After a somewhat long introduction to the relevance of Cantor sets in number theory and dynamical systems in the previous post, I hope we are sufficiently motivated to see a sketch of the proof of Moreira’s theorem 🙂 .
We begin our discussion with the precise definitions of some concepts introduced informally earlier.
Definition 1. A dynamically defined (or regular) Cantor set of class is a Cantor set of the real line such that there are disjoint compact intervals and an (uniformly) expanding function (i.e., for any ) from the disjoint union to its convex hull with
Moreover, by technical reasons, we assume also that is a Markov partition: for any , the interval is the convex hull of the union of some of the intervals and for some large .
Example 1. The usual ternary Cantor set is a regular Cantor set. Indeed, it is not hard to see that where is the (piecewise affine) expanding function defined by
Definition 2. Two -regular Cantor sets and are –close whenever the extremal points of the associated intervals and are close and the expanding functions and are -close.
In particular, this notion of “closeness” of regular Cantor sets means that the space of -regular Cantor sets admits a natural -topology.
Definition 3. A pair of -regular Cantor sets has –stable intersection whenever for every pair of -regular Cantor sets -close to .
After these definitions, let us recall that the main theorem of Moreira’s lecture is:
Theorem. The set of pairs of -regular Cantor sets without -stable intersection is -open and dense.
Before entering the proof of this result, let me make a few comments concerning previous known results about stable intersections of regular Cantor sets.
Firstly, Moreira and Yoccoz (2001) showed that the opposite situation occurs in the context: the set of pairs of Cantor sets with -stable intersections are -open and dense (in fact, Moreira and Yoccoz also show that usually the relative size of the intersection is large). The basic fact behind this difference between the and is the so-called bounded distortion property: small parts of a -Cantor set (or even for some ) resembles the whole Cantor set (so that it is hard to destroy intersections because they “propagate in all scales”), but this is no longer true in the context.
More precisely, given a -regular Cantor set , it is not hard to show the following lemma:
Lemma (Bounded distortion property). For any , there exists a constant (with as ) such that every pair of points and every integer satisfying
- for all
Proof. Since is expanding, we have from our assumptions that for all where is a lower bound of . Also, (plus bounded away from 0) implies that (i.e., is Hölder). In particular,
This completes the proof of the lemma.
Remark 1. A geometrical consequence of this lemma (which motivates the name “bounded distortion”) is the following: given any four points sufficiently close to each other (so that the conditions of the lemma are fulfilled), the mean value theorem ( for some ) and the previous lemma imply
for some constant independent of and . In other words, this says that at the -th stage of the construction of the Cantor set , the map distorts relative ratios between close points by a universally bounded factor! This justifies our assertion that small parts of are essentially reproductions of the whole whenever is a -regular Cantor set. Of course, as the proof of the bounded distortion lemma shows, the assimption is fundamental here.
In fact, a clear manifestation of the difference between the and cases is a result of R. Ures saying that a -typical pair of Cantor sets whose convex hulls intersects exactly at one point (i.e., they have an extremal intersection) do not have stable extremal intersection (i.e., even if one moves the convex hulls in the “correct direction”, the Cantor sets itselves will not intersect). This should be constrated with our previous discussion of Newhouse phenomena where we saw that thick Cantor sets with extremal intersection exhibit stable extremal intersection in view of the Gap Lemma.
In resume, the basic reason for the different behaviors between the Cantor sets and Cantor sets is the lack or presence of the bounded distortion property. Keeping this in mind, we are ready to present a rough sketch of the proof of Moreira’s theorem.
–Proof of Moreira’s theorem–
The strategy of the argument can be summarized as follows:
- firstly, we will show how one can destroy the bounded distortion property in a small piece of a Cantor set using a -small perturbation; this is the content of the lemma 1 below;
- secondly, we apply a result of Moreira and Yoccoz saying that the intersection of -typical Cantor sets are small; thus, using the lemma 1, we take advantage of the lack of bounded distortion (so that the gaps of the Cantor sets are large) in order to destroy completely the intersection; this is the content of lemma 2 and 3 below.
