Stable intersections of regular conformal Cantor sets with large Hausdorff dimensions
aa r X i v : . [ m a t h . D S ] F e b STABLE INTERSECTIONS OF REGULAR CONFORMAL CANTOR SETSWITH LARGE HAUSDORFF DIMENSIONS
ALEX ZAMUDIO, CARLOS GUSTAVO MOREIRA AND HUGO ARA ´UJO Introduction
The aim of this paper is to prove a complex version of the main theorem in [1]. We will provethat among pairs
K, K ′ ⊂ C of conformal dynamically defined Cantor sets with sum of Hausdorffdimension HD ( K ) + HD ( K ′ ) >
2, there is an open and dense subset of such pairs verifyingint( K − K ′ ) = ∅ . The analogous statement for dynamically defined Cantor sets in the real line wasproved by Moreira and Yoccoz in [1]. Here we adapt their argument to the context of conformalCantor sets, this requires the introduction of several new concepts and a more detailed analysis insome parts of the argument. The main new difficulties come from the fact that smooth real mapsare naturally conformal, in the sense that their derivative preserves angles, and this is not true formaps in dimension two. A rough version of our main theorem is the following. Theorem.
There is an open and dense set U ⊂ Ω Σ × Ω Σ ′ of pairs of Cantor sets ( K, K ′ ) , suchthat if ( K, K ′ ) ∈ U then I s ( K, K ′ ) = { λ ∈ C : ( K + λ, K ′ ) has stable intersection } is dense in I = { λ ∈ C : ( K + λ ) ∩ K ′ = ∅} . In particular, int ( K − K ′ ) = ∅ . The work of Moreira and Yoccoz was motivated by a conjecture of Palis, according to whichgeneric pairs of dynamically defined Cantor sets in the real line
K, K ′ either verify that theirarithmetic difference K − K ′ = { x − y : x ∈ K, y ∈ K ′ } has zero Lebesgue measure or non-empty interior. Palis conjecture emerged from his work with Takens ([2], [3]), where, in theirstudy of homoclinic bifurcations for surface diffeomorphisms, they used the crucial fact that if HD ( K ) + HD ( K ′ ) <
1, then K − K ′ has zero Lebesgue measure. Looking for a converse, Palisproposed its conjecture.The study of homoclinic bifurcations has proved to be fruitful in the understanding of dynamicsfor surface diffeomorphisms. Complicated dynamical phenomena arise from them. For example,arbitrarily close to any diffeomorphism exhibiting a generic homoclinic tangency, there are openregions in which any diffeomorphism belonging to a residual set has an infinite number of sinks-this is the so called Newhouse phenomenon. Looking for analogous results and using similar ideasto those of Newhouse, Buzzard [4] proved the existence of an open set of automorphisms of C with stable homoclinic tangencies. Buzzard’s strategy was very similar to the work of Newhouse[9], constructing a “very thick” horseshoe, such that the Cantor sets, this time living in the complexplane, associated to it would also be “very thick”. However, the concept of thickness does not havea simple extension to this complex setting and so the argument to guarantee intersections betweenthe Cantor sets after a small pertubation is different.Furthermore, Moreira and Yoccoz were able to use their solution to Palis conjecture in the studyof homoclinic bifurcations for surface diffeomorphisms (see [5]). They proved that given a surfacedifeomorphism F with a homoclinic quadratic tangency associated to a horseshoe with dimension larger than one, the set of diffeomorphisms close to F presenting a stable tangency has positivedensity at F .The authors of this paper have been trying to push all this theory to the context of Cantor setsin the complex plane and apply it to the study of homoclinic bifurcations of automorphisms of C . In [6], Araujo and Moreira introduced the concept of conformal Cantor sets and also extendedthe concept of recurrent compact sets to this context, which serves as a tool to obtain stableintersection. More importantly, Araujo and Moreira showed that given a complex horseshoe of anautomorphism of C , one can associate to it a conformal Cantor set. Using all this, they were ableto recast Buzzard’s example in terms of the theory of conformal Cantor sets.In the paper [7], Moreira and Zamudio proved a multidimensional version of the scale recurrencelemma for conformal Cantor sets. The real version of this lemma was a key step in the solution, byMoreira and Yoccoz, of Palis conjecture. Moreover, Moreira and Zamudio used the multidimensionalconformal version of the scale recurrence lemma to prove a dimension formula relating the Hausdorffdimension of the image of a product of conformal Cantor sets h ( K × ... × K n ), where h is a C function, and the sum of the Hausdorff dimensions HD ( K ) + ... + HD ( K n ).In this paper we intend to use the results in [6], [7] and obtain a version of Palis conjecturefor conformal Cantor sets. In a future paper we plan to use the concepts and ideas laid down inthis paper to study homoclinic bifurcations for automorphisms of C . We plan to obtain resultsanalogous to the ones in [5]. We expect that conformal regular Cantor sets in C will play a role inthe study of homoclinic bifurcations of automorphisms of C , similar to regular Cantor sets in R in the study of homoclinic bifurcations of surface diffeomorphisms.The paper is organized as follows. In section 2 we fix the notation, present basic concepts andresults which will be used later. Most of the proofs are omitted and references are given to theworks [6], [7], [8] and [9]. We also reduce our work to the proof of theorem 2.22; its proof is longand contained in the remaining sections. In section 3 we define a random family of Cantor setsand analyze some of its geometrical properties; it is from this family that we will find the pair ofCantor sets in the conclusion of 2.22. In section 4 we prove theorem 2.22, assuming propositions4.1 and 4.2. The proof of these propositions is postponed to the last two sections (5 and 6) of thiswork. 2. Notation and preliminaries
In this section we give the definitions of the objects appearing in this paper and recall someresults regarding them that have appeared on previous works [6], [7], [8]. We also present newresults and reduce the proof of the main theorem to the proof of theorem 2.22.2.1.
Notations.
