Multi-Model Cantor Sets

In this paper we define a new class of metric spaces, called multimodel Cantor sets. We compute the Hausdorff dimension and show that the Hausdorff measure of a multi-model Cantor set is finite and non-zero. We then show that a bilipschitz map from one multi-model Cantor set to another has constant Radon-Nikodym derivative on some clopen. We use this to obtain an invariant up to bilipschitz homeomorphism. Introduction A multi-model Cantor set is a metric space which has the following property. There is a partition of C into finitely many clopens A1, A2, . . . , An (called models) so that given any point x in C and ǫ > 0 there is a neighborhood U of x with diam(U) < ǫ and a metric similarity mapping U onto one of the models. Such a Cantor set is described by a map τ : C → C which is a piecewise metric similarity (an expanding map). One may consider the Cantor set as determined by the dynamical system τ . The middle third Cantor set is an example of a multi-model Cantor set where one model suffices. The middle third Cantor set is self-similar. At every scale it contains identical copies of itself. In a sense a multi-model Cantor set has finitely many local pictures which are replicated, at different scales, everywhere. This behaviour is similar to certain fractals, such as Julia Sets, which have a compact family of local geometries, (rather than a finite family). We associate to C = C(τ) and a constant d > 0, an n × n matrix Md with non-negative entries. Roughly speaking the i, j entry of Md is the sum 1 of the d-powers of the inverse scale factors of the similarities of the clopens (later called clones) of type i contained in model j. This matrix determines the Hausdorff measure and dimension as follows. We use Hδ to denote the d-dimensional Hausdorff measure. Theorem 1. Suppose d > 0 and C is a multi-model Cantor set with matrix Md and λd is the Frobenius eigenvalue for each Md. Let d be chosen such that λd = 1. Then the Hausdorff measure of C is finite and non-zero in this dimension. Therefore d is the Hausdorff dimension of C. Also, let ~v be the Frobenius eigenvector of Md such that n ∑ i=1 vi = Hd(C) then vi = Hd(Ai). This result is given at the end of section 4. Then in section 5 we investigate bilipschitz maps between multi-model Cantor sets. We prove that every bilipschitz map from one multi-model Cantor set to another is measure linear (has constant Radon-Nikodym derivative) on some clopen. This is a generalization of the results of Cooper [3], Pignataro [2] and Vu’o’ng [5], [6] to this wider class of Cantor set. We use this in corollary 4 to provide an invariant up to bilipschitz homeomorphism of such Cantor sets. These results are from the author’s Master’s thesis [1]. 1 Definitions We are concerned, in this paper, with the study of Cantor sets with particular metrics. In the following sections we shall show some results about the Hausdorff measure of these Cantor sets. In order to make these results clearer to the reader we include the definitions of similarity map, K-bilipschitz map, and diameter and state two elementary results concerning Hausdorff dimension. Definition. A surjective map f : X → Y between metric spaces is a similarity map if there is a constant K > 0 such that for all x1, x2 in X dY (f(x1), f(x2)) = KdX(x1, x2). Definition. A map f : X → Y between metric spaces is K-bilipschitz if there is a constant K > 0 such that for all x1, x2 in X 1 K dX(x1, x2) ≤ dY (f(x1), f(x2)) ≤ KdX(x1, x2).