A class of self-affine sets and self-affine measures
type: Item
instance of: scholarly article
title: A class of self-affine sets and self-affine measures
zbMATH Open document ID: 1091.28005
author: De-Jun Feng
author: Yang Wang
published in: The Journal of Fourier Analysis and Applications
publication date: 2005-05-25T00:00:00Z
review text: Let \({\mathcal I}=\{\phi_j\}_{j=1}^m\) be an iterated function system (IFS) consisting of a family of contractive affine maps on \(\mathbb R^d\), then there exists a unique compact set \(K=K({\mathcal I})\), called the self-affine set, or attractor of the IFS \({\mathcal I}\), such that \(K=\bigcup_{j=1}^m\phi_j(K)\). Moreover, for any given probability vector \({\mathbf p}=(p_1,\ldots ,p_m)\), i.e., \(p_j>0\) for all \(j\) and \(\sum_{j=1}^m p_j=1\),there exists a unique compactly supported probability measure \(\nu =\nu _{{\mathcal I},{\mathbf p}}\) such that \[ \nu =\sum_{j=1}^m{p_j\nu \circ \phi_j^{-1}}, \] the measure is called self-affine measure. When the maps are self-similar, and the open set condition is fulfilled, then the structure (containing the dimensions) of the self-similar sets and measures are rather clear, but for the general self-affine maps, this is a very difficult problem, even determining the Hausdorff dimension and box dimension under the open set condition. The following are partial results concerning the problem. McMullen and Bedford independently considered the type for \({\mathcal I}=\{\phi_j\}_{j=1}^m\) in which all \(\phi_j\) have the form \[ \phi_j(x)= \left[\begin{matrix} n^{-1}&0\\ 0&k^{-1}\\ \end{matrix}\right]x+ \left[\begin{matrix} a_j/h\\b_j/k\\ \end{matrix}\right] \] where all \(a_j,b_j\) are integers, \(0\leq a_j<n\) and \(0\leq b_j <m\). They determined the Hausdorff dimension and the box dimension which do not coincide in general. Lallay and Gatzouras, in a highly technical article along the same spirit of McMullen, obtained the dimension results for a broader class of IFS \({\mathcal I}=\{\phi_j \}_{j=1}^m\), in which all rectangles \(\phi_j((0,1)^2)\) are parallel to the axes and have longer sides parallel to the \(x\)-axis. Aside from a few other special cases such as the graph-directed McMullen class studied by Kenyon and Peres, the Lalley-Gatzouras class (which includes the McMullen class) remains the only determinist…
reviewed by: Hua Su
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 37C35
zbMATH DE Number: 2171866
zbMATH Keywords: self-similar sets and measures
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: box dimension
zbMATH Keywords: entropy dimension
zbMATH Keywords: \(L^q\) spectrum
MaRDI profile type: Publication
full work available at URL: s00041-004-4031-4
OpenAlex ID: W2166136302
DOI: 10.1007/S00041-004-4031-4
description: scientific article; zbMATH DE number 2171866