Here we introduce some of the notations we will use along the text: • We will work with the space of complex numbers C . We identify it with R in the usualway. In this space we will use the Euclidean metric, given by the norm | ( x, y ) | = p x + y . • We will also work with the space of non-zero complex numbers C ∗ . We will sometimesidentify C ∗ with J = R × T , where T = R / (2 π Z ), through the map ( t, v ) → e t + iv . Notethat J has the structure of a commutative group. We will endow J with the metric comingfrom the inclusion J = C ∗ ⊂ C . • Given a linear map A : R → R we denote its norm by | A | , its minimum norm by m ( A ).They are given by | A | = sup v =0 | Av || v | , m ( A ) = inf v =0 | Av || v | . We will say that A is conformal if | A | = m ( A ). • We will use Id to denote the identity matrix. TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 3 • Let (
X, d ) be a metric space and A ⊂ X . We will use the notation V δ ( A ) for the open δ neighborhood around A , i.e. V δ ( A ) = { x ∈ X : d ( x, A ) < δ } . For a point x ∈ X and apositive real number r we use the notation B ( x, r ) = V r ( { x } ). • We will have to deal with many inequalities and several parameters. In order to reduce thenumber of constants introduced along the text we will use the following notations: Givenexpressions f ( x ) and g ( x ) depending in the parameter x , when we write f ( x ) . g ( x ) thismeans that there is a constant C > f ( x ) ≤ Cg ( x ) for all x . The constant C can only depend on other constants when those have already been fixed. This ensures thatwe will not get any contradiction between the different constants that will appear. f & g will mean g . f , f ≈ g will mean f . g and g & f both hold. • Derivative of a C l function, we use two notations D j f ( x ) or D jx f , both mean derivative oforder j of f at the point x . They are j -linear functions. • The uniform norm of functions f : X → Y , where X and Y are subsets of normed vectorspaces, will be denoted by k f k := sup x ∈ X | f ( x ) | . In many occasions, f will be the derivativeof order j of a C l function.2.2. The space of conformal regular Cantor Sets.
We begin by the very concept of conformalregular Cantor set . First we remember that a C m regular Cantor set , also called dynamically definedCantor set , is given by the following data. • A finite set A of letters and a set B ⊂ A × A of admissible pairs. • For each a ∈ A a compact connected set G ( a ) ⊂ C . • A C m map g : V → C , for m >
1, defined on an open neighbourhood V of F a ∈ A G ( a ).These data must verify the following assumptions: • The sets G ( a ), a ∈ A , are pairwise disjoint • ( a, b ) ∈ B implies G ( b ) ⊂ g ( G ( a )), otherwise G ( b ) ∩ g ( G ( a )) = ∅ . • For each a ∈ A , the restriction g | G ( a ) can be extended to a C m diffeomorphism from anopen neighborhood of G ( a ) onto its image such that m ( Dg ) > µ for some constant µ > m ( A ) := inf v =0 | Av || v | is the minimum norm of the linear operator A on R . • The subshift (Σ , σ ) induced by B , called the type of the Cantor set,Σ = { a = ( a , a , a , . . . ) ∈ A N : ( a i , a i +1 ) ∈ B, ∀ i ≥ } ,σ ( a , a , a , . . . ) = ( a , a , a , . . . ), is topologically mixing.Once we have all these data we can define a Cantor set (i.e. a totally disconnected, perfectcompact set) on the complex plane: K = \ n ≥ g − n G a ∈ A G ( a ) ! . We say that such a set is conformal if, for all x ∈ K , the derivative of g at x , denoted by Dg ( x ) : R → R , is a conformal linear operator.Notice that we always consider the degree of differentiability of the map g , m , to be a realnumber larger than one. This means that g has derivatives up to order [ m ] and D [ m ] g is H¨olderwith exponent m − [ m ]. This hypothesis, as we will precise later in this section, allows us to controlthe geometry of small parts of the Cantor set K . All Cantor sets in this paper will be conformalregular Cantor sets.Besides, as we mentioned on the introduction, an important family of dynamically defined setsare contained in the class of C m conformal regular Cantor sets. Let G be an automorphism of C exhibiting a horseshoe Λ and p be a hyperbolic periodic point in it. Then, there is a subset U ⊂ W s ( p ), open in the topology of W s ( p ) as an immersed manifold, and some ε > ALEX ZAMUDIO, CARLOS GUSTAVO MOREIRA AND HUGO ARA ´UJO small such that Λ ∩ U is, after some parametrization, a C ε conformal regular Cantor set. See Theorem A of [6].We will usually write only K to represent all the data that defines a particular dynamicallydefined Cantor set. Of course, the compact set K can be described in multiple ways as a Cantorset constructed with the objects above, but whenever we say that K is a Cantor set we assumethat one particular set of data as above is fixed. In this spirit, we may represent the Cantor set K by the map g that defines it as described above, since all the data can be inferred if we know g .Notice that in our definition we did not require the pieces G ( a ) to have non-empty interior. Tocircumvent this, we introduce the following sets. Lemma 2.1.
There is δ > sufficiently small such that the sets G ∗ ( a ) := V δ ( G ( a )) satisfy:(i) G ∗ ( a ) is open and connected.(ii) G ( a ) ⊂ G ∗ ( a ) and g | G ∗ ( a ) can be extended to an open neighbourhood of G ∗ ( a ) , such that it isa C m embedding (with C m inverse ) from this neighbourhood to its image and m ( Dg ) > µ .(iii) The sets G ∗ ( a ) , a ∈ A , are pairwise disjoint.(iv) ( a, b ) ∈ B implies G ∗ ( b ) ⊂ g ( G ∗ ( a )) , and ( a, b ) / ∈ B implies G ∗ ( b ) ∩ g ( G ∗ ( a )) = ∅ . The sets G ∗ ( a ) could substitute the pieces G ( a ) in our definition as to make the hypothesis ofopen interiors be true. These changes do not enlarge the Cantor set. To see this, we introducemore notation and a previous result.Associated to a Cantor set K we define the setsΣ fin = { ( a , . . . , a n ) : ( a i , a i +1 ) ∈ B ∀ i, ≤ i < n } , Σ − = { ( . . . , a − n , a − n +1 , . . . , a − , a ) : ( a i − , a i ) ∈ B ∀ i ≤ } . Given a = ( a , . . . , a n ), b = ( b , . . . , b m ), θ = ( . . . , θ − , θ − , θ ) and θ = ( . . . , θ − , θ − , θ ), wewrite: • if a n = b , ab = ( a , . . . , a n , b , . . . , b m ); • if θ = a , θ a = ( . . . , θ − , θ − , a , . . . , a n ) • if θ = θ and θ = θ , θ ∧ θ = ( θ − j , θ − j +1 , . . . , θ ), in which θ − i = θ − i = θ − i for all i = 0 , . . . , j and θ − j − = θ − j − .For a = ( a , a , . . . , a n ) ∈ Σ fin we say that it has length n and define: G ( a ) = { x ∈ G a ∈ A G ( a ) , g j ( x ) ∈ G ( a j ) , j = 0 , , . . . , n } and the function f a : G ( a n ) → G ( a ) by: f a = g | − G ( a ) ◦ g | − G ( a ) ◦ . . . ◦ ( g | − G ( a n − ) ) | G ( a n ) . Notice that f ( a i ,a i +1 ) = g | − G ( a i ) . Furthermore, we can consider the sets G ∗ ( a ) defined in the sameway G ∗ ( a ) = { x ∈ G a ∈ A G ∗ ( a ) , g j ( x ) ∈ G ∗ ( a j ) , j = 0 , , . . . , n } but using the ∗ version of the pieces and consider the function f a to be defined in the larger set G ∗ ( a n ) having image equal to G ∗ ( a ).Now we have the following lemma. Lemma 2.2.
