STABLE INTERSECTIONS OF CANTOR SETS AND HOMOCLINIC BIFURCATIONS

CARLOS GUSTAVO T. de A. MOREIRA

IMPA

Résumé: Nous étudions des intersections d’ensembles de Cantor et ses consequences sur les systmes dynamiques. Le concept de intersection stable de deux ensembles de Cantor dynamiquement définis est ici introduit.Nos prouvons que si les ensembles de Cantor stable et instable associés a une bifurcation homoclinique ont une intersection stable, alors il y a des ensembles ouverts dans la ligne de paramtres avec densité positive dans la valeur initiale de bifurcation pour lesquels les diffeomorphismes ne sont pas hyperboliques. Nous présentons des conditions qui assurent des intersections stables, plusgénérales que le conditions préalablement connues. Nous présentonsaussi des conditions pour que l’hyperbolicité ait densité positive aux bifurcations homocliniques. Cela nous permet de présenter des examples persistentes de familles de bifurcations homocliniques a un paramtre qui présentent en même temps de l’hyperbolicité et des tangences homocliniques avec densité positive dans la valeur initiale de bifurcation dans la ligne de paramtre.

Abstract: We study intersections of dynamically defined Cantor sets and consequences to dynamical systems. The concept of stable intersection of two dynamically defined Cantor sets is introduced. We prove that if the stable and unstable Cantor sets associated to a homoclinic bifurcation have a stable intersection, then there are open sets in the parameter line with positive density at the initial bifurcating value, for which the corresponding diffeomorphisms are not hyperbolic. We present conditions more general than the ones previously known that assure stable intersections. We also present conditions for hyperbolicity to be of positive density at homoclinic bifurcations. This allow us to provide persistent one-parameter families of homoclinic bifurcations that present both hyperbolicity and homoclinic tangencies with positive density at the initial bifurcating value in the parameter line.

Introduction:

The objective of this work is to study intersections of Cantor sets on the real line which are typical of the Cantor sets that appear in dynamical systems: the dynamically defined Cantor sets. Indeed, they appear naturally in the study of hyperbolic sets, like Smale’s horseshoe, which is the product of two Cantor sets of this kind. They also appear in different mathematical contexts. For instance, for any m ∈ N bigger than one, the set of numbers in the interval [0,1] whose continued fractions have all its coefficients less than or equal to n is a Cantor set of this kind. Roughly speaking, dynamically defined or regular Cantor sets are Cantor sets that are defined by expanding functions (see the precise definition in Section I).

The original motivation for this work was a conjecture of J. Palis, according to which generically the arithmetic difference K_{1} − K_{2} = {x − y | x ∈ K_{1}, y ∈ K_{2}} of two dynamically defined Cantor sets either has measure zero or else contains intervals. The study of measure-theoretical and topological properties of arithmetic differences of subsets of the real-line is particularly delicate for Cantor sets. For other classes of subsets of R, the following results about arithmetic differences are known, and their proofs are quite simple:

• A has measure zero and B is countable ⇒ A − B has measure zero;

• A is of first category and B is countable ⇒ A − B is of first category;

• A and B have positive measure ⇒ A − B contains an interval;

• A and B are residual ⇒ A − B = R.

On the other hand, concerning dynamically defined Cantor sets, several relevant results have been obtained, such as:

• If HD(K_{1}) + HD(K_{2}) < 1 (where HD(K) denotes the Hausdorff dimension of K) then K_{1} − K_{2} has measure zero;

• If τ(K_{1}).τ(K_{2}) > 1 then K_{1} − K_{2} contains an interval. Here τ(K) denotes the thickness of K (see Section II).

These results can be easily proved, but involve subtle concepts with relevant applications in dynamics. The second is proved in Section II in a somewhat more general form, and the proof of the first one is sketched in Section I.1.

The concept of thickness was used by Newhouse ([N1]) to exibit open sets of diffeomorphisms with persistent homoclinic tangencies, therefore without hyperbolicity. It is possible ([N2]) to prove that in such an open set there is a residual set of diffeomorphisms which present infinitely many coexisting sinks. In [N3], it is proved that under generic hypotheses every family of surface diffeomorphisms that unfold a homoclinic tangency goes throught such an open set. A new and perhaps clearer proof is in [PT2].

In this work, we introduce yet another concept about Cantor sets related to dynamical systems: we say that two dynamically defined Cantor sets K_{1} and K_{2} have stable intersection if for any pair of dynamically defined Cantor sets (K̃_{1}, K̃_{2}), near (K_{1}, K_{2}) (in a certain topology), we have that K̃_{1} ∩ K̃_{2} ≠ ∅. The study of stable intersections of Cantor sets has close relations with the study of arithmetic differences of Cantor sets. For instance, if (K_{1}, K_{2}) has stable intersection then K_{1} − K_{2} contains an interval. Theorem I-1 presents a kind of converse to this fact. On the other hand, it is easy to show that having stable intersection for two regular Cantor sets is more general than having the product of their thicknesses bigger than one.

This work is essentially dedicated to the study of stable intersections of dynamically defined Cantor sets and its consequences about the study of homoclinic bifurcations.

In Section I, we initially present some basic concepts and explain the relationship between homoclinic bifurcations and intersections of Cantor sets. In simple terms, we consider a one-parameter family (φ_{μ}), μ ∈ R, of diffeomorphisms which unfold a homoclinic tangency at μ = 0, associated to a saddle point p belonging to a non trivial basic set Λ, i.e., to a hyperbolic maximal invariant set in which the periodic points are dense. We can associate to Λ_{μ}, the continuation of Λ_{o} = Λ, two dynamically defined Cantor sets (K_{1})_{μ} and (K_{2})_{μ}, the “factors” of Λ_{μ}, such that, if (K_{1})_{μ} ∩ (K_{2})_{μ} ≠ ∅, then the stable and unstable foliations of Λ_{μ} have a point of tangency. In turn, such a tangency can be approximated by homoclinic tangencies associated to the point p_{μ} (the continuation of the saddle point p_{o} = p). In particular, if (K_{1})_{μ} ∩ (K_{2})_{μ} ≠ ∅, the diffeomorphism φ_{μ} is not hyperbolic.

We prove (Theorem I.2), that if (K_{1})_{0} has a stable intersection with (K_{2})_{0} + t for some value of t ∈ R, then the set {μ > 0 | (K_{1})_{μ} intersects (K_{2})_{μ} stably} has positive density at μ = 0. We also introduce the concept of extremal stable intersection in Section I, and prove that if ((K_{1})_{0}, (K_{2})_{0}) has extremal stable intersection, then (K_{1})_{μ} ∩ (K_{2})_{μ} ≠ ∅ for every μ > 0 small.

In Section II, we introduce the concepts of thickness and lateral thicknesses, that allows us, among other things, to give examples of pairs of Cantor sets (K_{1}, K_{2}) which have stable or extremal stable intersections and yet the product of their thicknesses may be smaller than one. Indeed, these conditions we use in these examples concern instead the Hausdorff dimensions HD(K_{1}) and HD(K_{2}) of K_{1} and K_{2}: We can give such examples for Cantor sets K_{1} and K_{2}, with their Hausdorff dimensions having any value between zero and one and their sum being bigger than one.

In Section III, we discuss some examples about arithmetic differences and stable intersections.

In Section IV, we discuss more general criteria for stable and extremal stable intersections and provide examples of their applications.

In Section V, we exhibit conditions on the Cantor sets (K_{1})_{0} and (K_{2})_{0} which imply that the set {μ > 0 | φ_{μ} is persistently hyperbolic} has positive density at μ = 0. Besides, we give examples of open sets of one-parameter families of diffeomorphisms which unfold homoclinic tangencies, such that the phenomena of hyperbolicity and persistent tangencies between the stable and unstable foliations have both uniformly positive densities at μ = 0.

We also introduce the concept of extremal stable intersection, which is stronger than the concept of stable intersection, and have relevant application in dynamics: we say that (K_{1}, K_{2}) have extremal stable intersection if the right extreme of K coincides with the left one of K_{2}, and for any pair of two dynamically defined Cantor sets (K̃_{1}, K̃_{2}) near (K_{1}, K_{2}) which support intervals intersects we have K̃_{1} ∩K̃_{2} ≠ ∅. It is easy to prove that, in the case of homoclinic bifurcations as above, if ((K_{2})_{0}, (K_{2})_{0}) has an extremal stable intersection, then (K_{1})_{μ} ∩ (K_{2})_{μ} ≠ ∅ for every μ > 0 small. We shall study here conditions that imply extremal stable intersections.

In section V we shall prove a theorem that, under certain hypothesis about the Cantor sets (K_{1})_{0} and (K_{2})_{0} assures that the set {μ > 0 | φ_{μ} is persistently hyperbolic} has positive density at μ = 0. Besides, we give an example of an open set of 1-parameter families of diffeomorphisms which unfold homoclinic tangencies such that the phenomena of hyperbolicity and persistent tangencies between the stable and unstable foliations have both uniformly positive densities at μ = 0.

This work is divided as follows: Section I: Cantor sets and homoclinic bifurcations Section II: Thickness and lateral thickness Section II: Examples about arithmetic differences and stable intersections Section IV: Criteria for stable and extremal stable intersections and generalized thickness test Section V: Intersections of Cantor sets and hyperbolicity.

Acknowledgements: I would like to thank J. Palis, for posing to me the problem and for his constant support. Also to J.C. Yoccoz, M. Viana, P. Mendes, S. Plaza and A. Sannami for stimulating and helpful conversations. I want also to stress the fine mathematical ambiance of IMPA, where this work was done.

Section I: Cantor sets and homoclinic bifurcations

I.0 Basic definitions:

Given a diffeomorphism φ: M →^{C^{k}} M of a surface and p a fixed point of φ, we define: W^{s}(p): = {x ∈ M | lim_{n→∞}φ^{n}(x) = p} (the stable manifold of p)

W^{u}(p): = {x ∈ M | lim_{n→−∞}φ^{n}(x) = p} ( the unstable manifold of p).

It is possible to prove that if φ ∈ C^{k}, then W^{s}(p) and W^{u}(p) are C^{k} manifolds.

We say that x is a homoclinic point associated to p if x ∈ W^{s}(p) ∩ W^{u}(p), and that φ exibits a homoclinic tangency at x associated to p if W^{s}(p) is tangent to W^{u}(p) at x.

Let Λ ⊂ M. We say that Λ is an hyperbolic set for φ: M → M if φ(Λ) = Λ and there is a decomposition T_{Λ}M = E^{s} ⊕ E^{u} such that Dφ|_{E^{s}} is uniformly contractive and Dφ |_{E^{u}} is uniformly expansive. Given x ∈ Λ, we define:

W^{s}(x) = {y ∈ M | lim_{n→∞} d(φ^{n}(y), φ^{n}(x)) = 0} e

W^{u}(x) = {y ∈ M | lim_{n→−∞} d(φ^{n}(y), φ^{n}(x)) = 0} ,

which are C^{k} manifolds. The union of these manifolds forms foliations F^{s} and F^{u} which can be extended to a neighbourhood of Λ and which are leaf-wise φ-invariants.

The non-wandering set of φ, Ω(φ), is defined by:

Ω(φ): = {x ∈ M | ∀ neighbourhood U(x), ∃n ∈ Z_{+} tq φ^{n}(U(x)) ∩ U(x) ≠ ∅}.

We say that φ is hyperbolic if Ω(φ) is hyperbolic. A 1-parameter family {φ_{μ}} of diffeomorphisms of M^{2} has an homoclinic Ω- explosion at μ = 0 if i) for μ < 0, φ_{μ} is persistently hyperbolic; ii) Ω(φ_{0}) = Λ∪O , where Λ is hyperbolic and O is an homoclinic tangency orbit associated

to a fixed saddle point p, such that W^{s}(p) and W^{u}(p) have quadratic tangencies at the points of O. A Cantor set of the real line is a compact, with empty interior and whithout isolated

points subset of R.

