Posted by: matheuscmss | July 31, 2008

Gugu’s lecture on stable intersection of Cantor sets – part II

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 C^k is a Cantor set K\subset\mathbb{R} of the real line such that there are disjoint compact intervals I_1,\dots,I_l\subset\mathbb{R} and an (uniformly) expanding C^k function \psi:I_1\cup\dots\cup I_l\to I (i.e., |\psi'(x)|>1 for any x) from the disjoint union I_1\cup\dots\cup I_l to its convex hull I with

K=\bigcap\limits_{n\in\mathbb{N}}\psi^{-n}(I).

Moreover, by technical reasons, we assume also that \{I_1,\dots,I_l\} is a Markov partition: for any 1\leq j\leq l, the interval \psi(I_j) is the convex hull of the union of some of the intervals I_i and \psi^{n(j)}(I_j)\supset I_1\cup\dots\cup I_l for some large n(j).

Example 1. The usual ternary Cantor set K_0 is a regular Cantor set. Indeed, it is not hard to see that K_0=\bigcap\limits_{n\in\mathbb{N}}\psi^{-n}([0,1]) where \psi:[0,1/3]\cup [2/3,1]\to [0,1] is the (piecewise affine) expanding function defined by

\psi(x)=\left\{\begin{array}{ll}3x & \textrm{ if } x\in [0,1/3] \\ 3x-2 & \textrm{ if } x\in [2/3,1] \end{array}\right.

Definition 2. Two C^k-regular Cantor sets K and \widetilde{K} are C^kclose whenever the extremal points of the associated intervals I_1,\dots, I_l and \widetilde{I_1},\dots,\widetilde{I_l} are close and the expanding functions \psi and \widetilde{\psi} are C^k-close.

In particular, this notion of “closeness” of regular Cantor sets means that the space of C^k-regular Cantor sets admits a natural C^k-topology.

Definition 3. A pair (K_1,K_2) of C^k-regular Cantor sets has C^kstable intersection whenever \widetilde{K_1}\cap\widetilde{K_2}\neq\emptyset for every pair (\widetilde{K_1},\widetilde{K_2}) of C^k-regular Cantor sets C^k-close to (K_1,K_2).

After these definitions, let us recall that the main theorem of Moreira’s lecture is:

Theorem. The set of pairs of C^1-regular Cantor sets (K,\widetilde{K}) without C^1-stable intersection is C^1-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 C^2 context: the set of pairs of Cantor sets with C^2-stable intersections are C^2-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 C^1 and C^2 is the so-called bounded distortion property: small parts of a C^2-Cantor set (or even C^{1+\varepsilon} for some \varepsilon>0) 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 C^1 context.

More precisely, given a C^{1+\varepsilon}-regular Cantor set K, it is not hard to show the following lemma:

Lemma (Bounded distortion property). For any \delta>0, there exists a constant C(\delta) (with C(\delta)\to 0 as \delta\to 0) such that every pair of points x, y and every integer n\geq 1 satisfying

  • |\psi^n(x)-\psi^n(y)|\leq\delta
  • \left[ \psi^i(x),\psi^i(y) \right]\subset I_1\cup\dots\cup I_l for all 0\leq i\leq n-1

verifies

\Big|\log|(\psi^n)'(x)|-\log|(\psi^n)'(y)|\Big|\leq C(\delta).

Proof. Since \psi is expanding, we have from our assumptions that |\psi^i(x)-\psi^i(y)|\leq \sigma^{i-n}\cdot \delta for all 0\leq i\leq n where \sigma>1 is a lower bound of |\psi'|. Also, \psi\in C^{1+\varepsilon} (plus \psi' bounded away from 0) implies that \log|\psi'|\in C^{\varepsilon} (i.e., \log|\psi'| is Hölder). In particular,

\begin{array}{ll}\Big|\log|(\psi^n)'(x)|-\log|(\psi^n)'(y)|\Big|&= \Big|\sum\limits_{i=0}^{n-1}(\log|\psi'(\psi^i(x))|-\log|\psi'(\psi^i(y))|)\Big| \\ &\leq C\sum\limits_{i=0}^{n-1}|\psi^i(x)-\psi^i(y)|^{\varepsilon} \\ & \leq C\delta^{\varepsilon}\sum\limits_{i=0}^{n-1}\sigma^{\varepsilon(i-n)}\\ &\leq C\delta^{\varepsilon} \frac{\sigma^{-\varepsilon}}{1-\sigma^{-\varepsilon}} \\ & :=C(\delta).\end{array}

