On the arithmetic sum of regular Cantor sets
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 be made arbitrarily small. The proof proceeds by a direct construction of \(H\), based on estimates on the geometry of the product set \(C:= C_1\times C_2\). For \(\lambda >0\), the authors consider the projection from the plane onto \(\mathbb{R}\) given by \(\pi_\lambda (x,y) = \lambda x+y\). By considering ``average'' properties of this map with respect to \(\lambda\), the authors find a value for which the image of \(C\) by \(\pi_\lambda\) occupies a large portion of many small intervals, and this is used to construct \(H\).
scientific article; zbMATH DE number 1071469