Given K ⊂ R a Cantor set and U a finite covering of K by open intervals U = (U_{i})_{i∈I} we define diam(U) = max{|U_{i}| , i ∈ I}, where |U_{i}| is the lenght of U_{i}. Given α ∈ R_{+}, we

define H_{α}(U) = ∑_{i∈I} |U_{i}|^{α}.

The Hausdorff α-measure of K is m_{α}(K) = lim_{→0} ( inf_{UcoversK ; diam(U)<} H_{α}(U)).

It is easy to see that there is an unique real number, the Hausdorff dimension of K, which we name HD(K) such that, α < HD(K) ⇒ m_{α}(K) = +∞, and α > HD(K) ⇒ m_{α}(K) = 0.

I.1: Dynamically defined Cantor sets

Let us consider a homoclinic tangency associated to a saddle point p of a diffeomorphism φ of a surface. Suppose that p belongs to a basic set Λ of saddle-type (a horseshoe). The intersection of the local stable manifold of p and Λ is a Cantor set which we will name K^{s}, and the intersection of the local unstable manifold of p and Λ is another Cantor set which we will name K^{u}.

If q is a point of tangency, unfolding the tangency by a generic family φ_{μ} (φ_{0} = φ) of diffeomorphisms, we will have a “line of tangencies” ℓ_{μ} where the leaves of the foliations F_{μ}^{s} and F_{μ}^{u} are tangent. The projections of K_{μ}^{s} and K_{μ}^{u} (the continuations of K^{s} and K^{u}) along leaves of the foliations over ℓ_{μ} give Cantor sets diffeomorphic to K_{μ}^{s} and K_{μ}^{u} which we call (K_{1})_{μ} and (K_{2})_{μ}, respectively. If (K_{1})_{μ} ∩ (K_{2})_{μ} ≠ ∅, there will be a new tangency between W^{s}(Λ_{μ}) and W^{u}(Λ_{μ}), and φ_{μ} will not be persistently hyperbolic.

With some generic hypothesis we can conclude (perhaps reparametrizing the family (φ_{μ})) that the pair ((K_{1})_{μ}, (K_{2})_{μ}) is close to the pair (K_{1}, K_{2} + μ), and then is natural to ask about the size of the set {μ | K_{1} ∩ (K_{2} + μ) ≠ ∅} =: K_{1} − K_{2} = {x − y | x ∈ K_{1}, y ∈ K_{2}}. We say that K_{1} − K_{2} is the arithmetic difference between K_{1} and K_{2}.

The Cantor sets that we will consider (and in particular the ones above) are of a special kind: the dynamically defined Cantor sets. We say that a Cantor set K is dynamically defined if:

i) there are disjoint compact intervals K_{1}, K_{2}, … K_{r} such that K ⊂ K_{1} ∪ ⋯ ∪ K_{r} and the boundary of each K_{i} is contained in K;

ii) there is a C^{1+} expanding map ψ defined in a neighbourhood of K_{1} ∪ K_{2} ∪ ⋯ ∪ K_{r} such that ψ(K_{i}) is the convex hull of a finite union of some intervals K_{j} satisfying:

ii.1 For each i, 1 ≤ i ≤ r and n sufficiently big, ψ^{n}(K ∩ K_{i}) = K;

ii.2 K = ⋂_{n=0}^{∞} ψ^{−n}(K_{1} ∪ K_{2} ∪ ⋯ ∪ K_{r}).

We say that {K_{1}, K_{2}, … K_{r}} is a Markov partition for K, and that D = ⋃_{i=1}^{r} K_{i} is the Markov domain of K.

If K_{1} = [a, b] is the left interval of has of the Markov partition it is usual to suppose that ψ|_{K_{1}}: K_{1} → K_{0} is surjective, where K_{0} is the convex hull of K, and increasing. In this case, there is a diffeomorphism α: K_{0} → K̃_{0} (where K̃_{0} is an interval) with α′(0) = 1 such that α ∘ ψ_{1} ∘ α^{−1} is an affine map, because |ψ′_{1}(a)| > 1. In this case the set K̃ = α(K) defined by the functions ψ̃_{i} = α ∘ ψ_{i} ∘ α^{−1} is auto-similar (has ψ̃_{1} affine), and is called the linearized of K.

For a more careful discussion of the relationshp between dynamically defined Cantor sets see [PT2].

The proof of the fact that if HD(K) + HD(K_{2}) < 1 then K_{1} − K_{2} has measure zero, and the study of dynamical consequences of this result (prevalence of hyperbolicity near μ = 0) may be found in [PT1] and in [PT2]. Let we sketch the proof:

Consider the map f: R^{2} → R, f(x, y) = x − y. Then K_{1} − K_{2} = f(K_{1} × K_{2}), and it is possible to prove (see [PT2]) that if K_{1} and K_{2} are dynamically defined Cantor sets, then HD(K_{1} × K_{2}) = HD(K_{1}) + HD(K_{2}), which we suppose to be less than 1. Therefore, K_{1} − K_{2} = f(K_{1} × K_{2}) is a Lipschitz image of a set with Hausdorff dimension less than one, and so has Hausdorff dimension less than one, and therefore Lebesgue measure zero.

I.2: Intersections of Cantor sets and arithmetic differences

As we have seen in I.1, the study of homoclinic byfurcations leads naturally to the study of intersections of Cantor sets, and to the study of arithmetic differences of Cantor sets. About this, there is Palis’ conjecture mentioned before, according to which the arithmetic difference of two dynamically defined Cantor sets generically either have measure zero, or Contains an interval. There is an example by A. Sannami ([S]) of a dynamically defined Cantor set K such that K − K is a Cantor set with positive measure. Bamón, Plaza and Vera also discuss this kinds of examples in [BPV]. These examples are very rigid, and the Palis’ conjecture remains. For affine Cantor sets (which we will define later) there is no counter example.

In this work we shall study a concept as closely related to homoclinic bifurcations as that of arithmetic difference: the concept of stable intersections. We say that two dynamically defined Cantor sets K_{1} and K_{2} have stable intersection if there is a neighbourhood of the pair (K_{1}, K_{2}) (in some apropriate topology; here we shall generally work with the C^{1+} topology, that we shall define later) such that for every pair (K̃_{1}, K̃_{2}) in this neighbourhood we have K̃_{1} ∩ K̃_{2} ≠ ∅.

Here we will generally work with the topology C^{1+} in which a Cantor set K defined by an expansive function ψ with Markov partition {K_{1}, K_{2}, … K_{r}} is close to a Cantor set K̃ with Markov partition K̃_{1}, K̃_{2}, … K̃_{s}} if and only if r = s, the extremes of K_{i} are near the corresponding extremes of K̃_{i}, i = 1, 2, …, r and, supposing ψ ∈ C^{1+} with Hölder constant C, we must have ψ̃ ∈ C^{1+}˜ with Hölder constant C̃ such that (C̃,˜) is near (C,) and ψ̃ is close to ψ in the C^{1} topology.

This is a natural topology because, on one hand it assures the control of the distortion (see [PT2]). In particular, if K have small distortion, then so does K̃ (for instance,this holds when K is affine). On the other hand, the sets K^{s} and K^{u} associated to a diffeomorphism φ ∈ C^{2} depend continuously of φ in the C^{1+} topology.

There are relations between the concepts of aritmetic difference and stable intersection.

For instance, if K_{1} intersects K_{2} stably then t ∈ int(K_{1} − K_{2}). In the direction of the reciprocal we have the following result:

Theorem I.1. Suppose that there is an open set U of pairs of Cantor sets such that if (K_{1}, K_{2}) ∈ U and t ∈ R then K_{1} does not have stable intersection with (K_{2} + t). Then there is a residual set R ⊂ U such that (K_{1}, K_{2}) ∈ R ⇒ int(K_{1} − K_{2}) = ∅; that is the interior of K_{1} − K_{2} is empty.

Proof. Let {r_{1}, r_{2}, …, r_{n}, … } be an enumeration of the rational numbers, and let U_{n} = {(K_{1}, K_{2}) ∈ U | K_{1} ∩ (K_{2} + r_{n}) = ∅}. By this hypothesis, U_{n} is dense, and clearly

U_{n} is open. Thus, R: = ⋂_{n=1}^{∞} U_{n} is residual, and (K_{1}, K_{2}) ∈ R ⇒ (K_{1} − K_{2}) ∩ Q = ∅ ⇒

int(K_{1} − K_{2}) = ∅. □

Remark.. The results above are valid in any C^{k} topology, k ≥ 1.

We recall that a dynamically defined Cantor set is affine if the functions ψ_{i}: = ψ|_{K_{i}} that define K are affine and surjective. This implies that K is similar to K ∩ K_{i}, 1 ≤ i ≤ r. If we do not suppose the functions ψ_{i} to be surjective, we say that K is a generalized affine Cantor set.

Let us fix a notation for a certain type of affine Cantor set that will be often used in examples.

Frequently, when we mention an affine Cantor set of the type we mean the Cantor set with this Markov domain defined by increasing and surjective affine maps.

The study of stable intersections of Cantor sets has more immediate applications to dynamics than the study of arithmetic differences. As an example, there is a theorem by Marstrand that implies that if HD(K_{1}) + HD(K_{2}) > 1, then for almost every λ ∈ R, K_{1} − λK_{2} has positive Lebesgue measure (see [PT2]); and there is a theorem by Palis and Yoccoz, according to which if HD(K^{s}) + HD(K^{u}) > 1, then generically μ = 0 is not a point of total density of hyperbolicity of (φ_{μ}) (see [PY]). This theorem is, however, very difficult, and by no means a trivial consequence of Marstrand’s theorem.

We will now see an application of stable intersections to dynamics. Theorem I.2. In the case of he homoclinic bifurcation, if there is t ∈ R such that (K̃_{1})_{0} intersects ((K̃_{2})_{0} + t) stably, where (K̃_{1})_{0} and (K̃_{2})_{0} are linearized of (K_{1})_{0} and (K_{2})_{0}, respectively, then the set {μ > 0 | (K_{1})_{μ} ∩ (K_{2})_{μ} ≠ ∅} contains an open set with positive density at μ = 0. Proof. Let (K̃_{1})_{μ} and (K̃_{2})_{μ} be the linearized of (K_{1})_{μ} and (K_{2})_{μ}, respectively. Then (K̃_{1})_{μ} and (K̃_{2})_{μ} are diffeomorphic to K_{μ}^{s} and K_{μ}^{u}, and we have (λ_{1})_{μ}(K̃_{1})_{μ} ⊂ (K̃_{1})_{μ} and (λ_{2})_{μ}^{−1}(K̃_{2})_{μ} ⊂ (K̃_{2})_{μ}, where (λ_{1})_{μ} and (λ_{2})_{μ} are the eigenvalues of the diffeomorphism φ_{μ}. By the differenciability of the family (φ_{μ}) relative to μ there is a constant C > 0 such that |(λ_{i})_{μ} − λ_{i}| ≤ C |μ| , 1 = 1, 2, for |μ| small. Besides, there are C^{1+} diffeomorphisms (h_{1})_{μ} and (h_{2})_{μ} which depend continuously in the C^{1+} topology on μ such that (K_{1})_{μ} = (h_{1})_{μ}((K̃_{1})_{μ}) and (K_{2})_{μ} = (h_{2})_{μ}((K̃_{2})_{μ}), with (h_{1})′_{0}(μ) = (h_{2})′_{0} = 1. We suppose, after reparametrization of ℓ_{μ}, that (K̃_{1})_{μ} ⊂ (−∞, 0], (K̃_{2})_{μ} ⊂ [0, ∞), 0 ∈ (K̃_{1})_{μ} ∩(K̃_{2})_{μ}, h_{1}(0) = μ and h_{2}(0) = 0.