This completes the proof of the lemma. \square

Remark 1. A geometrical consequence of this lemma (which motivates the name “bounded distortion”) is the following: given any four points x,y,z,w\in K sufficiently close to each other (so that the conditions of the lemma are fulfilled), the mean value theorem (|f(a)-f(b)|=|f'(c)|\cdot |a-b| for some a<c<b) and the previous lemma imply

e^{-C}\frac{|x-y|}{|z-w|}\leq \frac{|\psi^n(x)-\psi^n(y)|}{|\psi^n(z)-\psi^n(w)|}\leq e^{C}\frac{|x-y|}{|z-w|}

for some constant C>0 independent of n and x,y,z,w. In other words, this says that at the n-th stage of the construction of the Cantor set K, the map \psi^n distorts relative ratios between close points by a universally bounded factor! This justifies our assertion that small parts of K are essentially reproductions of the whole K whenever K is a C^{1+\varepsilon}-regular Cantor set. Of course, as the proof of the bounded distortion lemma shows, the C^{1+\varepsilon} assimption is fundamental here.

In fact, a clear manifestation of the difference between the C^1 and C^2 cases is a result of R. Ures saying that a C^1-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 C^2 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 C^1 Cantor sets and C^2 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 C^1-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 C^1-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 C^1-Cantor sets (K,K'). Of course, Moreira’s theorem is proved once we show that (K,K') can be C^1-approximated by a pair of disjoint regular Cantor sets. Therefore, up to a initial C^1-perturbation, we can assume that (K,K') is a pair of C^2-regular Cantor sets. After doing this preliminary perturbation, we will keep K' fixed so that we will touch only on K.

Lemma 1. There exists a constant c(K)>0 (varying continuously with K in the C^{3/2}-topology) such that, given \varepsilon>0, \delta>0 and a subset X\subset K with

  • \psi^j|_X injective for all 0\leq j\leq n_0:=\lfloor c(K)\varepsilon^{-1}\log(\varepsilon^{-1})\rfloor;
  • \psi^i(X)\cap\psi^j(X)=\emptyset for 0\leq i\leq j\leq n_0,

one can find a covering of K by intervals (J_i) of its construction of size |J_i|<\delta and a Cantor set \widetilde{K} C^1-close to K verifying the following property: any J_i intersecting X is an interval of the construction of \widetilde{K} and, moreover, \widetilde{K}\cap J_i has a large gap U_i in the sense that |U_i|\geq (1-\varepsilon)|J_i|.

In particular, as we announced above, this lemma says that we can drastically deform the geometry of K near a “small” set X (with C^1-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 K is a generalized affine Cantor set, i.e., the dynamical partition induced by \psi 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 K and we denote by (J_i) the family of intervals of this stage intersecting X. Using our assumptions on X, it is not hard to see that for a sufficiently fine covering (J_i), we can suppose (up to another small perturbation) that \psi|_{\psi^j(J_i)} is affine for all 0\leq j\leq n_0. On the other hand, each interval J_i is associated to a gap U_i of K of proportion |U_i|/|J_i|\geq a:=a(K)>0. Take c(K):=2\lambda(K)/a(K) where \lambda:=\lambda(K) is an upper bound of |\psi'|. We make a C^1-perturbation of size \varepsilon of \psi inside the intervals \psi^j(J_i), 0\leq j\leq n_0 so that the two connected components of \psi^j(J_i)-\psi^j(U_i) are multiplied by 1-(2a\varepsilon/3\lambda). This is relatively easy to do because \psi is affine on each of these intervals. At the end of this process, we obtain a new Cantor set \widetilde{K} so that the proportion of the complement of its gap is

(1-\frac{2a\varepsilon}{3\lambda})^{n_0}(1-a)\leq\varepsilon^{4/3}<\varepsilon

since n_0=\lfloor c(K)\varepsilon^{-1}\log(\varepsilon^{-1})\rfloor. This ends the proof of the lemma 1. \square

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 k:=\lfloor (1-HD(K))^{-1}\rfloor where HD denotes the Hausdorff dimension.

Remark 2. We are implicitly using here that K is C^2: indeed, the definition of k makes sense only if HD(K)<1. It turns out that this is always the case for C^2 regular Cantor sets due to the bounded distortion property! See the book of Palis and Takens for further details.