Let me stress that a crucial point in this strategy is the fact that we are destroying the bounded distortion property only at small pieces of a given Cantor set. Indeed, a direct inspection of some examples (such as the ternary Cantor set) shows that it is not reasonable to destroy the bounded distortion property on the whole Cantor set (although this can always be done in small pieces)! In particular, this explains why the result of R. Ures deals only with extremal intersections: in this case, it suffices to destroy the bounded distortion near a single point! On the other hand, the second step of the strategy explains why the argument works: by the strong results of Moreira and Yoccoz, the problematic region (corresponding to the intersection of the Cantor sets) is a relatively small part of both Cantor sets (in the sense of the Hausdorff dimension). Now let us turn to the details.
We begin with a pair of -Cantor sets . Of course, Moreira’s theorem is proved once we show that can be -approximated by a pair of disjoint regular Cantor sets. Therefore, up to a initial -perturbation, we can assume that is a pair of -regular Cantor sets. After doing this preliminary perturbation, we will keep fixed so that we will touch only on .
Lemma 1. There exists a constant (varying continuously with in the -topology) such that, given , and a subset with
- injective for all ;
- for ,
one can find a covering of by intervals of its construction of size and a Cantor set -close to verifying the following property: any intersecting is an interval of the construction of and, moreover, has a large gap in the sense that .
In particular, as we announced above, this lemma says that we can drastically deform the geometry of near a “small” set (with -small perturbations) in order to destroy the bounded distortion property (since large gaps at small scales contradicts the geometrical property of remark 1).
Proof of lemma 1. By a general lemma of Moreira (see lemma II.2.1 of this article of Moreira), we can assume (up to a small perturbation) that is a generalized affine Cantor set, i.e., the dynamical partition induced by coincides with the partition of some (piecewise) affine expnding map (although the maps do not necessarily coincide). We look at a advanced step of the construction of and we denote by the family of intervals of this stage intersecting . Using our assumptions on , it is not hard to see that for a sufficiently fine covering , we can suppose (up to another small perturbation) that is affine for all . On the other hand, each interval is associated to a gap of of proportion . Take where is an upper bound of . We make a -perturbation of size of inside the intervals , so that the two connected components of are multiplied by . This is relatively easy to do because is affine on each of these intervals. At the end of this process, we obtain a new Cantor set so that the proportion of the complement of its gap is
since . This ends the proof of the lemma 1.
Once we are able to destroy the local geometry of a regular Cantor set, we will take advantage of these big gaps near the (typically small) intersections in order to get the main theorem. To do so, we fix where denotes the Hausdorff dimension.
Remark 2. We are implicitly using here that is : indeed, the definition of makes sense only if . It turns out that this is always the case for regular Cantor sets due to the bounded distortion property! See the book of Palis and Takens for further details.
Lemma 2. Given two -regular Cantor sets , denote by the distinct elements of the family
Then, for a -typical (that is, Baire generic) close to , we have
Here, is the integer introduced above.
Proof. It is not hard to show that the intersection typically doesn’t contain periodic points of . Hence, we can fix integer numbers and intervals such that are injective. Next, we consider families of perturbations of depending on small parameters such that
In other words, we consider families of perturbations of giving independent translations of the Cantor set near the intervals .
Observe that has limit capacity at most . Thus, the limit capacity of
is bounded by . In particular, it follows that the projections of this set into should have zero -dimensional Lebesgue measure. Since
where , we conclude that, for almost all (in the sense of the Lebesgue measure), the intersection
is empty. This ends the proof of lemma 2.
This lemma can be interpreted as follows: for a typical pair of Cantor sets, the intersection disappears at sufficiently small scales (i.e., when one takes sufficiently high iterates of ). Hence, the proof of the theorem of Moreira is complete if one can show that the intersection can be destroyed at all scales once this was done in a small scale. This is the content of the following lemma:
Lemma 3. Let
Then, residual implies residual.
Proof. It suffices to prove that is open and dense for fixed elements . Define . Observe that implies that satisfies the hypothesis of lemma 1 for all . Thus, we can apply this lemma to get a covering of by intervals of the construction of a Cantor set close to such that each contains a gap of of proportion . It follows that we can make small independent translations (of order in order to get for all . In particular, we have . This completes the proof of lemma 3.
At this point, it is a simple task to prove the desired theorem:
End of the proof of Moreira’s theorem. In the notation of lemma 3, it is easy to check that lemma 2 says that is residual for . Hence, we can apply the lemma 3 (in a kind of “backward induction”) to obtain that is residual. This finishes the proof since (by definition).