By the hypothesis, there is an > 0 such that if (K, K′) is near ((K̃_{1})_{0}, (K̃_{2})_{0} + t), with |t′ − t| < then K intersects K′ stably. Suppose |λ_{1}| < 1, |λ_{2}| > 1 and let m, n ∈ N be such that |λ_{1}^{m}λ_{2}^{n} − 1| is sufficiently small. Then, if μ ∈ (λ_{2}^{−n}(t−), λ_{2}^{−n}(t+)), where n is large enough, we will have (K_{1})_{μ} ∩ (K_{2})_{μ} ≠ ∅. (λ_{1})_{μ}^{m}(K̃_{1})_{μ} ⊆ (K̃_{1})_{μ} and (λ_{2})_{μ}^{−n}(K̃_{2})_{μ} ⊆ (K̃_{2})_{μ}. It is enough to prove that (h_{1})_{μ}((λ_{1})_{μ}^{m}(K̃_{1})_{μ}) ∩ (h_{2})_{μ}((λ_{2})_{μ}^{−n}(K̃_{2})_{μ}) ≠ ∅, or, equivalently, that (λ_{1})_{μ}^{−m}·(h_{1})_{μ}((λ_{1})_{μ}^{m}(K̃_{1})_{μ})∩(λ_{1})_{μ}^{−m}·(h_{2})_{μ}((λ_{2}))μ^{−m}(K̃_{2})_{μ}) ≠ ∅, but this follows from the fact that the first set is close to (K̃_{1})_{0} +λ_{1}^{−m}μ and the second is close to (K̃_{2})_{0}, because (h_{1})_{μ}(x) ∼ x+μ and (h_{2})_{μ}(x) ∼ x for x near 0, and also λ_{1}^{m}/(λ_{1})_{μ}^{m} and λ_{λ}^{n}/(λ_{2})_{μ}^{n} are close to 1. This last affirmation follows from |(λ_{i})_{μ} − λ_{i}| ≤ K |μ| ≤ K′λ_{2}^{−n} ≤ K′′λ_{1}^{n}, where K, K′ and K′′ are constants. Then, for instance, ((λ_{1})_{μ}/λ_{1})^{m} ≤ (1 + K′′λ^{m−1})^{m} ∼ 1, because m · λ_{1}^{m−1} → 0 when m → ∞.

To conclude the proof it is enough to show that: (*) there is N ∈ N such that for each K ∈ N there is m ∈ N with K ≤ m < K + N and

n ∈ N with |λ_{1}^{m}λ_{2}^{n} − 1| suficiently small.

And (*) is a consequence of: (**) there are N′, N′′ ∈ N such that λ_{1}^{N′}λ_{2}^{N′′} is very close to 1.

Note that (**) is equivalent to obtaining N′, N′′ ∈ N such that N′ log λ_{1} + N′′ log λ_{2} is close to 0,which it is possible if log λ_{1}/ log λ_{2} is irrational by Dirichlet’s theorem and if log λ_{1}/ log λ_{2} = −p/q, p, q ∈ N we can take N′ = q, N′′ = p. To deduce (*) from (**) observe that λ_{1}^{N′}λ_{2}^{N′′} = λ_{1}^{τ} with |τ| very small. We have 2 cases:

• τ = 0: Take N = N′, m = KN′, n = KN′′(K ∈ N)

• τ ≠ 0: take Ñ = [\frac{1}{|τ|}] + 1, N = N′Ñ, m = N′ |[\frac{K}{τ} ]| − K \frac{τ}{|τ|} and n = N′′ |[\frac{K}{τ} ]| (K ∈ N).

Remark. 1. If \frac{log λ_{1}}{log λ_{2}} ∉ Q, it is enough to suppose there exists λ > 0 and t ∈ R such that (K̃_{1})_{0} intersects (λ(K̃_{2})_{0} + t) stably.

Obs. 2. The hypothesis of the Theorem I.2 are open, and as a consequence of the demonstration we can obtain an uniform positive estimate in a neighbourhood of the family (φ_{μ})

to lim inf_{δ→0} \frac{m(\{μ∈[0,δ]|(K_{1})_{μ}∩(K_{2})_{μ}≠∅\})}{δ} .

We say that the pair of Cantor sets (K_{1}, K_{2}) have extremal stable intersection if the right extreme of K_{1} coincides with the left one of K_{2}, and if (K̃_{1}, K̃_{2}) is close to (K_{1}, K_{2}), and the support intervals of K̃_{1} and K̃_{2} intersects their K̃_{1} ∩ K̃_{2} ≠ ∅. This implies in particular that (K_{1}+, K_{2}) has stable intersection, for any > 0 small.

The importance of this concept to the study of homoclinic bifurcations is that if the pair ((K_{1})_{0}, (K_{2})_{0}) have extremal stable intersection then {μ > 0|(K_{1})_{μ} ∩ (K_{2})_{μ} ≠ ∅} contains an interval [0, δ] for some δ > 0.

We shall study in this work conditions that imply stable intersections and extremal stable intersections of dynamically defined Cantor sets.

We end this section with a problem that we consider to be very important in our context: are the families {φ_{μ}} satisfying the condition in Theorem I.2 dense in the region {(φ_{μ}) | HD(K^{s}) + HD(K^{u}) > 1}? This is, in a certain sense, a stronger version of Palis’ conjecture, and, if it is true, we will have a stronger version of the results of Palis-Yoccoz paper [PY].

Section II: Thickness and lateral thickness

II.1: Definitions, examples and consequences of stable intersections Def.. A gap of a Cantor set is a connected component of its complement.

Given an U gap of a Cantor set K, we associate to it the intervals L_{U} and R_{U}, which are the intervals to its left and its right that separate it from the closest larger gaps.

We define τ_{R}(U) = \frac{|R_{U} |}{|U|}, τ_{L}(U) = \frac{|L_{U} |}{|U|}, τ_{R}(K) = inf{τ_{R}(U)|U bounded gap of K}, the right thickness of K; τ_{L}(K) = inf{τ_{L}(U)|U bounded gap of K}, the left thickness of K, and τ(K) = min{τ_{R}(K), τ_{L}(K)}, the thickness of K. Remark.. Given a Cantor set K, a presentation of K is an enumeration {U_{1}, U_{2}, ⋯ } of its limited gaps. We can define intervals L_{U} and R_{U} as being the intervals between U and the nearest gaps of indexes smaller than U. In an case we are using the presentation by order of the gaps’size, which maximizes the thickness τ(K). See [PT2].

The importance of the thicknesses derives from in the following result, which is an adaptation of the “gap lemma” ([PT2]). Proposition II.1. Given K_{1} Cantor sets, if τ_{R}(K_{1})·τ_{L}(K_{2}) > 1 and τ_{L}(K_{1})·τ_{R}(K_{2}) > 1, then either K_{1} is contained in a gap of K_{2} or K_{2} is contained in a gap of K_{1} or K_{1}∩K_{2} ≠ ∅. Remark.. This holds in particular if τ(K_{1}) · τ(K_{2}) > 1, which is the hypothesis of the known gap lemma.

Proof. Consider the case in which none of the K_{i} is contained in a gap of the other. There is, then, a pair of linked gaps, such as in the figures below:

a)

Since τ_{R}(U_{1})τ_{L}(U_{2}) > 1, we have |R_{U_{1}}| > |U_{2}| or |L_{U_{2}}| > |U_{1}| ⇒ a new pair of gaps exists like in b) smaller than the pair (U_{1}, U_{2})

b)

As τ_{L}(U_{1})τ_{R}(U_{2}) > 1, we have |L_{U_{1}}| > |U_{2}| or |R_{U_{2}}| > |U_{1}| ⇒ a new pair of gaps exists like in a) smaller than the pair (U_{1}, U_{2}).

In any case, we obtain a smaller pair of gaps, for instance in the sense that the sum of the lenghts of its gaps decrease. Using repeatedely the same argument we obtain a seguence of pairs of gaps converging to a point that necessarily belongs to K_{1} ∩ K_{2}. □

We can define, for p ∈ K, τ_{loc}(K, p) = lim sup_{→0} τ(\overline{K ∩ (p−, p+)}) and similar (τ_{R})_{loc}(K, p) and (τ_{L})_{loc}(K, p). For dynamically defined Cantor sets, τ_{loc}(K, p) does not depend on p (see [PT2], and the same proof shows that (τ_{R})_{loc}(K, p) and (τ_{L})_{loc}(K, p) do not depend on p either. Let us therefore, call them τ_{loc}(K), (τ_{R})_{loc}(K) and (τ_{L})_{loc}(K).

Proposition II.1 implies that if (τ_{R})(K_{1}) · (τ_{L})(K_{2}) ≥ 1 and (τ_{L})(K_{1}) · (τ_{R})(K_{2}) ≥ 1 then K_{1} − K_{2} contains an interval.

As a result, if (τ_{R})_{loc}(K_{1}) · (τ_{L})_{loc}(K_{2}) > 1 and (τ_{L})_{loc}(K_{1}) · (τ_{R})_{loc}(K_{2}) > 1, then K_{1} − K_{2} contains an interval.

For this last result we need strict inequalities. Indeed, in Sannami’s example of a dynamically defined Cantor set K with λ(K −K) > 0 and int (K −K) = ∅ have τ_{loc}(K) = 1. (See example III.3).

Also as a result of proposition II.1, if τ_{R}(K_{1}) · τ_{L}(K_{2}) > 1 and τ_{L}(K_{1}) · τ_{R}(K_{2}) > 1, and if τ_{R} and τ_{L} are continuous in K_{1} and K_{2}, then, for every t ∈ R such that K_{1} and K_{2} + t are linked, K_{1} intersects K_{2} + t stably.

Thickness τ(K) is continuous in the C^{1+} topology (see [PT2]). The same thing does not occur with lateral thicknesses: as be seen in the following example. Ex. II.1.1. In the set

the lateral thicknesses are not continuous.

In fact we have, next to K, sets

K′ K′′

for which τ_{R}(K_{′)= \frac{1+}{1−}} (→ 1 when → 0) and τ_{L}(K_{′)= \frac{1}{1−} (→2} when → 0), τ_{R}(K_{′′)= \frac{2+}{1−} (→2} when → 0) and τ_{L}(K_{′′)= \frac{1}{1−} (→1} when → 0), which shows that τ_{R} and τ_{L} are discontinuous in K.

We shall prove in II.2 that there is an open and dense set U in the C^{1+} topology where the lateral thicknesses are continuous. We shall also prove that the lateral thicknesses are continuous in affine Cantor sets of the type

for which the right thickness τ_{R}(K) = \frac{c}{b} and the left τ_{L}(K) = \frac{a}{b}. Therefore, if \frac{a_{1}c}{b_{1}b} > 1 e \frac{c_{1}a}{b_{1}b} > 1, the sets

have stable intersection whenever they are linked, and its arithmetic difference will contain an interval. This phenomenon being stable among dynamically defined Cantor sets.

There is an inequality whose proof can be found in [PT2] which states that HD(K) ≥ log 2/ log (2 + \frac{1}{τ(K)}), which implies that if τ(K) ≥ 1, then HD(K) ≥ log 3/ log 2 > 0.6. Therefore, if K_{1} and K_{2} have Hausdorff dimension 0.6 each, the product of their thicknesses is smaller than 1, and consequently the test of thickness cannot guarantee the existence of stable intersection between K_{1} and K_{2}.

However, the notion of lateral thickness makes it possible to prove the next result. Theorem II.1.2. Given h_{1}, and h_{2} in the interval (0,1) with h_{1} + h_{2} > 1, there are affine Cantor sets K_{1} and K_{2} with Hausdorff dimensions h_{1} and h_{2}, respectively, that intersect stably. Besides we can obtain that (K_{1}, K_{2}) and (K_{2}, K_{1} + 1) have extremal stable intersection. Proof. We shall look for examples of the type

In a set

with a a + b + c = 1 we have HD(K) = λ where λ is the only real number such that a^{λ} + c^{λ} = 1 ([see [PT2]), τ_{R}(K) = c/b and τ_{L}(K) = a/b.

Hence, if we have \frac{a_{1}c_{2}}{b_{1}b_{2}} > 1 and \frac{a_{2}c_{1}}{b_{1}b_{2}} > 1, (K_{1}, K_{2}) will have stable intersection, provided they are linked. Besides, when in the position where the extremes touch, (K_{1}, K_{2}) well have extremal stable intersection.