Let K be a dynamically defined Cantor set and G ∗ ( a ) the sets defined above. Thereexists a constant C > such that: diam ( G ∗ ( a )) < Cµ − n . TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 5
As a consequence of this lemma we can see that K = \ n ≥ g − n G a ∈ A G ∗ ( a ) ! since G ( a ) ⊂ G ∗ ( a ) and diam( G ∗ ( a )) →
0, and so the Cantor set has not been enlarged. Anotherconsequence is that K is contained in the interior of the union of the pieces G ∗ ( a ). From now onwe will work with the assumption that the sets G ( a ) have non-empty interior and that they contain K in the interior of their union. We keep the definition G ∗ ( a ) := V δ ( G ( a )) as before because it willbe useful in the definition of limit geometries.The following definition will be useful in the future. Definition 2.3.
For every Cantor set K we define the homeomorphism H : K → Σthat carries each point x ∈ K to its itinerary along the pieces G ( a ), that is H ( x ) = ( a , a , . . . , a n , . . . )if and only if g i ( x ) ∈ G ( a i ) for all i ≥ g that is, at least, C ε for some ε > Definition 2.4. (The space Ω m Σ ) For a fixed symbolic space Σ and real number m > ∞ ). The set of all C m conformal regular Cantor sets K with the type Σ is defined as theset of all C m conformal Cantor sets described as above whose set of data includes the alphabet A and the set B of admissible pairs used in the construction of Σ. We denote it by Ω m Σ .The topology on Ω m Σ is generated by a basis of neighbourhoods U K,δ ⊂ Ω Σ where K is any C m Cantor set in Ω m Σ and δ >
0. The neighborhood U K,δ is the set of all C m conformal regular Cantorsets K ′ given by g ′ : V ′ → C , V ′ ⊃ F a ∈ A G ′ ( a ) such that G ( a ) ⊂ V δ ( G ′ ( a )), G ′ ( a ) ⊂ V δ ( G ( a ))(that is, the pieces are close in the Hausdorff topology) and the restrictions of g ′ and g to V ∩ V ′ are δ close in the C m metric. The topology on Ω ∞ Σ is the one such that a sequence of C ∞ maps g n converges to g if and only if the sequence converges to g in the topology of Ω m Σ for every m ∈ (1 , ∞ ).We also consider the union Ω Σ := S m> Ω m Σ , the topology in Ω Σ is the finest topology such thatthe inclusions Ω m Σ ⊂ Ω Σ are continuous maps, the so called inductive limit topology. Thus, a set U ⊂ Ω Σ is open if and only if U ∩ Ω m Σ is open in Ω m Σ for all m >
1. It is not difficult to prove thatan open set U ⊂ Ω Σ can be written as a union U = S m> U m , where each U m is open in Ω m Σ and U m ⊂ U m ′ if m > m ′ .2.3. Limit geometries.
To study the geometry of small parts of our Cantor sets, we introducemore objects. For each a = ( a , . . . , a n ) ∈ Σ fin , denote by K ( a ) the set K ∩ G ( a ). For each a ∈ A ,fix a point c a ∈ K ( a ). We will refer to these points as base points. Define c a ∈ K ( a ) by c a := f a ( c a n ) . Additionally, given θ = ( . . . , θ − n , . . . , θ ) ∈ Σ − we write θ n := ( θ − n , . . . , θ ) and r θ n := diam( G ( θ n )).Given θ ∈ Σ − and n ≥
1, define Φ θ n as the unique map in Af f ( C ) := { αz + β, α ∈ C ∗ , β ∈ C } such that Φ θ n ( c θ n ) = (cid:0) Φ θ n ◦ f θ n (cid:1) ( c θ ) = 0 and D (cid:0) Φ θ n ◦ f θ n (cid:1) ( c θ ) = Id.
ALEX ZAMUDIO, CARLOS GUSTAVO MOREIRA AND HUGO ARA ´UJO
The maps Φ θ n act as a normalization of small parts of the Cantor set K . For that purpose, wedefine the maps k θn by k θn := Φ θ n ◦ f θ n . Through them we have the first result that allows control over the sets G ( θ n ).In what follows we consider some m > K being a Cantor set in the space Ω m Σ . Proposition 2.5. (Limit Geometries) For each θ ∈ Σ − the sequence of C m embeddings k θn : G ∗ ( θ ) → C converges in the C [ m ] topology to a C m embedding k θ : G ( θ ) → C . The convergencealso happens in the C m ′ topology for every m ′ ∈ (1 , m ) . Moreover, the convergence is uniform overall θ ∈ Σ − and in a small neighbourhood of g in Ω m Σ . The k θ : G ( θ ) → C defined for any θ ∈ Σ − are called the limit geometries of K . Remark . Define Σ − a = { θ ∈ Σ − , θ = a } and consider in this set the topology given by themetric d ( θ , θ ) = diam( G ( θ ∧ θ )). Likewise, for m >
1, let Emb m ( G ∗ ( a ) , C ) be the space of C m embeddings from G ∗ ( a ) to C with C m inverse equipped with the topology given by the C metric d ( g , g ) = max {|| g − g || , || Dg − Dg ||} . For fixed 0 < ε < C ε Cantor set K , the map k : Σ − a → Emb ε ( G ∗ ( a ) , C ) , θ k θ is ε -H¨older, if we consider the metrics described above for both spaces. In case the Cantor set K is C m , for m ≥
2, then there is a constant
C > d ( k θ , k θ ) ≤ Cd ( θ , θ ). The constant C can be chosen uniformly in a neighborhood of the Cantor set.Since the convergence is uniform with respect to θ and in a neighborhood of Ω m Σ , the limitgeometries k θ depend continuously in θ and the Cantor set K .We also remark that the derivative Dk θ ( x ) is conformal for all x ∈ K ( θ ). Remark . It is important to observe that limit geometries depend on the choice of base points.Nonetheless, different choice of base points do not change the resultant limit geometries by much,only by an affine transformation that is bounded by some constant C depending on K . Here wemean that such transformations are given by maps A ( z ) = αz + β , where | α | , | β | < C . This boundis, as before, uniform for Cantor sets ˜ K sufficiently close to K . See the paragraph after Corollary3.2 of [6].For reasons that will become more clear in the future, from now on we assume that for each a ∈ A the corresponding base point c a is a pre-periodic point.Before proceeding, we fix some more notation. For θ ∈ Σ − and a ∈ Σ fin we write G θ ( a ) := k θ ( G ( a )) , K θ ( a ) := k θ ( K ( a )) , c θa := k θ ( c a ) . Furthermore, to establish stable intersections, we are going to analyse very small parts of theCantor sets, whose size will be controlled by a real number ρ . This number should be regardedas a variable that we are going to assume in various instances to be very small, as to make thevarious estimates we are going to find in the future to fit all together. This being said, let c be asufficient large constant as required in the statement of the scale recurrence lemma (lemma 2.26)in subsection 2.7. Definition 2.8.
