NONEMPTY INTERSECTIONS OF MIDDLE α CANTOR SETS

Gregory J. Davis

Periodically, one is lucky enough to be able to find an interesting result from current mathematical research that is accessible to undergraduate students. This paper describes such a result from the field of Dynamical Systems. Here we will be exploring the nonempty intersections of two middle α Cantor sets as they are translated across one another. We will present criteria for which the intersection between two such Cantor sets is always nonempty as they are translated across one another. This fact has generated much interest and discussion in some of my upper division classes for mathematics majors and perhaps it can do the same for others.

In the past, most students of mathematics have been introduced to Cantor sets in an introductory course of real analysis. In introductory real analysis, the middle third Cantor set is explored as an example of an uncountable, closed set which contains no interior points or isolated points and has Lebesgue measure zero. More recently, with the popularization of fractals, the middle third Cantor set has become a standard example of a self–similar set. The middle third Cantor set is merely a specific example of a middle α Cantor set, where α has been set to 1/3.

While the result that we will present here is interesting in its own right, it is also interesting to know some of the motivation for studying intersecting Cantor sets. Earlier we noted that this problem is connected to the discipline of Dynamical Systems. A brief history of why intersections of Cantor sets are important in Dynamical Systems starts in the late 1800’s with Poincaré. Poincaré had identified a problem common to understanding many nonlinear dynamical systems, i.e, how to describe changes in the system as a homoclinic bifurcation takes place. It is known that as such a bifurcation takes place, the behavior of a deterministic dynamical system can change dramatically from an easy to understand stable system to a completely chaotic system.

Over the past 20 years, there has been much work done in understanding homoclinic bifurcations (see [9] for a recent overview of the subject). Several major theories have been built around homoclinic bifurcations: omega explosions [8], infinitely many co–existing sinks [1, 7, 10], and antimonotonicity [4]. Each of these theories rely on knowledge of how certain stable and unstable manifolds intersect as homoclinic bifurcations take place. It turns out that these manifolds intersect in the shape of Cantor sets. It is here that the problem of understanding the intersections of Cantor sets as they are translated across one another has become of importance in Dynamical Systems theory.

Recall the standard construction for a middle α Cantor set. First, we let C_{0} be the closed interval I of length 1 and 0 < α < 1. Although we identify I with [0, 1], a middle α Cantor set construction can be carried out in any closed interval of length 1 (or any closed interval in general).

Now, remove from C_{0} the open middle interval of length α to obtain C_{1} which consists of two closed intervals, each of length β (where β = (1 − α)/2).

Next, remove the open middle interval (which is of length αβ) from the two closed intervals in C_{1} to obtain C_{2} which consists of four closed intervals each of length β^{2}.

This process is continued indefinitely, creating a nested collection of closed nonempty sets C_{0} ⊃ C_{1} ⊃ C_{2} ⊃ C_{3} ⊃ ⋯, where each C_{n} is the union of 2^{n} intervals, each of length β^{n}. We define the middle α Cantor set C to be the infinite intersection:

C = ⋂_{n=0}^{∞} C_{n}.

Notice that at every stage in the construction of a middle α Cantor set, each closed interval in C_{i} is divided into three parts – two closed intervals which move on to the next construction stage and a gap (open interval) which is removed. Let c be a closed interval from C_{i}. If c has length l(c) prior to being divided, then when c is divided into three parts, the length of each of the two subsequent closed intervals will be βl(c) and the length of the subsequent gap will be αl(c). The ratio of the length of the closed intervals to the length of the gap which separates them is always \frac{βl(c)}{αl(c)} = \frac{β}{α}. This common ratio, denoted by τ(C), will represent the thickness of the middle α Cantor set. The concept of thickness for Cantor sets was first introduced by Newhouse in [6] and was refined in [7].

We will now begin our study of intersecting Cantor sets. Suppose that two (possibly different) middle α Cantor sets C and D are positioned in such a way that they are constructed on top of each other. Then C ∩ D ≠ ∅, because they would have at least points 0 and 1 in common. A more challenging situation is the following one. Suppose that the initial construction intervals C_{0} and D_{0} are arbitrarily positioned such that C_{0} overlaps D_{0}, that is, C_{0} ∩ D_{0} ≠ ∅.

We can think of C_{0} as having been translated to the right x units (0 ≤ x ≤ 1) in relation to D_{0}. Notice that the intersection of C_{0} and D_{0} can vary anywhere from the complete interval [0, 1] when x = 0, to a single point when x = 1.