We shall try to obtain examples like the one above, with a_{1} + b_{1} + c_{1} = a_{2} + b_{2} + c_{2} = 1 and a_{1}c_{2} = a_{2}c_{1} (that is, \frac{c_{1}}{a_{1}} = \frac{c_{2}}{a_{2}} =: r), so that the conditions \frac{a_{1}c_{2}}{b_{1}b_{2}} > 1 and \frac{a_{2}c_{1}}{b_{1}b_{2}} > 1 are equivalent. Given h_{1} and h_{2} in the interval (0,1) with h_{1} + h_{2} > 1, we shall obtain these examples with HD(K_{1}) = h_{1} and HD(K_{2}) = h_{2}, that is, a_{1}^{h_{1}} + c_{1}^{h_{1}} = 1 and a_{2}^{h_{2}} + c_{2}^{h_{2}} = 1.

For a Cantor set

with c = ra, a + b + c = 1 and HD(K) = h we must have a^{h} + (ra)^{h} = 1 ⇒ a = \frac{1}{(1+r^{h})^{1/h}} ,

c = \frac{r}{(1+r^{h})^{1/h}} and b = \frac{(1+r^{h})^{1/h}−1−r}{(1+r^{h})^{1/h}}. In our case, we want a_{1}c_{2} > b_{1}b_{2}(⇔ a_{2}c_{1} > b_{1}b_{2}), and thus our problem is to find

r > 0 satisfying

\frac{1}{(1 + r^{h_{1}})^{1/h_{1}}} · \frac{r}{(1 + r^{h_{2}})^{1/h_{2}}} > \frac{((1 + r^{h_{1}})^{1/h_{1}} − 1 − r)}{(1 + r^{h_{1}})^{1/h_{1}}} · \frac{((1 + r^{h_{2}})^{1/h_{2}} − 1 − r)}{(1 + r^{h_{2}})^{1/h_{2}}} ⇔

(∗) r > ((1 + r^{h_{1}})^{1/h_{1}} − 1 − r) ((1 + r^{h_{2}})^{1/h_{2}} − 1 − r). We have

lim_{r→0_{+}} \frac{((1 + r^{h})^{1/h} − 1 − r)}{r^{h}} = \frac{1}{h}, lim_{r→0_{+}} \frac{((1 + r^{h_{1}})^{1/h_{1}} − 1 − r) ((1 + r^{h_{2}})^{1/h_{2}} − 1 − r)}{r} =

= lim_{r→0_{+}} \frac{((1 + r^{h_{1}})^{1/h_{1}} − 1 − r)}{r^{h_{1}}} · lim_{r→0_{+}} \frac{(1 + r^{h_{2}})^{1/h_{2}} − 1 − r)}{r^{h_{2}}} · lim_{r→0_{+}} \frac{r^{h_{1}+h_{2}}}{r} = \frac{1}{h_{1}} · \frac{1}{h_{2}} · 0 = 0

since h_{1} + h_{2} > 1, and thus, if r is sufficiently small, the right side of (∗) is less than the left one. Thus, for every r sufficiently small, we obtain examples as desired

a_{i} = \frac{1}{(1+r^{h_{i}})^{1/h_{i}}} , c_{i} = \frac{r}{(1+r^{h_{i}})^{1/h_{i}}} , b_{i} = 1 − a_{i} − c_{i}, for i = 1, 2 □ Remark. In the example IV.6 we will see another proof of this theorem.

II.2: Continuity of the Lateral Thicknesses

We saw in the example 1 of this section that the lateral thicknesses are not always continuous. However, there is an open and dense subset, in the C^{1+} topology of dynamically defined Cantor sets whose elements have these thicknesses varying continuously. To see this we shall prove that there is an open and dense subset of dynamically defined Cantor sets U in this topology such that if K ∈ U then there exists δ > 0 such that if U_{1} ≠ U_{2} are gaps of K and if there is no gaps V between U_{1} and U_{2} with |V | > 2 max{|U_{1}| , |U_{2}|} then:

(*)  1 + δ < |U_{2}| / |U_{1}| or |U_{2}| / |U_{1}| < 1 − δ

Such sets are points of continuity of the lateral thickness. In fact, in this case, given a gap U of K and a gap Ũ corresponding to U in a K̃ close to K, the gaps near U large than U which appear in the definition of thickness correspond dynamically to the gaps near Ũ larger than Ũ. This, together with the arguments of distortion of used in the proof of the continuity of thickness in [PT2], implies that the lateral thicknesses are continuous in a dense open set.

To prove the density of condition (*), we shall use the Lemma II.2.1. Generalized affine Cantor sets are “dense” in the topology C^{1+} in the sense that any K can be approximated by a K̃ that coincides as a subset of the straight line with a generalized affine Cantor set (but it does not have to be defined by the same functions.)

Proof. Consider an advanced step in the construction of K where K is a given Cantor set. The functions that define K will be replaced by functions that coincide with the preceding in the extremities of the bridges of this stage, being affine in these bridges, remaining in class C^{1}.

we claim that, if we do this in an advanced stage of the construction of K we can obtain a K̃ set close to K in the topology C^{1+}, and which coincides with an generalized affine set. For that purpose observe that if I_{1} and I_{2} are consecutive bridges and ψ ∈ C^{1+}, |ψ′(x) − ψ′(y)| < C |x − y|^{∀} x, y, and if U is the gap between I_{1} and I_{2} then there is a constant A with |U| ≥ A |I_{j}| j = 1, 2, by limited distortion. By the mean value theorem, when we linearize (render affine) the function ψ in I_{1} and I_{2} we shall obtain segments of inclination ψ′(c_{j}), c_{j} ∈ I_{j}, j = 1, 2. We want to extend the function ψ̃ thus obtained to U. We shall do this by extending the derivative of ψ̃. Let U = (a, b), c = \frac{a+b}{2}. Suppose ψ̃ ≡ λ_{j}: = ψ′(c_{j}) in I_{j}. We make ψ̃′ affine in (a, c) and (c, b), assuming the value λ_{3} in c.

where λ_{3} is such that ∫_{a}^{b} ψ̃′(t) dt = ψ(b) − ψ(a) ⇒

(\frac{b − a}{2}) (\frac{λ_{1} + λ_{2}}{2} ) + (\frac{b − a}{2}) (\frac{λ_{2} + λ_{3}}{2} ) = (b − a)ψ′(ξ), ∃ ξ ∈ (a, b)

⇒ λ_{3} = 2ψ′(ξ) − \frac{λ_{1} + λ_{2}}{2} = 2ψ′(ξ) − \frac{ψ′(c_{1}) + ψ′(c_{2})}{2}.

Therefore, ψ̃′ satisfies a condition of Hölder with the same and constant C̃ = 2(A + 2)C, and if we exchange for an˜ a little smaller, we have proved the nearness between ψ̃ and ψ in the C^{1+} topology.

Given the lemma, the result follows from the following observations:

i) the set of ratios {|U_{2}| / |U_{1}| | U_{1}, U_{2} are gaps of K such that there is no gap V of K with |V | > 2 max\{|U_{1}| , |U_{2}|\} between U_{1} and U_{2} and \frac{1}{2} < |U_{2}| / |U_{1}| < 2\} is finite when K is generalized affine, being equal to the set of ratios between the gaps of the first steps of the construction of K that are between 1/2 and 2; ii) given a finite number of gaps in a generalized affine Cantor set it is possible to modify

a little the initial partition that defines the set by keeping it generalized affine of the same type so that the ratio two of these gaps never be equal to 1. Since for generalized affine Cantor sets the distortion starting from a certain stage of

the construction is null, it is enough to make the ratio between two gaps crated up to this step (a finite number) never equal to 1, so that (*) be satisfied, and we have a point of continuity of lateral thicknesses.

Observe that the affine Cantor sets of type

are such that the ratio between the lenght of a gap and the lenghts of the nearest gaps larger than it are always far from 1, and are therefore points of continuity of the lateral thicknesses, as we have stated.

Section III: Examples about arithmetic differences and stable intersections

Example III.1. Let us consider the sets

The Hausdorff dimmension of K_{α} is equal to: −\frac{log 2}{log( \frac{1−α}{2} )} (see [PT2]), and τ(K_{α}) = τ_{L}(K_{α}) = τ_{R}(K_{α}) = \frac{1−α}{2α}. If α > \frac{1}{2} we shall have HD(K_{α}) < \frac{1}{2}, and (K_{α} − λK_{α}) will have measure zero for every λ ∈ R.

If α < \frac{1}{3}, τ(K_{α}) > 1 and (K_{α}, λK_{α}) will have stable intersection, therefore K_{α} − λK_{α} will contain an interval, ∀ λ ∈ R^{∗}.

We shall prove hereby that, when \frac{1}{3} < α < \frac{1}{2}, K_{α} − K_{α} has measure zero, but there are be intervals of values of λ (that will depende on α) for which K_{α} − λK_{α} contains an interval.

Proof. Consider the set K = K_{α} × K_{α} ⊂ R^{2} · K_{α} − K_{α} is its projection over R × {0} at an angle of 45^{o} (Fig.1).

At the first step of the construction of K_{α} × K_{α}, four congruent squares remain and the lower left square projects itself exactly over the one at the right higher corner, (Fig.2).

Therefore, by the self-similarity of K_{α}, the construction continues as if they were one only square. In case α > \frac{1}{2}, the projection of the first stage of the construction is not all the interval [0, 1], but the set [−1, −α] ∪ [\frac{α−1}{2}, \frac{1−α}{2}] ∪ [α, 1] consisting of three connected components. From then on, the difference is constructed as an affine Cantor set, being equal to the set

which has measure zero (and even Hausdorff dimension less than one).

The usual Cantor set K_{\frac{1}{3}} can be approximated, on one hand, by sets K_{\frac{1}{3} −}, > 0 that have stable self-intersection (and even (K_{\frac{1}{3} −}, K_{\frac{1}{3} −} + 1 has extremal stable intersection) and, on the other, by K_{\frac{1}{3} +}, > 0, that has arithmetic self-difference with measure zero and Hausdorff dimension < 1.

If the projection of the first step of the construction of K_{α} × (λK_{α}) is surjective over the interval [−λ, 1], the projections of all the stages will be surjective by self-similarity, and we shall have K_{α} − λK_{α} = [−λ, 1]. We shall determine when this happens:

= [−λ, \frac{1−α}{2} −λ(\frac{1+α}{2})]∪[\frac{1+α}{2} −λ, 1−λ(\frac{1+α}{2})]∪[−λ(\frac{1−α}{2}), \frac{1−α}{2}]∪[\frac{1+α}{2} −λ(\frac{1−α}{2}), 1], which is equal to the interval [−λ, 1] if (\frac{1−α}{2} −λ(\frac{1+α}{2}) ≥ \frac{1+α}{2} −λ, 1−λ(\frac{1+α}{2}) ≥ −λ(\frac{1−α}{2}) and \frac{1−α}{2} ≥ \frac{1+α}{2} − λ(\frac{1−α}{2})) or (\frac{1−α}{2} − λ(\frac{1+α}{2}) ≥ −λ(\frac{1−α}{2}), \frac{1−α}{2} ≥ \frac{1+α}{2} − λ, and 1 − λ(\frac{1+α}{2}) ≥ \frac{1+α}{2} − λ(\frac{1−α}{2})), that is, if (λ ≥ \frac{2α}{1−α} and λ ≤ 1) or (λ ≤ \frac{2α}{1−α} e λ ≥ α).

As α > \frac{1}{3}, \frac{2α}{1−α} > 1 and the first possibility never occurs. However, as α < 1/2, the interval [α, \frac{1−α}{2α}] is nonempty, and it is an interval of values of λ for which

K_{α} − λK_{α} = [−λ, 1]. For self-similarity, if λ ∈ ⋃_{k=0}^{∞} [(\frac{1 − α}{2})^{k}α, (\frac{1 − α}{2})^{k} · \frac{1 − α}{2α}],

K_{α} − λK_{α} will contain an interval.