For 0 < ρ <
1, the set Σ( ρ ) is defined as the set of words a ∈ Σ fin such that c − ρ ≤ diam( G ( a )) ≤ c ρ. We say that the set G ( a ) has an approximate size ρ .Using standard techniques (see [2] and [8]), one can prove that there is a constant C dependingonly in the Cantor set and the parameter c such that C − ρ − d ≤ ρ ) ≤ Cρ − d . TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 7
Remark . Notice that, because the set of limit geometries represent a compact subset of ∪ a ∈ A Emb m ( G ( a )), every piece of approximate size ρ also contains the ball B ( c a , ( c ′ ) − ρ ) for some c ′ > K . The result remains valid for perturbations ˜ K sufficiently close to K .Even more, by maybe enlarging c ′ a little bit, because of Corollaries 3.3 and 3.4 of [6], it followsthat for any a = ( a , . . . , a n ) ∈ Σ fin (1) ( c ′ ) − ≤ (cid:12)(cid:12) Df a ( c a n ) (cid:12)(cid:12) diam( G ( a )) ≤ c ′ . This allows us to control the approximate size of the sets G ( a ) through the derivative of the map f a at c a n .2.4. Recurrent compact criterion for stable intersections.
Our next objects are called con-figurations . They are a way of moving a Cantor set in the plane without changing its internalstructure.
Definition 2.10. A C m - configuration of a piece G ( a ) of a Cantor set is a C m , m >
1, diffeomor-phism h : G ( a ) → U ⊂ C . The space of all C m configurations of a piece G ( a ) is denoted by P m ( a ) and we equip it with the C m topology. The space of all configurations is denoted by P ( a ) = ∪ m> P m ( a ) and we equip itwith the inductive limit topology. This is, U ⊂ P ( a ) is open if and only U ∩ P m ( a ) is open in thetopology of P m ( a ), for all m > h is an affine map, we call it an affine configuration . Observe that a limit geometry is aconfiguration of a piece. Configurations of the type A ◦ k θ , where A ∈ Af f ( C ) and θ ∈ Σ − , are ofgreat importance to our work and so are called affine configurations of limit geometries .The renormalization operators represent a way of looking into smaller parts of the Cantor set. Definition 2.11.
Let K and K ′ be two Cantor sets. Choose any pair of words a = ( a , a , . . . , a n ) ∈ Σ fin and a ′ = ( a ′ , a ′ , . . . , a ′ m ) ∈ Σ ′ fin . Then, the renormalization operator T a T a ′ acts on any pairof configurations h : G ( a ) → C and h ′ : G ( a ′ ) → C by T a T a ′ ( h, h ′ ) := ( h ◦ f a , h ′ ◦ f a ′ ) . The notation above clearly indicates that we can consider the operators T a and T a ′ as separate,each acting on configurations of K and K ′ respectively. In a very similar way to the lemma 2.5,the one defining limit geometries, one can show (see Lemma 3.11 of [6]) that the set of affineconfigurations of limit geometries form an attractor in the space of configurations under the actionof renormalizations. Even more, see lemma 3.8 of [6], the renormalization operators act in a verysimple manner over limit geometries: Lemma 2.12.
For any θ ∈ Σ − and a ∈ Σ fin , a = ( a , ..., a m ) , there is an affine transformation F θa ∈ Af f ( C ) such that k θ ◦ f a = F θa ◦ k θa . Moreover, this transformation can be calculated by DF θa = lim n →∞ (cid:0) Df θ n ( c θ ) (cid:1) − · Df ( θa ) n + m ( c a m ) and F θa (0) = c θa = k θ ( c a ) . Now we properly establish the notion of stable intersection between Cantor sets. For any pair ofconfigurations ( h a , h ′ a ′ ) ∈ P ( a ) × P ′ ( a ′ ) we say that it is: • linked whenever h a ( G ( a )) ∩ h ′ a ′ ( G ( a ′ )) = ∅ . • intersecting whenever h a ( K ( a )) ∩ h ′ a ′ ( K ′ ( a ′ )) = ∅ . ALEX ZAMUDIO, CARLOS GUSTAVO MOREIRA AND HUGO ARA ´UJO • has stable intersections whenever ˜ h a ( ˜ K ( a )) ∩ ˜ h ′ a ′ ( ˜ K ′ ( a ′ )) = ∅ for any pairs of Cantor sets( ˜ K, ˜ K ′ ) ∈ Ω Σ × Ω Σ ′ in a small neighbourhood of ( K, K ′ ) and any configuration pair (˜ h a , ˜ h ′ a ′ )that is sufficiently close to ( h a , h ′ a ′ ) in the topology of P ( a ) × P ′ ( a ′ ).The set I s ( K, K ′ ) in the statement of theorem 1 represents the set of all λ ∈ C such that ( τ λ , Id )is a pair of configurations having stable intersections in the sense just described, where τ z is thetranslation by z on C .The main theorem in the introduction is a particular case of theorem 2.23, it guarantees stableintersections for affine configurations of limit geometries of Cantor sets. To state it, we need to fixsome more notation. First, notice that the space of affine configurations of limit geometries of aCantor set can be seen as the image of the continuous association I : A := Af f ( C ) × Σ − → P ( A, θ ) A ◦ k θ . Definition 2.13.
