On the arithmetic sum of regular Cantor sets
type: Item
instance of: scholarly article
title: On the arithmetic sum of regular Cantor sets
zbMATH Open document ID: 0895.58020
DOI: 10.1016/S0294-1449(97)80135-7
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1998-09-14T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78418
review text: In his study of bifurcations of diffeomorphisms, Sheldon Newhouse came across the question of what conditions might guarantee that two Cantor sets intersect. He gave an answer in terms of certain geometric ``thickness'' which, together with the obvious requirement that one Cantor set is not contained in a gap of the other one, implies that the intersection was non-empty. From this lemma, a theory eventually developed. A typical question is about the algebraic difference of two Cantor sets. If one of the sets, say \(C_2\), is fixed whilst the other, say \(C_1\), is shifted by the parameter \(t\), then the algebraic difference is the set of \(t\) for which \(C_1+t\) and \(C_2\) have non-empty intersection. In many cases the difference contains an interval; for example, when Newhouse's condition is satisfied. Moreover, for each fixed \(t\) the intersection is typically more than one point. For a review of this stage of the theory, see \textit{R. Kraft} [Mem. Am. Math. Soc. 468, 119 p. (1992; Zbl 0753.28009)]. In the paper under review, the question is posed differently. Can we perturb one Cantor set, say \(C_1\), mapping it by some diffeomorphism \(H\), so that the algebraic sum \(H(C_1)+C_2\) contains intervals? The distinction between the algebraic sum and difference is only in the wording, of course. To talk about perturbations of Cantor sets is natural in the context of applying the results to generic bifurcating systems. The answer the authors give is simple and elegant. The Cantor sets must be regular, and the essential condition for such a perturbation to be feasible is that the sum of Hausdorff dimensions of the sets be greater than 1. This is not too far from a sharp estimate, because if the Hausdorff dimension of the product of two Cantor sets is below 1, the measure of the sum must be 0. The diffeomorphism \(H\) is \(C^\infty\), can be found arbitrarily near the identity, and the Hausdorff distance between \(H(C_1) +C_2\) and the closure of its interior can …
Mathematics Subject Classification ID: 37E99
zbMATH DE Number: 1071469
zbMATH Keywords: Cantor sets
zbMATH Keywords: perturbation
zbMATH Keywords: Hausdorff dimension
author: Jacob Palis
author: Jean-Christophe Yoccoz
reviewed by: Grzegorz Świątek
MaRDI profile type: Publication
cites work: Q3714302
cites work: Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms
cites work: Q4727077
cites work: Q4003408
cites work: Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension
OpenAlex ID: W2084392819
description: scientific article; zbMATH DE number 1071469