respect to Lebesgue measure (dim denotes Hausdorff dimension). They also discuss the Hausdorff dimension behaviour of exceptional fibres. Finally, the relations of Hausdorff and Minkowski dimension of \(F_ x\) are considered.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 11K55
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 32240
zbMATH Keywords: fractal reconstruction
zbMATH Keywords: fibre of a set
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Minkowski dimension
reviewed by: Hermann Haase
MaRDI profile type: Publication
cites work: Q5674726
cites work: Q5515914
cites work: Random walks on a tree and capacity in the interval
cites work: Billingsley dimension in probability spaces
cites work: Superbranching processes and projections of random Cantor sets
cites work: Projections of random Cantor sets
cites work: Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation
cites work: Q5610469
cites work: Intersecting random translates of invariant Cantor sets
cites work: Random walks and percolation on trees
cites work: Some Fundamental Geometrical Properties of Plane Sets of Fractional Dimensions
cites work: Hausdorff dimension and capacities of intersections of sets in n-space
cites work: Random Recursive Constructions: Asymptotic Geometric and Topological Properties
cites work: Dimension, entropy and Lyapunov exponents
full work available at URL: https://doi.org/10.1007/bf02775791
OpenAlex ID: W193419487
description: scientific article; zbMATH DE number 32240