Lemma 2. Given two C^2-regular Cantor sets K, K', denote by A_j the distinct elements of the family

\{\psi^r((K\cap K')\cap I): I \textrm{ is a maximal interval with } \psi^{r}|_I \textrm{ is injective}\}.

Then, for a C^1-typical (that is, Baire generic) \widetilde{K}' close to K', we have

\bigcap\limits_{j=1}^k A_j=\emptyset.

Here, k:=\lfloor (1-HD(K))^{-1}\rfloor is the integer introduced above.

Proof. It is not hard to show that the intersection K\cap K' typically doesn’t contain periodic points of \psi. Hence, we can fix r_1,\dots,r_k\in\mathbb{N} integer numbers and I_1,\dots,I_k intervals such that \psi^{r_j}|_{I_j} are injective. Next, we consider families \psi_{t_1,\dots,t_k} of perturbations of \psi depending on k small parameters t_1,\dots,t_k\in\mathbb{R} such that

\psi_{t_1,\dots,t_k}^{r_j}|_{I_j}=\psi^{r_j}|_{I_j}+t_j.

In other words, we consider families of perturbations of \psi giving independent translations of the Cantor set K near the intervals I_j.

Observe that \psi^{r_j}(K\cap K'\cap I_j) has limit capacity at most HD(K). Thus, the limit capacity of

\prod\limits_{j=1}^k\psi^{r_j}(K\cap K'\cap I_j)

is bounded by k\cdot HD(K)<k-1. In particular, it follows that the projections of this set into \mathbb{R}^{k-1} should have zero (k-1)-dimensional Lebesgue measure. Since

\begin{array}{l}\{(t_1,\dots,t_k)\in\mathbb{R}^k: \bigcap\limits_{j=1}^k(A_j+t_j)\neq\emptyset\} = \\ \{(t,t+s_1,\dots,t+s_{k-1}): (s_1,\dots,s_{k-1})\in g(\prod\limits_{j=1}^k A_j)\}\end{array}

where g(x_1,\dots, x_k)=(x_2-x_1,\dots,x_k-x_1), we conclude that, for almost all (t_1,\dots,t_k) (in the sense of the Lebesgue measure), the intersection

\bigcap\limits_{j=1}^k\psi_{t_1,\dots,t_k}^{r_j}(K\cap K'\cap I_j):= \bigcap\limits_{j=1}^k\psi^{r_j}(K\cap K'\cap I_j)+t_j:= \bigcap\limits_{j=1}^k A_j+t_j

is empty. This ends the proof of lemma 2. \square

This lemma can be interpreted as follows: for a typical pair (K,K') of Cantor sets, the intersection K\cap K' disappears at sufficiently small scales (i.e., when one takes sufficiently high iterates of \psi). 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

\mathcal{F}:=\{\psi^j|_{I_r}: j\in\mathbb{N}, I_r \textrm{ is a maximal interval with } \psi^j|_{I_r} \textrm{ injective }\}

and

\mathcal{R}_m:=\{(K,K'): \bigcap\limits_{j=1}^m\phi_j(K\cap K')=\emptyset,\,\forall \phi_1,\dots,\phi_m\in\mathcal{F}\textit{ distinct}\}.

Then, \mathcal{R}_{m+1} residual implies \mathcal{R}_m residual.

Proof. It suffices to prove that \{(K,K'): \bigcap\limits_{j=1}^m\phi_j(K\cap K')=\emptyset\} is open and dense for fixed elements \phi_1,\dots,\phi_m\in\mathcal{F}. Define X:=\bigcap\limits_{j=1}^m\phi_j(K\cap K'). Observe that (K,K')\in\mathcal{R}_{m+1} implies that X satisfies the hypothesis of lemma 1 for all n_0. Thus, we can apply this lemma to get a covering J_{i} of X by intervals of the construction of a Cantor set \widetilde{K} close to K such that each J_i contains a gap of \widetilde{K} of proportion 1-\varepsilon. It follows that we can make small independent translations (of order O(\varepsilon|J_i|)=o(|J_i|) in order to get (\widetilde{K}\cap J_i)\cap K=\emptyset for all i. In particular, we have (\widetilde{K},K')\in\mathcal{R}_m. This completes the proof of lemma 3. \square

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 \mathcal{R}_k is residual for k=\lfloor (1-HD(K))^{-1}\rfloor. Hence, we can apply the lemma 3 (in a kind of “backward induction”) to obtain that \mathcal{R}_1 is residual. This finishes the proof since \mathcal{R}_1=\{(K,K'): K\cap K'=\emptyset\} (by definition). \square

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