On the Structure of the Intersection of Two Middle Third Cantor Sets

Motivated by the study of planar homoclinic bifurcations, in this paper we describe how the intersection of two middle third Cantor sets changes as the sets are translated across each other. The resulting description shows that the intersection is never empty; in fact, the intersection can be either finite or infinite in size. We show that when the intersection is finite then the number of points in the intersection will be either 2n or 3 · 2n. We also explore the Hausdorff dimension of the intersection of two middle third Cantor sets as the sets are translated across one another. We show that the Hausdorff dimension of the intersection can take on any value from 0 to ln 2/ ln 3; in addition, we show that for each Hausdorff dimension, between 0 and ln 2/ ln 3, there is a dense set of translation parameters for which the intersections have that particular Hausdorff dimension. 1. Dynamical systems and intersecting Cantor sets Our motivation to study the intersection of Cantor sets comes from the discipline of Dynamical Systems. In the late 1800’s, Poincaré identified a problem common to many nonlinear dynamical systems how to describe the changes in a dynamical system when a homoclinic bifurcation takes place. This problem is still the center of much work in the theory of dynamical systems (see the recent monograph [9], for example). As a homoclinic bifurcation takes place, the behavior of a deterministic dynamical system can change from being very robust and predictable (with respect to initial conditions) to being completely chaotic. Over the past 15 years, work has accelerated in the area of homoclinic bifurcations. Several majour theories have been explored in conjunction with the creation and destruction of homoclinic bifurcations. Along with the possibility of strange attractors, some of the phenomena 44 G. J. Davis, T.-Y. Hu associated with homoclinic bifurcations include omega explosions [8], infinitely many coexisting sinks [7], [10], [1] and antimonotonicity [4]. At this time, there is no single theory which integrates and predicts the order of occurrence of these phenomena. However, the development of the theories related to each of these phenomena requires the understanding of how certain stable and unstable manifolds intersect as the homoclinic bifurcation takes place. It is known that the intersections of stable and unstable manifolds have the shape of intersecting Cantor sets; because of this fact, it seems that in order to create a theory unifying omega explosions, infinitely many coexisting sinks and antimonotonicity, it is necessary to understand how Cantor sets intersect in general. The theories of infinitely many sinks and antimonotonicity rely heavily on knowing when stable and unstable manifolds cannot be separated as they slide across one another, while the theory of omega explosions requires that the stable and unstable manifolds seldom intersect as they slide across one another. Each of these theories hold for parameter values close to a given homoclinic bifurcation in a dissipative planar diffeomorphism. The criterion used for showing that the stable and unstable manifolds cannot be separated is that the product of the thicknesses of the manifolds is greater than one. Newhouse [6] defines the concept of thickness and show that two Cantor sets which have the product of their thicknesses greater than one cannot be separated. A Cantor set C in the line is represented as the difference of an interval C0 and an infinite collection {Uj} of disjoint open subintervals (also known as gaps) contained in C0. More precisely, C = ∞ ⋂ i=0 Ci, where C0 is the smallest interval containing C, and Ci = C0 − i−1 ⋃ j=0 Uj . Such a sequence of sets {Ci} is called a defining sequence of C. Let Iij for j = 1, 2 be the two components of Ci on either side of the gap Ui and let l(J) be the length of an interval J . The thickness of a defining sequence is defined by: τ({Ci}) = inf{l(Iij)/l(Ui) : i ≥ 1, where j = 1, 2}. The thickness of a Cantor set is then defined by τ(C) = sup{τ({Ci}) : {Ci} is a defining sequence for C}. Using this definition, Newhouse proved the following striking lemma: Lemma. Let C and C be two Cantor sets in R such that τ(C)τ(C) > 1. If C is not contained in a gap of C and C is not contained in a gap of C, then C ∩ C = ∅. On the intersection of two Cantor sets 45 On the other hand, the theory of omega explosions uses the criterion that the sum of the limit capacities of the stable and unstable manifolds is less than one. The limit capacity, d(C), of a Cantor set C will be defined as: lim sup →0 (lnn(C, )/ ln(1/ )), where n(C, ) is the minimum number of open intervals of radius needed to cover C. The limit capacity of a Cantor set is closely related to its Hausdorff dimension (see [6]). Williams [11], Kraft [5] and Hunt, Kan, and Yorke [3] have explored the size of the intersections resulting from two Cantor sets being translated across one another. All of these explorations have assumed that the Cantor sets in question have product of thickness greater than one. Williams produced examples to show that the intersection of two Cantor sets can vary from being one point to containing another Cantor set. Independently, Kraft and Hunt et. al. further developed these ideas and determined all pairs of thicknesses for which the intersection may be a single point and all pairs of thicknesses which must contain another Cantor set. They also consider the problem of how often —as one Cantor set is being translated over another one— does the intersection contain another Cantor set. In this paper it is our goal to completely describe how the intersection of two middle third Cantor sets change as they are translated across each other. Recall the standard construction for the middle third Cantor set. First we let C0 be the closed interval I of length 1. Although we identify I with [0, 1], we note that this construction can be carried out in any closed interval of length 1 (or any closed interval in general).