The space of relative affine configurations of limit geometries will be denoted by C . It is the quotient of A × A ′ by the action of the affine group by composition on the left, that is,(( A, θ ) , ( A ′ , θ )) (cid:0) ( B ◦ A, θ ) , ( B ◦ A ′ , θ ′ ) (cid:1) , where B ranges in Af f ( C ).The concepts of linking , intersection and stable intersection were well defined for pairs of affineconfigurations of limit geometries, and since they are invariant by the action of Af f ( C ), they arealso defined for relative configurations in C .Also, since the renormalization operator acts by composition on the right on ( A, θ ), its actioncommutes with the multiplication on the left by affine transformations and so it can be natu-rally defined on C . This space can be identified with Σ − × Σ ′− × Af f ( C ) by the identification[( A, θ ) , ( A ′ , θ ′ )] ≡ ( θ, θ ′ , A ′− ◦ A ) and, in this manner, the topology on C is the product topologyon Σ − × Σ ′− × Af f ( C ). The action of the renormalization operator on a relative configuration canbe described by T a T a ′ ( θ, θ ′ , A ) = ( θa, θ ′ a ′ , (cid:16) F θ ′ a ′ (cid:17) − ◦ A ◦ F θa ) , and if A = sz + t , then(2) (cid:16) F θ ′ a ′ (cid:17) − ◦ A ◦ F θa ( z ) = DF θa DF θ ′ a ′ sz + 1 DF θ ′ a ′ (cid:16) sc θa + t − c θ ′ a ′ (cid:17) . It is more convenient to see the space C through one more identification:Σ − × Σ ′− × Af f ( C ) ≡ Σ − × Σ ′− × C ∗ × C ( θ, θ ′ , A ) ≡ ( θ, θ ′ , s, t ) , where A ( z ) = sz + t . We will call s the scale part of the relative configuration and t the translation part. The equation (2) provides to us a formula for the renormalization under this identification ifwe analyse the scale and translation parts separately: s DF θa DF θ ′ a ′ s and t DF θ ′ a ′ (cid:16) sc θa + t − c θ ′ a ′ (cid:17) . The space of relative scales is given by S = Σ − × Σ ′− × J , where J = C ∗ . We identify J with R × T through the map ( t, v ) → e t + iv . It acts on C by complex multiplication. The space C ofrelative configurations projects to S by the map C → S [( θ, A ) , ( θ ′ , A ′ )] → ( θ, θ ′ , DA/DA ′ ) , TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 9 where DA means derivative of the affine map A (which is an element in J ). We trivialize C → S in the following way: we map [( θ, A ) , ( θ ′ , A ′ )] ∈ C to ( θ, θ ′ , s, λ ) such that s = DA/DA ′ and λ = ( A ′ ) − ◦ A (0). In this sense we can think of C as S × C . Most of the time we will work withscales which are bounded away from zero and infinity, for this purpose we introduce the notation J R = { s ∈ J : e − R ≤ | s |≤ e R } , where R is a positive real number.The object which we present in the following definition will play a central role in the proof ofour main theorems. It is a useful tool to get stable intersection between pair of Cantor sets. Definition 2.14 (Recurrent compact) . Let K and K ′ be a pair of Cantor sets. Let L be a compactset in C . We say that L is recurrent (for the pair ( K, K ′ )) if for any relative affine configurationof limit geometries v ∈ L , there are finite words a , a ′ such that u = T a T ′ a ′ ( v ) satisfies u ∈ int L ,where the T a T ′ a ′ are renormalization operators associated to the pair of Cantor sets K and K ′ .If such a renormalization can be done using words a and a ′ such that their total size combinedis equal to one, we say that such a set is immediately recurrent . Theorem B of [6] states that if u belongs to a recurrent compact set associated to a pair ofCantor sets K and K ′ , then it represents pairs of affine configurations of limit geometries of theseCantor sets that have stable intersections. For the convenience of the reader, we copy its statementbelow. Theorem.
The following properties are true:(1) Every recurrent compact set is contained in an immediately recurrent compact set.(2) Given a recurrent compact set L (resp. immediately recurrent) for g , g ′ , for any ˜ g , ˜ g ′ ina small neighbourhood of ( g, g ′ ) ∈ Ω Σ × Ω Σ ′ we can choose base points ˜ c a ∈ ˜ G ( a ) ∩ ˜ K and ˜ c a ′ ∈ ˜ G ( a ′ ) ∩ ˜ K ′ respectively close to the pre-fixed c a and c a ′ , for all a ∈ A and a ′ ∈ A ′ , ina manner that L is also a recurrent compact set for ˜ g and ˜ g ′ .(3) Any relative configuration contained in a recurrent compact set has stable intersections.Remark . For each pair of maps (˜ g, ˜ g ′ ) in the small neighbourhood of ( g, g ′ ) in the theoremabove, let ˜ H and ˜ H ′ be the corresponding homeomorphisms defined in 2.3. The base points˜ c a ∈ ˜ G ( a ) ∩ ˜ K and ˜ c a ′ ∈ ˜ G ( a ′ ) ∩ ˜ K ′ in the theorem above are chosen so that ˜ H (˜ c a ) = H ( c a ) and˜ H ′ (˜ c a ′ ) = H ′ ( c a ′ ) for all a ∈ A and all a ′ ∈ A ′ , meaning that their itineraries under the action of ˜ g and ˜ g ′ are the same for all pairs of maps in this neighbourhood. In subsequent contexts, the basepoints will be chosen in the same way.2.5. Perturbation of Conformal Cantor Sets.
Let K be a conformal Cantor set defined bya C m map g . We show that if K is non-essentially real then arbitrarily close to K , in the C [ m ] topology, there is a C ∞ conformal Cantor set ˜ K defined by a map ˜ g that is holomorphic on a smallopen neighbourhood of ˜ K . This is an important property that will allow us to perturb more freelythe conformal Cantor sets and adapt the random perturbation argument from [1] to our context.We begin with the following lemma: Lemma 2.16.
Let K be a C m conformal Cantor set given by g . For x ∈ K consider the set K dirx := \ δ> (cid:26) y − x | y − x | : y ∈ B δ ( x ) ∩ ( K \ { x } ) (cid:27) . Assume that, for all x ∈ K , K dirx has two linearly independent vectors (over R ). Then, for all ≤ l ≤ [ m ] the l -linear map D lx g : R × ... × R → R is conformal, i.e. there is a complex number c lx such that D lx g ( z , ..., z l ) = c lx · z · z · . . . · z l . The operation · in the right hand side of the last equality corresponds to complex multiplication. Proof.