An interesting observation is that the second part of this example is not as rigid as it looks. In fact, if λ ∈ (α, \frac{1−α}{2α}), \frac{1}{3} < α < \frac{1}{2} and t ∈ (−λ, 1) then K_{α} has an intersection with λK_{α} + t that we cannot prove to be stable, and (K_{α}, λK_{α} + 1) and (λK_{α} − λ, K_{α}) have no extremal stable intersections, as we shall prove in the next sections. However, these intersections cannot be destroyed by applying diffeomorphisms φ and ψ C^{1}-near the identity to K_{α} and λK_{α} + t. More precisely, given t ∈ (−λ, 1) there exists δ > 0 such that if ∥φ − Id∥_{1} < δ and ∥ψ − Id∥_{1} < δ then φ(K_{α}) ∩ ψ(λK_{α} + t) ≠ ∅. Besides, (K_{α}, λK_{α} + 1) and (λK_{α} − λ, K_{α}) have extremal intersections such that there exists δ > 0 so that if ∥φ − Id∥_{1} < δ and ∥ψ − Id∥_{1} < δ then φ(K_{α}) ∩ ψ(λK_{α} + 1) ≠ ∅ and φ(K_{α})∩ψ(λK_{α}−λ) ≠ ∅, provided that their support intervals have nonempty intersections.

In fact, the ratio between the lenghts of a gap of λK_{α} and a gap of K_{α} is equal to λ(\frac{1−α}{2})^{k}, for some k ∈ Z. If λ ∈ (α, \frac{1−α}{2α}) and δ is small enough then the ratio between a gap of φ(K_{α}) and a gap of ψ(λK_{α}) will be larger than \frac{2α}{1−α} or smaller than \frac{1−α}{2α}, for all the φ, ψ ∈ C^{1} with ∥φ − Id∥_{1} < δ and ∥ψ − Id∥_{1} < δ. Therefore, as in the proof of “gap lemma”, if we have linked gap pair (U, Ũ), for instance as below

where we suppose U > Ũ ⇒ U > \frac{2α}{1−α} Ũ, but we have τ_{L}(ψ(λK_{α})) ≃ τ_{L}(K_{α}) = \frac{1−α}{2α} ⇒ |L| > |Ũ| ⇒ we obtain a smaller gap pair. We proceed as in the proof of “gap lemma” and obtain the desired result (that is, if φ(K_{α}) and ψ(λK_{α} + t) have a linked gap pair therefore they have non-empty intersection.) Exemple III.2. We shall show here examples of afinne Cantor sets whose Hausdorff dimension is very close to 1 and whose arithmethic self-difference has measure zero. Let \overline{K}_{k} =

The projection of the first step of the construction of \overline{K}_{k} × \overline{K}_{k} is ⋃_{j=−k}^{k} [\frac{4j − 1}{4k + 1}, \frac{4j + 1}{4k + 1}],

which is not all the interval [−1, 1] and by the same argument of self-similitude applied to K_{α} − K_{α}, \overline{K}_{k} − \overline{K}_{k} will have measure zero (and even Hausdorff dimmension < 1: it will be an affine Cantor set) for the first step of the construction of \overline{K}_{k} × \overline{K}_{k} will consist of several congruent squares such that the projection of two of them either coincide or have empty intersection.

Another way of seeing this is to observe that \overline{K}_{k} is the set of numbers from [0, 1] in whose representation in base 4k + 1 only the numbers 0, 4, 8, …, 4k appear ⇒

32

\overline{K}_{k} − \overline{K}_{k} = \overline{K}_{k} + \overline{K}_{k} − 1, where \overline{K}_{k} + \overline{K}_{k} is equal to affiine Cantor set \{x ∈ [0, 2] all digits of x/2 in base 4k + 1 are even }.

The Hausdorff dimension of \overline{K}_{k} is \frac{log(k+1)}{log(4k+1)} (it is the only λ such that (k+1)·(\frac{1}{4k+1})^{λ} = 1; see [PT2]).

Therefore, as lim_{k→∞} \frac{log(k+1)}{log(4k+1)} = 1, we obtain the desired examples. Example III.3. Let K = \{ ∑_{i=1}^{∞} \frac{δ_{i}}{9^{i}}; δ_{i} ∈ \{0, 4, 6, 8\}\}. K is an affine Cantor set. We have:

K + K = \{x + y | x ∈ K, y ∈ K\} = \{ ∑_{i=1}^{∞} \frac{μ_{i}}{9^{i}} , μ_{i} ∈ \{0, 4, 6, 8\} + \{0, 4, 6, 8\} = \{0, 4, 6, 8, 10, 12, 14, 16\}\}

K − K = \{x − y | x ∈ K, y ∈ K\} = \{ ∑_{i=1}^{∞} \frac{λ_{i}}{9^{i}} , λ_{i} ∈ \{0, 4, 6, 8\} − \{0, 4, 6, 8\} = \{−8, −6, −4, −2, 0, 2, 4, 6, 8\}\}.

Therefore, K + K = \{x ∈ R | \frac{x}{2} = ∑_{i=1}^{∞} \frac{μ̃_{i}}{9^{i}} , μ̃_{i} ∈ \{0, 2, 3, 4, 5, 6, 7, 9\}\}, which is an affine

Cantor set, of measure zero and Hausdorff dimension smaller than on, while

K − K = \{x ∈ R | \frac{x + 1}{2} = ∑_{i=1}^{∞} \frac{λ̃_{i}}{9^{i}} , λ̃_{i} ∈ \{0, 1, 2, 3, 4, 5, 6, 7, 8\}\} = [−1, 1].

Thus,

K + K = \{x ∈ R | \frac{x}{2} = ∑_{i=0}^{∞} \frac{μ̃_{i}}{9^{i}} , μ̃_{i} ∈ \{0, 2, 3, 4, 5, 6, 7, 9\}\},

which is an affine Cantor set, with measure zero and Hausdorff dimension less than 1, while

K − K = \{x ∈ R | \frac{x + 1}{2} = ∑_{i=1}^{∞} \frac{λ̃_{i}}{9^{i}} , λ̃_{i} ∈ \{0, 1, 2, 3, 4, 5, 6, 7, 8\}\} = [−1, 1].

Therefore, K + K has measure zero, and K − K is an interval.

Example III.4. Let us consider the examples of Sannami ([S]) and of Bamón-Plaza-Vera ([BPV]), of dynamically define Cantor sets whose arithmetic self-difference is a Cantor set of positive measure. In these examples, the Cantor sets are central: a sequence \underline{α} = (α_{1}, α_{2}, α_{3}, …) of real numbers is given (in these cases we have lim_{i→∞}α_{i} = \frac{1}{3} and α_{i} > \frac{1}{3}, ∀ i) and the central Cantor set K\underline{α} is defined in the following manner: we take the interval [0,1], make a gap with the proportion α_{1}, at the center of the interval, and at the first step two connected components are left out. At the k-th step 2^{k} connected components will be left out, and gaps of proportion α_{k} in each connected component resulting from step k − 1 will be made. The intersection of the sets resulting from each step is the set K\underline{α}. The differentiability of K\underline{α} is related to the speed of convergence of sequence (α_{i}). We can aproximate \underline{α} by \underline{α}+: = (α_{i}+)_{i∈N}, obtaining sets K_{\underline{α}+} as diferentiable as K\underline{α}. From the calculation of the measure of the arithmetic self-difference of central Cantor sets (which is made in the referred articles) we can deduce that, if > 0 then λ(K_{\underline{α}+} − K_{\underline{α}+}) = 0. On the other hand, if < 0 we have τ_{loc}(K_{\underline{α}+}) > 1, and thus (K_{\underline{α}+}, K_{\underline{α}+}) will have stable intersection, and (K_{\underline{α}+}, K_{\underline{α}+} + 1) will have extremal stable intersection. In our last considerations we are referring to the sequence \underline{α} that appear in the mentioned examples (for which lim_{i→∞} α_{i} = \frac{1}{3}). In this cases we have τ_{loc}(K\underline{α}) = 1, as we observed in the comments after the proof of Proposition II.1.

IV: Criteria for stable and extremal stable intersections and generalized thckness test

We have seen in the preceding sections some examples of stable and extremal stable intersections of dynamically defined Cantor sets. In these examples the product of the right thickness of each of these sets by the left thickness of the other was always greater than one, so that it was possible to apply Proposition II.1.

In this section we shall give examples of stable and extremal stable intersections of Cantor sets with small lateral thicknesses, and we shall give more general criteria for stable and extremal stable intersections that generalize the idea of thickness. In the case of affine Cantor sets, it is possible to test these criteria by a finite algorithim.

We shall also prove a theorem that characterizes when an arithmetic difference K_{2} −K_{1}, of two dynamically defined Cantor sets, such that the right extreme of K_{1} coincides with the left one of K_{2}, contains an interval of the type [0,], > 0. This theorem will give us several examples of pairs of Cantor sets (K_{1}, K_{2}) which do not have extremal stable intersection. Example IV.1. Consider the affine Cantor sets

We have τ(\overline{K}) = 7/6 > 1, and thus \overline{K} intersects itself stably. Observe that K contains \overline{K} (we simply forget the two central intervals of the Markov partition of K, and consider the Cantor set defined by ψ|_{K_{1}} and ψ|_{K_{4}}, which coincides with \overline{K}).

For the same reason, a Cantor set K_{1} near K contains a Cantor set \overline{K}_{1} near \overline{K}. This implies that (K, K) have stable intersection, but τ_{R}(K) = τ_{L}(K) = \frac{2}{5} < 1. This example shows that the lateral thickness test of Proposition II.1 does not characterize stable intersections, and also shows that we can have \overline{K} ⊂ K with τ(\overline{K}) > τ(K) (same for τ_{R} and τ_{L}).

We shall see now a criteria for stable and extremal stable intersections which we will study first for affine Cantor sets, and later for the general cases.

Let us consider the affine Cantor sets

With Markov domains respectively P = ⋃_{j=1}^{n}[a_{j}, b_{j}] and P̃ = ⋃_{j=1}^{m}[ã_{j}, b̃_{j}]. We say that (K, K̃)

(or (P, P̃)) satisfies the Generalized Thickness Test (GTT) if for every t ∈ R and every λ ∈ R_{+}^{∗} either λP + t is contained in a gap of P̃ or P̃ is contained in a gap of λP + t or (λP + t) ∩ P̃ ≠ ∅.

This is equivalent to the condition that the arithmetic difference P̃ − λP contains the set of the t ∈ R such that, neither P̃ is contained in a gap of λP +t nor λP +t is contained in a gap of P̃, for every λ ∈ R_{+}^{∗}.

Geometrically, this means that every line that intercepts [a_{1}, b_{n}]×[ã_{1}, b̃_{m}] and makes an angle between 0 and π/2 with ther horizontal axis has an intersection with [a_{1}, b_{n}]×[ã_{1}, b_{m}], which is either contained in a vertical strip (b_{i}, a_{i+1}) × [ã_{1}, b̃_{m}] or in an horizontal strip [a_{1}, b_{n}] × (b̃_{i}, ã_{1+1}), or then intercepts P × P̃.

Algebraically, the set of the t such that λP +t is contained in some gap (b̃_{i}, ã_{i+1}) of P̃ is

⋃_{i=1}^{m−1} {t ∈ R|[λa_{1} + t, λb_{n} + t] ⊂ (b̃_{i}, ã_{i+1})} = ⋃_{i=1}^{m−1} (b̃_{i} − λa_{1}, ã_{i+1} − λb_{n}), and the set of the t

such that P̃ is contained in some gap (λb_{i}+t, λa_{i+1}+t) of λP +t is ⋃_{i=1}^{n−1} (b̃_{m}−λa_{i+1}, ã_{1}−λb_{i}).

Since the arithmetic difference P̃ − λP is equal to

⋃_{\begin{matrix}1≤i≤m \\ 1≤j≤n \end{matrix}} [ã_{i} − λb_{j}, b̃_{i} − λa_{j}],

we have that (K, K̃) satisfies the GTT ⇔ for any λ ∈ R_{+}^{∗},

(*)  ⋃_{\begin{matrix}1≤i≤m \\ 1≤j≤n \end{matrix}} [ã_{i} − λb_{j}, b̃_{i} − λa_{j}] ⊃ [ã_{1} − λb_{n}, b̃_{m} − λa_{1}] \ (A ∪ B),

where

A = ⋃_{i=1}^{m−1} (b̃_{i} − λa_{1}, ã_{i+1} − λb_{n}) and B = ⋃_{i=1}^{n−1} (b̃_{m} − λa_{i+1}, ã_{1} − λb_{i}).