Now, will C ∩ D ≠ ∅? If C is a middle α_{c} Cantor set, D is a middle α_{d} Cantor set, and C ∩ D is nonempty, then we must have C_{1} ∩ D_{1} ≠ ∅. The only way that C_{1} ∩ D_{1} = ∅, when C_{0} ∩ D_{0} ≠ ∅, is if α_{d} > β_{c} and α_{c} > β_{d}, or \frac{β_{c}}{α_{d}} < 1 and \frac{β_{d}}{α_{c}} < 1. Therefore, (\frac{β_{c}}{α_{d}} )(\frac{β_{d}}{α_{c}} ) < 1. If, on the other hand, we have that (\frac{β_{c}}{α_{d}} )(\frac{β_{d}}{α_{c}} ) ≥ 1, then either α_{d} ≤ β_{c} or α_{c} ≤ β_{d} and C_{1} ∩ D_{1} ≠ ∅. Note that (\frac{β_{c}}{α_{d}} )(\frac{β_{d}}{α_{c}} ) ≥ 1 is equivalent to (\frac{β_{c}}{α_{c}} )(\frac{β_{d}}{α_{d}} ) = τ(C)τ(D) ≥ 1. Hence, C_{0} ∩ D_{0} ≠ ∅ together with τ(C)τ(D) ≥ 1 implies C_{1} ∩ D_{1} ≠ ∅. Newhouse used the concept of thickness for Cantor sets in [7] to prove a technical result which has had far reaching consequences in the field of Dynamical Systems [4, 6, 8]. It has also spurred further study in the structure of intersecting Cantor sets (e.g., [2, 3, 5, and 11]). The following interesting result, accessible for undergraduates, is a special case of Newhouse’s lemma, which is the focus of our study.

\underline{Proposition}. If C and D are two middle α Cantor sets which satisfy C_{0}∩D_{0} ≠ ∅ and τ(C)τ(D) ≥ 1, then C ∩ D ≠ ∅.

We will use an inductive proof to show that if C_{0} ∩ D_{0} ≠ ∅ and τ(C)τ(D) ≥ 1, then C_{n} ∩ D_{n} ≠ ∅ for all n ≥ 0.

We notice that each of the intersections C_{n} ∩ D_{n} (n ≥ 0) is a closed and bounded set and that (C_{0} ∩ D_{0}) ⊃ (C_{1} ∩ D_{1}) ⊃ (C_{2} ∩ D_{2}) ⊃ · · ·. Thus, the Closed Nested Interval Theorem implies the desired result – the following infinite intersection is nonempty:

⋂_{n=0}^{∞}(C_{n} ∩ D_{n}) = C ∩ D ≠ ∅.

Earlier we saw that C_{0}∩D_{0} ≠ ∅, together with τ(C)τ(D) ≥ 1, forces C_{1}∩D_{1} ≠ ∅. This is the base case for our induction argument. Now suppose C_{i} ∩ D_{i} ≠ ∅; we need to prove that C_{i+1} ∩ D_{i+1} ≠ ∅. If C_{i} ∩ D_{i} ≠ ∅, then at least one of the 2^{i} closed intervals in C_{i} must intersect one of the 2^{i} closed intervals in D_{i}. Let c and d be a pair of closed intervals (c ⊂ C_{i} and d ⊂ D_{i}) which satisfy the condition c ∩ d ≠ ∅. Then the closed intervals c and d will satisfy one of the following two cases.

\underline{Case 1}. One closed interval is a subset of the other; that is, c ⊂ d or d ⊂ c. For example:

\underline{Case 2}. Each closed interval is not a subset of the other; c − (c ∩ d) ≠ ∅ and d − (c ∩ d) ≠ ∅. For example:

If c is a component of C_{i}, then in the next or (i + 1)st stage, c will be divided into two closed intervals denoted by c_{L} and c_{R} and a gap denoted by g_{c} (similarly, d will be divided into d_{L}, d_{R}, and g_{d}):

In the (i)th stage for Case 1, we will assume (without loss of generality) that c ⊂ d. In the (i+1)st stage of the Cantor sets’ constructions, we will have one of the two following situations: or

That is, either c ⊄ g_{d} or c ⊂ g_{d}. If c ⊄ g_{d}, then clearly C_{i+1} ∩ D_{i+1} ≠ ∅. In the situation where c ⊂ g_{d}, we need to recall that there is a bounded gap g adjacent to c (on either the left or right) which satisfies \frac{l(c)}{l(g)} = τ(C) (where l(c) is the length of c and l(g) is the length of the gap). Suppose that g is on the left of c. Recall that \frac{l(d_{L})}{l(g_{d})} = τ(D) and, as we have assumed, τ(C)τ(D) ≥ 1. We now have

1 ≤ τ(C)τ(D) = (\frac{l(c)}{l(g)})(\frac{l(d_{L})}{l(g_{d})}) = (\frac{l(d_{L})}{l(g)})(\frac{l(c)}{l(g_{d})}) < \frac{l(d_{L})}{l(g)}.