Notice that the case l = 1 is just the definition of conformality for the Cantor set. Now weproceed by induction, assume the result for l −
1. Let w ∈ K dirx , then there are sequences t n → w n → w such that x + t n w n ∈ K . Hence D lx g ( w, z ..., z l − ) = lim n →∞ D l − x + t n w n g ( z , ..., z l − ) − D l − x g ( z , ..., z l − ) t n = lim n →∞ c l − x + t n w n · z · · · z l − − c l − x · z · · · z l − t n = lim n →∞ c l − x + t n w n − c l − x t n ! · z · · · z l − . This shows that the limit lim n →∞ c l − x + tnwn − c l − x t n exists, denote it by c lx ( w ). Moreover D lx g ( w, z ..., z l − ) = c lx ( w ) · z · · · z l − . If we take another vector ˜ w ∈ K dirx , and using the symmetry of the operator D lx g , we would have c lx ( w ) ˜ w = D lx g ( w, ˜ w, , ...,
1) = D lx g ( ˜ w, w, , ...,
1) = c lx ( ˜ w ) w. This shows that c lx ( w ) w does not depend on w , denote it by c lx . Since we can choose w, ˜ w generating R , we conclude that D lx g ( z ..., z l ) = c lx · z · · · z l . (cid:3) To use this lemma we need to consider Cantor sets that are indeed two dimensional. This conceptis precised by the following definition.
Definition 2.17.
We will say that a Cantor set K is essentially real if there exists θ ∈ Σ − such thatthe limit Cantor set K θ ( θ ) is contained in a straight line. Otherwise, we say it is non-essentiallyreal .It is not difficult to prove that K is essentially real if and only if for every θ ∈ Σ − the limitCantor set K θ ( θ ) is contained in a straight line. Moreover, one can prove that K being essentiallyreal is equivalent to K being contained in a C one dimensional manifold embedded on the plane.Lemma 1.4.1 from [8] can be adapted to our context and it can be used to prove that every point x belonging to a non-essentially real Cantor set K verifies that K dirx has two linearly independentvectors (over R ). We now show that being non-essentially real is an open property. Lemma 2.18.
Let K be a C m non-essentially real conformal Cantor set. Every conformal Cantorset, close enough to K in the C m topology, is also non-essentially real.Proof. If the lemma does not hold, we would have a sequence K n of conformal Cantor sets con-verging to K and such that all K n are essentially real. Let θ ∈ Σ − , denote by k θ,n the limitgeometry associated to θ and the Cantor set K n . Since all K n are essentially real then, for all n , K θ,n ( θ ) = k θ,n ( K n ) is contained in a line passing through the origin. Taking a subsequence wecan assume that, as n goes to infinity, K θ,n ( θ ) converges to a a set contained in a line passingthrough the origin. Using the fact that limit geometries depend continuously on the Cantor set,we conclude that K θ ( θ ) is contained in a line and therefore K is essentially real. Contradictingthe hypothesis in the lemma. (cid:3) Lemma 2.19.
Let ( K, g ) be a C m non-essentially real conformal Cantor set. Arbitrarily closeto K , in the C [ m ] topology, we can construct a C ∞ conformal Cantor set ( ˜ K, ˜ g ) such that ˜ g isholomorphic on a neighbourhood of ˜ K . TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 11
Proof.
Since K is non-essentially real then the l -derivative D lx g , at a point x in the Cantor set, isdetermined by a complex number, which we denote by g ( l ) ( x ), i.e. D lx g ( z ..., z l ) = g ( l ) ( x ) · z · · · z l . In this situation, the Taylor approximation of g at the point x is g ( x + z ) = g ( x ) + [ m ] X j =1 j ! (cid:0) D jx g (cid:1) ( z, . . . , z ) ++ Z [0 , [ m ] t [ m ] − t [ m ] − · · · t [ m ] − (cid:16)h D [ m ] x + t t ...t [ m ] z − D [ m ] x i g (cid:17) ( z, . . . , z ) dt . . . dt [ m ] = g ( x ) + [ m ] X j =1 j ! g ( j ) ( x ) z j ++ Z [0 , [ m ] t [ m ] − t [ m ] − · · · t [ m ] − (cid:16)h D [ m ] x + t t ...t [ m ] z − D [ m ] x i g (cid:17) ( z, . . . , z ) dt . . . dt [ m ] . Hence, g is approximated (close to x ) by a complex polynomial, which is an holomorphic function.Now, to globally aproximate g by a function ˜ g which is holomorphic in a neighborhood of itsCantor set ˜ K , we are going to take many of the previous polynomial approximations and glue themtogether.Choose any real number ρ larger than zero. Consider Λ ⊂ Σ( ρ ) such that { G ( a ) ∩ K } a ∈ Λ is apartition of K . For each a ∈ Λ we choose a point x a ∈ G ( a ) ∩ K and define the polynomial p a ( z ) = [ m ] X j =0 j ! g ( j ) ( x a )( z − x a ) j . We can also consider C ∞ functions φ a : C → R , a ∈ Λ, with the following properties: • φ a ( z ) = 1 for all z ∈ G ( a ). • supp ( φ a ) ⊂ V ˜ cρ ( G ( a )), for a constant ˜ c , independent of ρ . • supp ( φ a ) ∩ supp ( φ b ) = ∅ , for all a = b . • k D j φ a k≤ ˜ Cρ − j , for a constant ˜ C independent of ρ .Indeed, to be able to construct these bump functions, we only need to show that given a = a both in Σ( ρ ), the distance between the pieces G ( a ) and G ( a ) is at least ˜ cρ for some constant ˜ c > ρ . For that, we can suppose that a = aa and a = aa for some a = a ∈ A ,since this would be the worst scenario. If θ ∈ Σ − ends with a , the distance between these sets iscomparable to diam ( G ( a )) · d (cid:16) k θ ( G ( a )) , k θ ( G ( a )) (cid:17) . Hence the existence of ˜ c follows from the compactness of the space of limit geometries.Now, let ˆ g be C ∞ and very close, in the C [ m ] topology, to g . Define ˜ g , with the same domain as g , by ˜ g ( z ) = X a ∈ Λ φ a ( z ) p a ( z ) + − X a ∈ Λ φ a ( z ) ˆ g ( z ) . Notice that ˆ g − ˜ g = X a ∈ Λ φ a · (ˆ g − p a ) . Therefore, the C [ m ] norm of ˆ g − ˜ g will be small provided k D j φ a k·k D k − j (ˆ g − p a ) | V ˜ cρ ( G ( a )) k≤ k D j φ a k· h k D k − j ( g − p a ) | V ˜ cρ ( G ( a )) k + k D k − j ( g − ˆ g ) | V ˜ cρ ( G ( a )) k i is small, for all 0 ≤ k ≤ [ m ], 0 ≤ j ≤ k (remember that support of φ a is contained in V ˜ cρ ( G ( a ))).We already know that k D j φ a k≤ ˜ Cρ − j . On the other hand, Taylor approximation implies thatlim ρ → sup a ∈ Λ k D k − j ( g − p a ) | V ˜ cρ ( G ( a )) k ρ j = 0 . We conclude that taking ρ small enough and ˆ g close enough to g , we get ˜ g as C [ m ] close to g as wewant. Notice that thanks to the way in which we defined ˜ g , it is C ∞ and by lemma 2.18 we cansuppose it is non-essentially real. Moreover, in the set ⊔ a ∈ Λ G ( a ) the function ˜ g is holomorpic. Wecan also guarantee that ˜ g verifies the hypothesis necessary to define a dynamically defined Cantorset (with the same sets G ( a ), a ∈ A ), we just need to take ˜ g C -close enough to g . Even more, theCantor set ˜ K , generated by ˜ g , is contained in ⊔ a ∈ Λ G ( a ). (cid:3) To prove our main theorems we will use the scale recurrence lemma (see subsection 2.7). Touse this lemma we need that our Cantor sets are non-essentially affine. A C m Cantor set K , with m ≥
2, is said to be non-essentially affine when there is a pair of limit geometries θ and θ in Σ − such that θ = θ and a point x ∈ K θ ( θ ) such that D h k θ ◦ ( k θ ) − i ( x ) = 0 . The following lemma allow us to perturb and get a non-essentially affine Cantor set.