Note that we have a finite number of possibilities for the relative position of the intervals [ã_{i} − λb_{j}, b̃_{i} − λa_{j}], (b̃_{i} − λa_{1}, ã_{i+1} − λb_{n}) and (b̃_{m} − λa_{i+1}, ã_{1} − λb_{i}), and to each possibility there corresponds a set of values of λ determined by a finite number of inequalities of the type A − Bλ ≤ C − Dλ, or A − Bλ < C − Dλ, which determine a real interval of values of λ whose extremes are functions of the a_{i}, ã_{j}, b_{i} and b̃_{j}. The GTT means that for each relative position of the intervals such that (*) does not hold we should have the corresponding interval of values of λ empty, which can be translated in a finite number of inequalities involving the a_{i}, ã_{j}, b_{i} and b̃_{j}. If these inequalities are strict, we say that (K, K̃) satisfies the GTT strictly.

If (K, K̃) satisfies the GTT, then (λK +t, K̃) also satisfies the GTT, ∀ t ∈ R, ∀ λ ∈ R_{+}^{∗}. Therefore, if (K, K̃) satisfies the GTT, I is an interval of the i^{th} step of the construction of K (that is, I is a connected component of ψ^{1−i}(P)) and J is an interval of the j^{th} step of the construction of K̃ then (K ∩ I, K̃ ∩ J) satisfies the GTT. If (U, Ũ) is a pair of linked gaps, U being a gap of the i\underline{th} step of the construction of K (ψ^{i}(U) = U′, U′ gap of P) contained in some interval I of the i^{th} step, and Ũ being a gap of the j^{th} step of the construction of K̃, contained in some interval J, then, applying the GTT to (K ∩I, K̃ ∩J), we will obtain a gap pair (U_{1}, Ũ_{1}), where U_{1} is a gap of the i^{th} step of the construction of K and Ũ_{1} is a gap of K̃ of a step superior to the j\underline{th}, or a gap pair (U_{1}, Ũ_{1}) where U_{1} is a gap of K of a step superior to the i\underline{th} and Ũ_{1} is a gap of K̃ of the j\underline{th} step. (See the figure below). Consequently, if (K, K̃) satisfies the GTT, then either K is contained in a gap of K̃, or K̃ is contained in a gap of K or K ∩ K̃ ≠ ∅, as in Proposition II.1.

If (K, K̃) satisfies the GTT strictly and there is a pair of linked gaps of K and K̃, then K intersects K̃ stably, and if the right extreme of K coincides with the left extreme of K̃, then (K, K̃) has extremal stable intersection.

Note that the conditions of the GTT are closed, while those of the Strict GTT are open. To apply the GTT to generalized affine Cantor sets K and K̃ defined by expansive

functions ψ and ψ̃, and with Markov partitions {K_{1}, K_{2}, …, K_{n}} and {K̃_{1}, K̃_{2}, …, K̃_{m}}, we must apply GTT to any pair (ψ(K_{i}) ∩ P, ψ̃(K̃_{j}) ∩ P̃), where P and P̃ are the Markov domains of K and K̃.

We shall see later how to apply the GTT when K and K̃ are not affine, and when some ψ_{i} or some ψ̃_{i} is decreasing.

Let us see now to apply the GTT when m = n = 2: Let

Let I_{1}: = [ã_{1} − λb_{2}, b̃_{1} − λa_{2}], I_{2}: = [ã_{1} − λb_{1}, b̃_{1} − λa_{1}], I_{3}: = [ã_{2} − λb_{2}, b̃_{2} − λa_{2}] and I_{4}: = [ã_{2} − λb_{1}, b̃_{2} − λa_{1}].

Since (b̃_{1}−λa_{1}, ã_{2}−λb_{2})∪(b̃_{2}−λa_{2}, ã_{1}−λb_{1}) = conv(I_{2}∪I_{3})\(I_{2}∪I_{3}), where conv(I_{2}∪I_{3}) is the convex hull of I_{2} ∪ I_{3}, we will have (*) satisfied if and only if I_{1} ∩ (I_{2} ∪ I_{3}) ≠ ∅ and I_{4} ∩ (I_{2} ∪ I_{3}) ≠ ∅, ∀λ ∈ R_{+}^{∗} (note that I_{1} ∩ I_{4} = ∅ is always empty).

We have:

I_{1} ∩ I_{2} ≠ ∅ ⇔ b̃_{1} − ã_{1} ≥ λ(a_{2} − b_{1}), I_{1} ∩ I_{3} ≠ ∅ ⇔ ã_{2} − b̃_{1} ≤ λ(b_{2} − a_{2})

I_{4} ∩ I_{2} ≠ ∅ ⇔ ã_{2} − b̃_{1} ≤ λ(b_{1} − a_{1}), I_{4} ∩ I_{3} ≠ ∅ ⇔ b̃_{2} − ã_{2} ≥ λ(a_{2} − b_{1})

Thus,

I_{1} ∩ (I_{2} ∪ I_{3}) = ∅ ⇔ \frac{b̃_{1} − ã_{1}}{a_{2} − b_{1}} < λ < \frac{ã_{2} − b̃_{1}}{b_{2} − a_{2}}

and

I_{4} ∩ (I_{2} ∪ I_{3}) = ∅ ⇔ \frac{b̃_{2} − ã_{2}}{a_{2} − b_{1}} < λ < \frac{ã_{2} − b̃_{1}}{b_{1} − a_{1}}

The TEG will be satisfied ⇔

\frac{ã_{2} − b̃_{1}}{b_{2} − a_{2}} ≤ \frac{b̃_{1} − ã_{1}}{a_{2} − b_{1}} e \frac{ã_{2} − b̃_{1}}{b_{1} − a_{1}} ≤ \frac{b̃_{2} − ã_{2}}{a_{2} − b_{1}} ,

that is, if and only if

\frac{b̃_{1} − ã_{1}}{ã_{2} − b̃_{1}} · \frac{b_{2} − a_{2}}{a_{2} − b_{1}} ≥ 1 and \frac{b̃_{2} − ã_{2}}{ã_{2} − b̃_{1}} · \frac{b_{1} − a_{1}}{a_{2} − b_{1}} ≥ 1 ⇔ τ_{R}(K_{1})τ_{L}(K_{2}) ≥ 1 and

τ_{L}(K_{1})τ_{R}(K_{2}) ≥ 1.

We also have that (K_{1}, K_{2}) satisfies the GTT strictly if and only if τ_{R}(K_{1})τ_{L}(K_{2}) > 1 and τ_{L}(K_{1})τ_{R}(K_{2}) > 1.

We shall see now how to apply an adaptation of the GTT to dynamically defined Cantor sets, not necessarily affine. Observe first that applying the GTT to a pair of affine Cantor sets (K_{1}, K_{2}) is equivalent to applying them to (λK_{1}, μK_{2}), ∀ λ, μ ∈ R_{+}^{∗}. Consequently, we can suppose that the support intervals of K_{1} and K_{2} are equal to [0, 1]. We shall have in this case the following normalized representations of K_{1} and K_{2}:

where c_{1} +d_{1} +c_{2} +d_{2} +⋯+d_{n−1} +c_{n} = 1 and c̃_{1} +d̃_{1} +c̃_{2} +d̃_{2} +⋯+d̃_{m−1} +c̃_{m} = 1. If \underline{c}: = (c_{1}c_{2}, ⋯, c_{n}), \underline{d}: = (d_{1}, …, d_{n−1}), \underline{c̃}: = (c̃_{1}, …, c̃_{m}) and ˜ \underline{d} = (d̃_{1}, …, d̃_{m−1}) then the conditions of the GTT applied to the pair (K_{1}, K_{2}) can be translated in a finite number of inequalities f_{i}(\underline{c}, \underline{d}, ˜ \underline{c}, ˜ \underline{d}) ≥ 0, for 1 ≤ i ≤ ℓ. As we saw before, if I and I_{2} are intervals of any steps of the constructions of K_{1} and K_{2} then applying the GTT to (K_{1}, K_{2}) is equivalent of applying them to (I_{1} ∩ K_{1}, I_{2} ∩ K_{2}).

For a pair of dynamically defined Cantor sets (K_{1}, K_{2}) choose I_{1} and I_{2} intervals of some steps of the constructions of K_{1} and K_{2}, and apply the GTT to (ψ_{1}^{−n_{1}}(P_{1})∩I_{1}, ψ_{2}^{−n_{2}}(P_{2})∩ I_{2}) (as for affine Cantor sets), where P_{i} is the Markov domain of K_{i}, ψ_{i} is the expansive function that defines K_{i} and n_{i} is the step of I_{i}, i = 1, 2. This test, after normalization, can be translated in inequalities f_{i}(\underline{c}(P_{1}^{∗}), \underline{d}(P_{1}^{∗}), ˜ \underline{c}(P_{2}^{∗}), ˜ \underline{d}(P_{2}^{∗})) ≥ 0, 1 ≤ i ≤ ℓ, where P_{i}^{∗} = ψ_{i}^{−n_{i}}(P_{i}) ∩ I_{i} = 1, 2, which do not depend on the choice of I_{1} and I_{2} if K is affine. In the general case, we say that (K_{1}, K_{2}) satisfies the GTT if

A_{j}(K_{1}, K_{2}): = inf_{I_{1},I_{2}}f_{j}(\underline{c}(P_{1}^{∗}), \underline{d}(P_{1}^{∗}), ˜ \underline{c}(P_{2}^{∗}), ˜ \underline{d}(P_{2}^{∗})) ≥ 0 for 1 ≤ j ≤ ℓ,

and that (K_{1}, K_{2}) satisfies the GTT strictly if A_{j}(K_{1}, K_{2}) > 0, 1 ≤ j ≤ ℓ. Note that, as in the affine case, the strict GTT implies stable intersection and extremal stable intersection of (K_{1}, K_{2}) if there is a gap pair or if the extremes coincide, respectively. Note also that ℓ depends only on the combinatorial type of K_{1} and K_{2}. The first statement follows from the continuity of the A_{j}(K_{1}, K_{2}) in the C^{1+} topology, which follows from the bounded distortion property for dynamically defined Cantor sets (see the proof of the continuity of τ(K) in [PT2]). As for τ(K) we can define (A_{j})_{loc}(K_{1}, K_{2}), 1 ≤ j ≤ ℓ and if (A_{j})_{loc}(K_{1}, K_{2}) > 0, 1 ≤ j ≤ ℓ then (K_{1}, K_{2}) have extremal stable intersection, provided that the extremes coincide. Remark. If any of the functions that define K_{1} or K_{2} is decreasing,

then we must apply the GTT for (λP_{1} + t, P_{2}), ∀λ ∈ R^{∗}, and not only for λ positive, that is, it is necessary to make sure that either λP_{1} + t is contained in a gap of P_{2} or P_{2} is contained in a gap of λP_{1} + t or (λP_{1} + t) ∩ P_{2} ≠ ∅, for every t ∈ R, λ ∈ R^{∗}. This also applies to the dynamically defined case. Example IV.2. If then (K_{1}, K_{2} + 8) have extremal stable intersection by the GTT. However, if K̃_{1}, = K̃_{2} is defined as below, by affine functions, then (K̃_{1}, K̃_{2} + 8) do not have extremal stable intersection. Inded, K̃_{1} = K̃_{2} has gaps near the left extreme like:

. and near the right extreme like ⇒ [−8, −8+] ⊄ K̃_{1} − K̃_{2} ∀ > 0.

