Structural instability of cookie-cutter sets
type: Item
instance of: scholarly article
title: Structural instability of cookie-cutter sets
zbMATH Open document ID: 1128.28011
author: Lifeng Xi
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-05-30T00:00:00Z
review text: An invariant set \(E\) of the IFS \(\{f_i\}\) is said to be structurally instable in \(C^1\) topology if one can always perturb \(\{f_i\}\) arbitrarily small in \(C^1\) topology to provide an IFS \(\{g_i\}\) with its invariant set \(F\) such that \(\dim E = \dim F\), but \(E\) and \(F\) are not Lipschitz equivalent. In this paper the author proves that the cookie-cutter set in \({\mathbb R}\) is structurally instable in \(C^1\) topology. For proving this theorem, he considers a homotopic deformation between two IFSs and uses the continuity of Hausdorff dimensions in \(C^1\) topology proved by himself [J. Lond. Math. Soc. (2) 70, No. 2, 369--382 (2004; Zbl 1160.28305)].
reviewed by: Y. Xiao
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 5159600
zbMATH Keywords: iterated function system
zbMATH Keywords: Lipschitz equivalence
zbMATH Keywords: instability
MaRDI profile type: Publication
full work available at URL: j.na.2006.06.052
OpenAlex ID: W1978101970
cites work: Dimension and Dynamics for Fractal Recurrent Sets
cites work: On the shape of Cantor sets
cites work: Q4368752
cites work: Q3998464
cites work: Q3714302
cites work: Techniques in fractal geometry
cites work: On the Lipschitz equivalence of Cantor sets
cites work: Classification of a quasi-circles by Hausdorff dimension
cites work: Q3731958
cites work: Asymptotic differentiable structure on Cantor set
cites work: Q5691144
cites work: Hausdorff Dimension in Graph Directed Constructions
cites work: Uniformly perfect sets and the Poincaré metric
cites work: Q4528988
cites work: Q3816752
cites work: Relations among Whitney sets, self-similar arcs and quasi-arcs
cites work: LIPSCHITZ EQUIVALENCE OF SELF-CONFORMAL SETS
cites work: Uniform perfectness of self-affine sets
DOI: 10.1016/J.NA.2006.06.052
description: scientific article; zbMATH DE number 5159600