Lemma 2.20.
Let K be a C m non-essentially real conformal Cantor set. Arbitrarly close to K , inthe C [ m ] topology, there is a C ∞ Cantor set ˜ K which is non-essentially real, non-essentially affineand such that its expanding function ˜ g is holomorphic in a neighborhood of ˜ K .Proof. Let ( ˆ K, ˆ g ) be the perturbed Cantor set from lemma 2.19. If ˆ K is non-essentially affine weare done. Otherwise, choose a piece G ( a ), a ∈ A , and let c a be the corresponding base point. Aspreviously mentioned, it is pre-periodic. Claim. If ρ > is sufficiently small, we can chose a ∈ Σ( ρ ) ending with a so that no word in Σ( ρ / ) appears more than once in a . Given any word a with this property, if ρ > is sufficientlysmall, there is θ ∈ Σ − ending with a and such that a does not appear elsewhere in θ . Furthermore,there is θ ∈ Σ − ending with a such that no subword of a belonging to Σ( ρ / ) appears in it.Proof. Since the shift is mixing over Σ, there must be at least two sequences of distinct lengths(both larger than 1) a , a ∈ Σ fin such that both end and begin with a and no other letter in thesewords is a . Now construct a as a = a a . . . a | {z } N times a a . . . a a | {z } N times a a a . . . a a a | {z } N times a . . . a a | {z } N times . If ρ is sufficiently small, a suitable choice of N , N , N and N can be done so that no subwordin Σ( ρ / ) appears more than once. This can be seen by analysing the behaviour of the distancebetween two consecutive letters a in any two subwords of a .Now, choose a subword a ′ ∈ Σ( ρ / ) such that a begins with a ′ . If a ′ never appears in θ ′ ∈ Σ − ,then we can make θ = θ ′ a . Indeed, for a to appear more than once, the word a \ a ′ must becontained in a but in another position. This implies that a subword in Σ( ρ / ), corresponding tothe beginning of a \ a ′ for example, appears more than once in a . But this does not happen.Now, suppose we are given a word b ∈ Σ( ρ ). We want to prove there is some θ ∈ Σ − suchthat b never appears in θ . Choose a beginning b ∈ Σ( ρ / ) of b and an ending b ∈ Σ( ρ / ). Let TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 13 b ′ ∈ Σ( ρ ) be such that b and b never appear in it and also suppose that the first letter of b ′ is thesame as its last letter. Then, similarly to the analysis before, we can make θ = . . . b ′ . . . b ′ ∈ Σ − .The existence of b ′ comes from a counting argument, in which we show that the words in which b or b appear do not account for all possible words b ′ ∈ Σ( ρ ). Remember that if d is the Hausdorffdimension of K , then (mas isso s´o aparece muito depois no texto) ρ ) ≈ ρ − d .The number of words in Σ( ρ ) ending with b is . ρ − d/ . More than that, if we fix a startingposition for the appearance of b , such as it beginning in the 1000 th letter of b ′ (remember ρ is verysmall), then the same estimate remains true. Notice however that the number of letters of b ′ is . log ρ − , and so the number of words in Σ( ρ ) that fail our requirements is . ρ − d/ log ρ − , thus,for ρ small enough, there must be b ′ ∈ Σ( ρ ) that satisfies our requirements.To construct θ , we can use the same argument, all we need to observe is that the number ofsubwords of a in Σ( ρ / ) is also . log ρ . (cid:3) By maybe shrinking ρ even further, we can assume that the periodic part of the itinerary of c a is a word very small when compared to a . The combinatorial conditions above imply that f θ n ( c a )belongs to G ( a ) only when θ n = a . Besides, f θ n ( c a ) never belongs to this set.Let φ : G ( a ) → C be an holomorphic map C [ m ] close to the identity, and suppose it fixesthe point c a and has derivative equal to the identity at this point. Similar to the construction onlemma 2.19, define a new Cantor set given by an expanding map ˜ g that is equal to ˆ g outside a smallneighborhood of G ( a ) and equal to ˆ g ◦ φ in G ( a ). Note that the perturbed base point ˜ c a is equal to c a , thanks to the pre-periodicity of c a and the combinatorial properties of a . Moreover, the limitgeometry corresponding to θ stays the same close to c a , that is, we can choose a neighbourhood V = G ( b ) of the base point ˜ c a = c a , with b ∈ Σ fin sufficiently large, such that ˜ k θ | V = ˆ k θ | V . Onthe other hand, for θ , ˜ k θ | V = ˆ k θ | V ◦ f − a ◦ φ − ◦ f a , since the affine reescalings Φ θ n stay the same. Notice that the map ˜ g is still holomorphic in aneighbourhood of ˜ K and, because of lemma 2.18, it is non-essentially real if φ is C sufficientlyclose to the identity. However, we can still choose φ so that D (cid:0) f − a ◦ φ − ◦ f a (cid:1) ( c a ) = 0 . Hence D ˜ k θ ( c a ) = D ˆ k θ ( c a ). This implies that D h ˜ k θ ◦ (˜ k θ ) − i (0) = 0and so the new Cantor set is also non-essentially affine. (cid:3) Remark . We observe that the C ∞ Cantor set ˜ K constructed in lemma 2.19 can be alsoassumed to be close to K in the C m ′ topology for all m ′ ∈ (1 , m ). All one needs to do is tochoose ˆ g close to g in this topology. This can be done using a mollifier ϕ supported in a very smallneighbourhood of the origin and making ˆ g = g ∗ ϕ . Notice that the C m proximity between themaps p a and g in supp ( φ a ) comes from the fact that g is C m and supp ( φ a ) has diameter of order ρ .Moreover, using the fact that D [ m ] g is ǫ -Holder, for ǫ = m − [ m ], one gets the improved estimate k D m − j ( g − p a ) | V ˜ cρ ( G ( a )) k = O ( ρ j + ǫ ) for all 0 ≤ j ≤ [ m ]. This can be seen analysing the Taylorapproximation of the derivatives of g in this domain.In particular, given any non-essentially real Cantor set K ∈ Ω Σ , arbitrarily close to K in thetopology of this space, there is a Cantor set ˜ K ∈ Ω ∞ Σ ⊂ Ω Σ that satisfies the conclusion of lemma2.20. Main theorems.