K̃_{1} = K̃_{2}:

So, we see that it is really necessary to consider all the values of λ ∈ R^{∗} if there is some decreasing function. Exemple IV.3. Let

we have τ_{R}(K_{1})τ_{L}(K_{2}) = \frac{2}{3} · \frac{2}{3} = \frac{4}{9} < 1, hence we cannot apply the lateral thicknesses test of proposition II.1 to find stable intersections between K_{1} and K_{2}. However, (K_{1}, K_{2}) satisfies the GTT strictly. Indeed, if a gap U of K_{1} is of the type of V_{1} (there exists n such that ψ^{n}(U) = V_{1}, where ψ is the function that defines K_{1}) then τ_{R}(U) = \frac{2}{3} and τ_{L}(U) = \frac{7}{3}. If Ũ is a gap of K_{2} of the kind of W_{1} then τ_{R}(Ũ) = 3 and τ_{L}(Ũ) = \frac{7}{4}. If Ũ is of the type of W_{2} then τ_{R}(Ũ) = \frac{7}{3} and τ_{L}(Ũ) = \frac{2}{3}.

Thus, a gap pair (U, Ũ) satisfies the conditions of the gap lemma except if U =: U_{0}^{1} is of the kind of V_{1} and Ũ =: U_{0}^{2} is of the kind of W_{2}. In this case, we will have a situation like:

If |U_{0}^{1}| > |L_{U_{0}^{2}} | then |L_{U_{0}^{1}} | = \frac{7}{3} |U_{0}^{1}| > |U_{−1}^{2}|, because |U_{−1}^{2}| = 2 |L_{U_{0}^{2}} | and 2 < \frac{7}{3}. Thus, necessairly we will have an intersection between (λP_{1} + t) and P_{2}, and (K_{1}, K_{2}) satisfies the GTT. Exemple IV.4. If α ∈ (\frac{1}{3}, \frac{1}{2}) then (K_{α}, K_{α}) does not satisfy the GTT, the same ocurring in a neighbourhood of (K_{α}, K_{α}). We will prove later that if (K, K̃) is close to (K_{α}, K_{α} + 1) then (K, K̃) will not have extremal stable intersection. Note that if α ∈ (\frac{1}{3}, \frac{1}{2}), then HD(K_{α}) > \frac{1}{2} and HD(K_{α}) + HD(K_{α}) > 1 and thus extremal stable intersections are not dense in {(K_{1}, K_{2}) | HD(K_{1}) + HD(K_{2}) > 1}. Exemple IV.5. Let us give now an example of a pair of Cantor sets (K_{1}, K_{2}) which does not satisfy the GTT (and such that (K_{1}, K_{2} + t) do not have extremal stable intersection, ∀ t ∈ R) such that if we substitute K_{2} by an K̃_{2} aparently “smaller”, (K, K̃_{2}) does satisfy the GTT. We will also exibit an K̃_{1} such that (K̃_{1}, K̃_{2}) does not satisfy the GTT (and does not have extremal stable intersection), but (K̃_{1}, K_{2}) satisfies the GTT. This ilustrates the combinatorial character of the GTT.

Let

Surely (K_{1}, K_{2}) and (K̃_{1}, K̃_{2}) do not satisfy the GTT (and do not have extremal stable intersections, as we shall see later) by the pictures below:

We have τ(K̃_{1})τ(K_{2}) = 2 · \frac{2}{3} = \frac{4}{3} > 1, and thus (K̃_{1}, K_{2}) satisfies the GTT. For (P_{1}, P̃_{2}), the only pair of gaps that does not satisfy the condition of the gap lemma is (U, V). In this case, we have:

Also, to avoid intersections between λP_{1} + t and P̃_{2} we must have 2 + 7 + 3 < 3λ + 2λ + 4λ ⇒ 9 < 8λ ⇒ the gap of P̃_{2} of lenght 9 ends in the interval of λP_{1} + t of lenght 8λ ⇒ (λP_{1} + t) ∩ P̃_{2} ≠ ∅ ⇒ (K_{1}, K̃_{2}) satisfies the GTT. Clearly, by the arguments above, (K̃_{1}, K_{2}) and (K_{1}, K̃_{2}) satisfy the GTT stably.

Example IV.6. Let us give another proof of Theorem II.I.2, in which the sets K_{1} and K_{2} that we obtain are be simmetric, and can be seen as stable and unstable Cantor sets associated to a horseshoe in the sphere (in what follows, if n_{0} = 2^{k}, K_{1} and K_{2} have the same combinatories as the k^{th} power of the usual horseshoe).

Let h_{1}, h_{2} in the interval (0,1) such that h_{1} + h_{2} > 1, and let a < h_{1} and b < h_{2} such that a + b > 1. Consider for each λ ∈ (0, 1), the affine Cantor set

(with [n^{λ} log n] gaps of lenght n^{1−λ})

Since the Hausdorff dimension of a Cantor set normalized such that ∑_{k=1}^{n} a_{k} + ∑_{j=1}^{n−1} b_{j} = 1, is the only μ ∈ (0, 1) such that ∑_{k=1}^{n} a_{k}^{μ} = 1

(see [PT 2]), we have HD(K_{n}^{λ}) = μ, where μ satisfies

2 (\frac{n}{2n + 2([n^{λ} log n] − 1) + n^{1−λ}[n^{λ} log n]})^{μ} +

+ ([n^{λ} log n] − 1) (\frac{2}{2n + 2([n^{λ} log] − 1) + n^{1−λ}[n^{λ} log n]})^{μ} = 1.

If we denote by g_{λ}(n, μ) the left side of this equation, and observe that 2n + 2([n^{λ} log n] − 1) + n^{1−λ}[n^{λ} log n] is of the order of n log n, we conclude that

μ > λ ⇒ lim_{n→∞}g_{λ}(n, μ) = 0, and μ < λ ⇒ lim_{n→∞}g_{λ}(η, μ) = +∞. Thus, lim_{n→∞}HD(K_{n}^{λ}) = λ,

and then there exists n_{0} ∈ N such that HD(K_{n_{0}}^{a}) < h_{1} and HD(K_{n_{0}}^{b}) < h_{2}. We are

where M_{1} ≥ n_{0} and M_{2} ≥ n_{0} are choosen such that HD(K_{1}) = h_{1} and HD(K_{2}) = h_{2} (HD(K) depends continuously on K).

(K_{1}, K_{2}) satisfies the GTT: We have \frac{M_{1}}{n_{0}^{1−a}} · \frac{2}{n_{0}^{1−b}} ≥ \frac{2·n_{0}}{n_{0}^{2−a−b}} ≥ 2 > 1 and \frac{M_{2}}{n_{0}^{1−b}} · \frac{2}{n_{0}^{1−a}} ≥ 2 > 1, Thus, studying intersections of pairs (P_{1}, λP_{2} + t), where P_{i} is the Markov domain of K_{i} let us suppose, for instance, that the right interval I_{1} of P_{1} (but not all the P_{n}) is contained in a gap U = (a, b) of λP_{2} + t. Let Ũ be the gap of P_{1} that contains a and

I_{2} = [c, a] the interval of λP_{2} + t at the right of (a, b). We have \frac{|I_{1}|}{\left|{ \atop Ũ}\right|} · \frac{|I_{2}|}{|U|} ≥ \frac{M_{1}}{n_{0}^{1−b}} · \frac{2}{n_{0}^{1−b}} > 1,

and |I_{1}| < |U| ⇒ |I_{2}| > |Ũ| ⇒ P_{1} intersects λP_{2} + t. The other possible positions of (P_{1}, λP_{2} + t) have similar analysis. By the arguments above it is clear that (K_{1}, K_{2}) satisfies the GTT stably.

In the example IV.1, we have constructed a Cantor set \overline{K} ⊂ K with τ(\overline{K}) > τ(K) by simply forgetting some intervals of K, and considering the Cantor set defined by the same function restricted to the remaining intervals. This can be done even if some of the forgotten intervals are extremal, provided that the Markov partition of K has more than two intervals. In this case the description of \overline{K} is slightly more complicated: Indeed, let K be a Cantor set with Markov partitions {K_{1}, K_{2}, …, K_{n}} defined by an expanding function ψ, such that ψ|_{K_{1}} is increasing and surjective. If ψ|_{K_{n−1}} is increasing and surjective, then ψ|_{K_{n−1}} has an unique fixed point b. If ψ|_{K_{n−1}} is decreasing and surjective, we take b equal to the right extreme of K_{n−1}. Then we have a dynamically defined Cantor set \overline{K} contained in K with Markov domain \{\overline{K}_{1}, \overline{K}_{2}, … \overline{K}_{n−1}\} where \overline{K}_{i} = ψ^{−1}([0, b])∩K_{i}, i = 1, 2, …, n−1, defined by the (the restriction of) ψ.

This Cantor set does not depend on K_{n} or on ψ|_{K_{n}}.

As an example, if

\begin{matrix}\frac{14}{5}(b − 6) = b ⇒= \frac{28}{3} \\ \frac{14}{5}x = \frac{28}{3} ⇒ x = \frac{10}{3} \end{matrix}

If K̃ = then clearly the pair (K, K̃) does not satisfy the GTT, as showed in the picture below:

However, τ(\overline{K})τ(K̃) = \frac{5}{4} 1 = \frac{5}{4} > 1, therefore (K̃, \overline{K} + 3) has extremal stable intersection, and so does (K̃, K + 3), since \overline{K} depends continuously on K.

On the other hand, (K, K̃ + 14) does not have extremal stable intersection, by Theorem IV.1 , and so the existence of extremal stable intersection depends on the order of the pair of Cantor sets.

Let us make a definition: Given compact sets C_{1} and C_{2} with convex hulls [a, b] and [c, d], respectively, we say that C_{1} → C_{2}: ⇔ (a < c, b < d and c < b).

Theorem IV.1. Let (K, \overline{K}) be a pair of dynamically defined Cantor sets whose Markov partitions are P = \{K_{1}, K_{2}, …, K_{n}\} and \overline{P} = \{\overline{K}_{1}, \overline{K}_{2}, …, \overline{K}_{2}\} respectively, such that the right extreme of K coincides with the left one of \overline{K}. Suppose that ψ|_{K_{n}} and \overline{ψ}′|_{\overline{K}_{1}} are affine, increasing and surjective, and that log(ψ′|_{K_{n}})/ log(\overline{ψ}′|_{\overline{K}_{1}}) ∉ Q, where ψ and \overline{ψ} are the functions that define K and \overline{K}. Then: \overline{K} − K ⊅ [0, δ], ∀δ > 0 ⇔ ∃λ > 0, t ∈ R s.t. (λK + t) → \overline{K} and (λK + t) ∩ \overline{K} = ∅. Remarks.

• This last condition is equivalent to the existence of a straight line ℓ which crosses only the upper and the right sides of K_{0} × \overline{K}_{0} and does not intersect some step P_{i} × \overline{P}_{i} of the construction of K × \overline{K}, where P_{i} = ψ^{1−i}(P), \overline{P}_{i} = \overline{ψ}^{1−i}(\overline{P}), and K_{0} and \overline{K}_{0} are respectively the convex hulls of K and \overline{K}

• We can supress the condition log(ψ′|_{K_{n}})/ log(ψ′/\overline{K}_{1}) ∉ Q if we change the last condition by ∃t ∈ R such that (K + t) → \overline{K} and (K + t) ∩ \overline{K} = ∅.

• If ψ|_{K_{n}} and ψ|_{\overline{K}_{1}} are increasing and surjective but not affine, then the results are also valid, provided that we test the hypothesis for the linearized of K and \overline{K} (for which the analogs of ψ|_{K_{n}} and \overline{ψ}|_{\overline{K}_{1}} are affine).