(Since c ⊂ g_{d}, we have l(c) < l(g_{d}), which implies \frac{l(c)}{l(g_{d})} < 1.) Hence, l(g) ≤ l(d_{L}), which in turn implies that d_{L} intersects the component c′ of C_{i} just to the left of the gap g:

Finally, we see that d_{L} intersects the right component of c′ in C_{i+1}. Therefore, C_{i+1} ∩ D_{i+1} ≠ ∅ and the verification of Case 1 is complete.

In the (i)th stage of Case 2, where each closed interval is not a subset of the other, we will assume (without loss of generality) that c and d have the following orientation.

Here, we claim that in the (i+1)st stage, C_{i+1} ∩ D_{i+1} ≠ ∅. Assume for a moment that our claim is incorrect. Then we would have the following situation.

Here, \frac{l(c_{L})}{l(g_{d})} < 1 and \frac{l(d_{R})}{l(g_{c})} < 1. The last two inequalities imply that

1 > (\frac{l(c_{L})}{l(g_{d})})(\frac{l(d_{R})}{l(g_{c})}) = (\frac{l(c_{L})}{l(g_{c})})(\frac{l(d_{R})}{l(g_{d})}) = τ(C)τ(D).

Since we have assumed τ(C)τ(D) ≥ 1, we have a contradiction. Hence, we must have that C_{i+1} ∩D_{i+1} ≠ ∅. We have now completed our verification of Case 2. This also completes the induction argument and therefore the proof of the proposition is now complete.

\underline{Note 1}. Cantor sets C and D, which satisfy the hypothesis τ(C)τ(D) ≥ 1, occupy about 38% of the α_{c} − α_{d} parameter space. The area of the shaded region in the figure below represents the region where τ(C)τ(D) ≥ 1 and has area 2 ln 2−1.

\underline{Note 2}. If the hypothesis τ(C)τ(D) ≥ 1 is replaced with τ(C)τ(D) > 1 and we further assume that C_{0}∩D_{0} contains more than a single point, then it can be shown that the nonempty intersection C ∩ D includes at least one interior point, that is, C and D will intersect at a point which is not an endpoint of the construction intervals.

\underline{References}

1. G. J. Davis, “Infinitely Many Coexisting Sinks From Degenerate Homoclinic Tangencies,” AMS Trans., 323 (1991), 727–748.

2. G. J. Davis and T.–Y. Hu, “On the Structure of the Intersection of Two Middle Third Cantor Sets,” Publications Matemàtiques, 39 (1995), 43–60.

3. B. R. Hunt, I. Kan, and J. Yorke, “When Cantor Sets Intersect Thickly,” AMS Trans., 339 (1993), 869–888.

4. I. Kan and J. Yorke, “Antimonotonicity: Concurrent Creation and Annihilation of Periodic Orbits,” AMS Bull., 23 (1990), 469–476.

5. R. L. Kraft, “Intersections of Thick Cantor Sets, Memoirs of the AMS, 468 (1992), American Mathematical Society, Rhode Island.

6. S. E. Newhouse, “Nondensity of Axiom A(a) on S^{2},” Proc. Sympos. Pure Math., 14 (1970), 191–202.

7. S. E. Newhouse, “The Abundance of Wild Hyperbolic Sets and Nonsmooth Stable Sets for Diffeomorphisms,” IHES Publ. Math., 50 (1979), 101–151.

8. J. Palis and F. Takens, “Hyperbolicity and the Creation of Homoclinic Orbits,” Annals of Math., 125 (1987), 337–374.

9. J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Studies in Advanced Mathematics, Cambridge, 1993.

10. C. Robinson, “Bifurcation to Infinitely Many Sinks,” Comm. Math. Phys., 90 (1983), 433–459.

11. R. F. Williams, “How Big is the Intersection of Two Thick Cantor Sets?,” Contemporary Mathematics, M. Brown ed., Proc. of the 1989 Joint Summer Research Conference on Continua and Dynamical Systems, AMS, Providence, Rhode Island, 1991.

Gregory J. Davis Department of Mathematics University of Wisconsin – Green Bay Green Bay, WI 54311-7001 email: davisg@uwgb.edu