Proof. Let λ_{1} = ψ′|_{K_{n}} and log λ_{1}/ log λ_{2} ∉ Q, there are, for each δ > 0, an infinity of pairs of natural numbers (m, n) such that |λ_{2}^{m}/λ_{2}^{n} − λ| < δ Since t ∉ \overline{K} −λK, there is δ > 0 such that t′ ∉ \overline{K} − λ′K ∀ λ′, t′ with |t − t′| < δ, and |λ − λ′| < δ ⇒ t ∉ \overline{K} − (λ_{2}^{n}/λ_{1}^{m})K ⇒ λ_{1}^{−n}t ∉ λ_{2}^{−n}\overline{K} − λ_{1}^{−m}K, with (λ_{1}^{m}K + λ_{2}^{−n}t) → λ_{2}^{−n}\overline{K}, and, since \overline{K}_{0} ∩ (K_{0} + λ_{2}^{−n}t) = (λ_{1}^{−m}K_{0} + λ_{2}^{−n}t) ∩ (λ_{2}^{−n}K_{0}), λ_{2}^{n}t ∉ \overline{K} − K. Therefore, \overline{K} − K ⊅ [0,], ∀ > 0. The other implication of this theorem is clear. Remark 1. The above demonstration also shows that, as in Theorem I.2, in the hypothesis of Theorem IV.1, (K_{2} −K_{1})^{c} has positive density at 0. We shall adapt this result in section V, to show that in the case of homoclinic bifurcations, if K = K^{s} and \overline{K} = K^{u} satisfy the hypothesis of the Theorem IV.1, then we shall have φ_{μ} persistently hyperbolic for a set of positive values of μ with positive density at μ = 0. Remark 2. If K_{1} and K_{2} are affine Cantor sets satisfying the hypothesis of Theorem IV.1, if there are λ > 0 and t ∈ R such that K_{1} → (λK_{2} + t) and K_{1} ∩ (λK_{2} + t) = ∅ then, by this theorem, (K_{1}, K_{2}) will not have extremal stable intersection, since (Perhaps by changing slightly the pair (K_{1}, K_{2}) to satisfy the hypothesis about λ_{1} and λ_{2}) its arithmetic difference will not contain [0,], ∀ > 0. Consequently, if

then (K, \overline{K} + 8) will not have extremal stable intersection, as we stated before. The statements in of the examples IV.4 and IV.5 also follow. Example IV.8. Let

where a + b + c = 1. we remember that HD(K) = HD(\overline{K}) = λ, where λ is the only real number such that a^{λ} + c^{λ} = 1. Therefore, even if we chose b small, we can choose a so small that HD(K) = HD(\overline{K}) < \frac{1}{2}. In this case, since HD(K) + HD(\overline{K}) < 1, we have m(K − \overline{K}) = 0, and so (K, \overline{K} + 1) will not have extremal stable intersection, and K − \overline{K} will not contain any interval. However, to apply Theorem IV.1, we must consider very advanced steps of the construction of K × \overline{K}, even in this case, which is combinatorially very simple.

Example IV.9. Let

The first step of the construction of K × \overline{K} does not assure that (K, \overline{K} + 10) does not have a stable intersection, but the second step does:

Section V: Intersections of Cantor sets and hyperbolicity

In [PT1], Palis and Takens proved the following theorem:

Let {φ_{μ} | μ ∈ R} be a family of diffeomorphisms of M^{2} with an homoclinic Ω-explosion at μ = 0. Suppose that HD(K^{s}) + HD(K^{u}) < 1 Then

lim_{δ→0} \frac{m(\{μ ∈ [0, δ] | φ_{μ} is persistently hyperbolic \})}{δ} = 1,

where m denotes the Lebesgue measure.

In [PY], Palis and Yoccoz proved that if HD(K^{s}) + HD(K^{u}) > 1 then generically

lim sup_{δ→0} \frac{m(\{μ ∈ [0, δ] | φ_{μ} is not persistently hyperbolic )}{δ} > 0,

because we have new tangencies between W^{s}(Λ) and W^{u}(Λ) with positive upper density at μ = 0, where Λ is the basic set of φ_{0} associated to the homoclinic tangency.

Here we shall prove that if K^{s} and K^{u} satisfy the hypothesis of Theorem IV.1, then we have hyperbolicity with positive density at μ = 0, even if HD(Λ) > 1. The proof of this result will be a combination of the arguments in [PT1] and Theorems I.2 and IV.1. We shall also give an example of an open set of families (φ_{μ}) where the phenomena of hyperbolicity and of new tangencies between W^{s}(Λ) and W^{u}(Λ) have both positive density at μ = 0.

Theorem V.1. Let (φ_{μ}) be a family of diffeomorphisms of M^{2} that presents a homoclinic Ω-explosion at μ = 0 associated to a fixed saddle point p that belongs to a basic set Λ, and let (K̃_{1})_{0} and (K̃_{2})_{0} be as in Theorem I.2. Suppose further that there is t ∈ R such that ((K̃_{1})_{0} + t) → (K̃_{2})_{0} and ((K̃_{1})_{0} + t) ∩ (K̃_{2})_{0} = ∅.

Then

lim inf_{δ→0} \frac{m(\{μ ∈ [0, δ] | φ_{μ} is persistently hyperbolic \})}{δ} > 0.

Remark. In the hypothesis of this theorem, as in the hypothesis of Theorem I.2, we suppose that (K̃_{1})_{0} ⊂ R_{−}, (K̃_{2})_{0} ⊂ R_{+} and 0 ∈ (K̃_{1})_{0} ∩ (K̃_{2})_{0}.

Proof. Let O the orbit of homoclinic tangency, q^{s} ∈ O ∩ W_{loc}^{s}(p) and q^{u} ∈ O ∩ W_{loc}^{u}(p). Let q_{μ}^{s} ∈ W_{loc}^{s,μ}(p_{μ}) and q_{μ}^{u} ∈ W_{loc}^{u,μ}(p_{μ}) depending continuously on μ such that q_{0}^{s} = q^{s} and q_{0}^{u} = q^{u} and q_{0}^{u} = q^{u}, and let O_{μ}^{s} and O_{μ}^{u} their φ_{μ} − orbits. Let A(μ) = K_{μ}^{s}∪O_{μ}^{s} and B(μ) = K_{μ}^{u}∪O_{μ}^{u}, where K_{μ}^{s} = W_{loc}^{s,μ}(p_{μ})∩Λ_{μ} and K_{μ}^{u} = W_{loc}^{u,μ}(p_{μ})∩ Λ_{μ}. (Λ_{μ} is the continuation of Λ). Choose parametrizations for W_{loc}^{s,μ}(p_{μ}) and W_{loc}^{u,μ}(p_{μ}) (for which φ_{μ}|_{W_{loc}^{s,μ}}(p_{μ}) and φ_{μ}|_{W_{loc}^{s,μ}}(p_{μ}) are linear maps) in such a way that K_{μ}^{s} and K_{μ}^{u} coincides with (K̃_{1})_{μ} and (K̃_{2})_{μ}, and that A(μ) and B(μ) are scaled sets, with scaling factors λ_{1}(μ) and λ_{2}(μ), respectively, where λ_{1}(μ) and λ_{2}(μ) are the eigenvalues of φ_{μ}. For a set A ⊂ R we define _{δ}A = {x ∈ R | ∃x′ ∈ A such that |x − x′| ≤ δ |x′|}.

Let Ã^{K·μ}(μ′) = π_{s,μ′}^{−1}(_{A}(μ′)∩[−Kμ, Kμ]),and_{B̃^{K·μ}=π_{u,μ′}^{−1} (_{B}(μ′)∩[−Kμ,Kμ])} where π_{s,μ′} and π_{u,μ′} are the projection associated to the stable and unstable foliations of φ_{μ′} over the line of tangencies ℓ_{μ}. Let C_{μ,} = {μ′ ∈ (0, μ) | the distance between _{B̃^{Kμ}(μ′)} and _{Ã^{Kμ}(μ′)} is greater or equal to μ\}. If we could prove that lim inf_{μ→0} \frac{m(C_{μ,})}{μ} > 0 for some > 0 then the result would follow exactly as in [PT1].

Let _{Õ_{μ′}^{s,Kμ} =π_{s,μ′}^{−1} (_{O} _{μ′}^{s} (μ′)∩[−Kμ,Kμ])}, and _{Õ_{μ′}^{u,K·μ}=π_{u,μ′}^{−1} (_{O} _{μ′}^{u})∩[−Kμ,Kμ])} then C′_{μ,} = {μ′ ∈ (0, μ) | d(_{A}^{Kμ}(μ′),_{O}_{μ′}^{u,Kμ}) ≥ μ and d(_{O}_{μ′}^{s,kμ}, _{B}^{Kμ}(μ′)) ≥ μ} is such that given δ > 0 there is > 0 for which we have lim inf_{μ→0} \frac{m(C′_{μ,})}{μ} > 1 − δ, by the arguments of [PT1], since O_{μ}^{s} and O_{μ}^{u} are countable, and thus have Hausdorff dimension zero.

Let _{K̃_{μ′}^{u,μ}=π_{u,μ′}^{−1} (_{K̃{u \atop μ′} ∩[−Kμ,Kμ])}}, and C′′_{μ′,} = \{μ ∈ (0, μ) | d(K̃_{μ′}^{u,μ}_{K̃{s,μ \atop μ′})≥μ\}}. Then we can prove, by arguments similar to the arguments of Theorem I.2, that if > 0 is sufficiently small then lim inf_{μ→0} \frac{m(C′′_{μ,})}{μ} > 0 (the essence of this argument is that “non-intersections” are always stable). Thus, if > 0 is sufficienlty small then is lim inf_{μ→0} \frac{m(C′′_{μ,})}{μ} > 0, and the theorem is proved. □

Remark. If log(λ_{1}(0))/ log(λ_{2}(0)) ∉ Q, it is enough to suppose that there are λ > 0, t ∈ R such that (λ(K̃_{1})_{0} + t) → (K̃_{2})_{0} and (λ(K̃_{1})_{0} + t) ∩ (K̃_{1})_{0} = ∅.

Example. Suppose that K_{0}^{s} and K_{0}^{u} are affine Cantor sets as in the pictures B and C below, associated to a horseshoe as in pinture A, which is combinatorially equivalent to the square

56

of the usual horseshoe.

Let us suppose that (φ_{μ}) presents a homoclinic Ω-explosion at μ = 0 associated to the fixed point p in the left lower extreme of Λ. Then, since the hypothesis of the Theorem V.1 are satisfacted, we shall have hyperbolicity with positive density at μ = 0.

On the other hand, as the picture D shows, K_{0}^{s} contains an affine Cantor set with thickness 17/10 (we “forget” the right interval). Similarly, we can show that K_{0}^{u} contains an affine Cantor set with thickness greater than 1 ⇒ there is a t ∈ R such that (K_{0}^{s} + t) intersects K_{0}^{u} stably, and so, according to Theorem I.2, there will be tangencies between W^{s}(Λ_{μ}) and W^{u}(Λ_{μ}) with positive density at μ = 0. The same phenomena occur in a neighbourhood of (φ_{μ}), since the hypothesis are open.

REFERENCES

[BPV] R. Bamón, S. Plaza and J. Vera: On Central Cantor Sets with self-arithmetic difference of positive Lebesgue measure, to appear in J. London Math. Soc.

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[MO] Mendes, P. and Oliveira, F.: On the topological structure of the arithmetic sum of two Cantor sets, Nonlinearity 7 (1994), 329-343.

[N1] S. Newhouse: Non density of Axiom A(a) on S^{2}, Proc.A.M.S. Symp. Pure Math. 14 (1970), 191-202.

[N2] S. Newhouse: Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9-18.

[N3] S. Newhouse: The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. IHES 50 (1979), 101-151.

[P] Palis, J., Homoclinic bifurcations, sensitive chaotic dynamics and strange attractors, Dynamical Syst. and Related Topics, World Scientific (1991), p. 466-473.

[PT] Palis, J. and Takens, F., Cycles and measure of bifurcation sets for two-dimensional diffeomorfisms, Invent.Math. 82, 1985, p. 379-442.

[PT1] Palis, J. and Takens, F., Hyperbolicity and the creation of homoclinic orbits, Annals of Math. 125 (1987), p. 337-374.

[PT2] Palis, J. and Takens, F., Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations: fractal dimensions and infinitely many attractors, Cambridge Univ.Press [PY] Palis, J. and Yoccoz, J.C., Homoclinic Tangencies for Hyperbolic sets of large Hausdorff Dimension Bifurcations, Acta Mathematica 172 (1994), p. 91-136.

[S] Sannami, A., An example of a regular Cantor set whose difference set is a Cantor set with positive measure, Hokkaido Math. Journal, vol. XXI (1), 1992, p.7-23.