Here we state our main theorems. The proof of theorem 2.22 will be giventhroughout the remaining sections. Using this theorem we will prove theorem 2.23. In particular,we will get that there is an open and dense set, among pairs of conformal Cantor sets K , K ′ with HD ( K ) + HD ( K ′ ) >
2, such that all elements in this set verify int( K − K ′ ) = ∅ . Before statingthe theorems, we remark that the Hausdorff dimension varies continuously with the Cantor set.This is proven in [9] and the argument there can be adapted to our context. Theorem 2.22.
Given a pair of non-essentially real conformal Cantor sets ( K, K ′ ) in Ω ∞ Σ × Ω ∞ Σ ′ ,such that HD ( K ) + HD ( K ′ ) > . Arbitrarily close to K , K ′ , in the C ∞ topology, we can findconformal Cantor sets ˜ K , ˜ K ′ in Ω ∞ Σ , Ω ∞ Σ ′ respectively, such that ˜ K , ˜ K ′ has a non empty recurrentcompact set. Define the set U as the pairs of conformal Cantor sets ( K, K ′ ) in Ω Σ × Ω Σ ′ such that for everyrelative configuration ( θ, θ ′ , s, t ), there is ˜ t ∈ C such that the configuration ( θ, θ ′ , s, ˜ t ) has stableintersection. Theorem 2.23.
The set U is open in Ω Σ × Ω Σ ′ and U ∩ Ω ∞ Σ × Ω ∞ Σ ′ is dense, in the C ∞ topology,in { ( K, K ′ ) ∈ Ω ∞ Σ × Ω ∞ Σ ′ : HD ( K ) + HD ( K ′ ) > , K, K ′ are non-essentially real } . Moreover,for any ( K, K ′ ) in U and ( h, h ′ ) ∈ P × P ′ , such that Dh ( z ) and Dh ′ ( z ′ ) are conformal for all ( z, z ′ ) ∈ K × K ′ , the set I s = { λ ∈ C : ( h + λ, h ′ ) has stable intersection } is dense in I = { λ ∈ C : ( h + λ, h ′ ) is intersecting } . In particular, int ( K − K ′ ) = ∅ .Proof. The proof is very similar to the one for the corresponding result in [1], except for theuse of lemma 2.25. For the openness of U , one observes that, if R is big enough, then any relativeconfiguration ( θ, θ ′ , s, t ) can be transported, using a renormalization operator, to the set Σ − × Σ ′− ×{ e − R ≤ | s |≤ e R } × C . Given ( K, K ′ ) in U , from compactness of the set Σ − × Σ ′− × { e − R ≤ | s |≤ e R } one sees that there is a neighborhood of ( K, K ′ ) such that for any pair in this neighborhood, wehave ( θ, θ ′ , s, t ) ∈ Σ − × Σ ′− × { e − R ≤ | s |≤ e R } × C implies there is ˜ t ∈ C such that ( θ, θ ′ , s, ˜ t ) hasstable intersection. Thus the same happens for the whole Σ − × Σ ′− × C ∗ × C , and the neighborhoodis contained in U .Furthermore, in the same context of the previous paragraph, from compactness of the set Σ − × Σ ′− × { e − R ≤ | s |≤ e R } , for each r > δ > h and ˜ h ′ are maps δ -close tothe identity in the C r metric, then (˜ h ◦ B ◦ k θ , ˜ h ′ ◦ k θ ′ ) has stable intersections, where B ( z ) = sz + ˜ t and e − R ≤ | s |≤ e R . We will need this later.For the denseness we use theorem 2.22. Given ( ˜ K, ˜ K ′ ) ∈ Ω ∞ Σ × Ω ∞ Σ ′ , with HD ( K ) + HD ( K ′ ) > K, K ′ ) ∈ Ω ∞ Σ × Ω ∞ Σ ′ havinga non empty recurrent compact set L . Perturbing, we may assume that ( K, K ′ ) also has periodicpoints p , p ′ , associated to finite words a , a ′ , that is f a ( p ) = p , f a ′ ( p ′ ) = p ′ , such that if we write a i,j = Df a i ( p ) /Df a ′ j ( p ′ ) = DF θ a i /DF θ ′ a ′ j ∈ C ∗ , where a i is concatenation of a with itself i times, similarly for a ′ , θ = ( ..., a, a ) and θ ′ = ( ..., a ′ , a ′ ),then the set { a i,j } i,j ∈ Z + is dense in C ∗ (see lemma 2.25).More precisely, in order to get the density of { a i,j } , we need to perturb ( g, g ′ ) such that Dg m ( p ) , Dg ′ m ′ ( p ′ ) ∈ C ∗ ≈ R × T have the property in lemma 2.25, where m , m ′ are the periods of p , p ′ , respectively. To do this, we define a family of conformal Cantor sets given by expanding functions( g x , g ′ y ) depending in complex parameters x , y , such that the pairs ( Dg mx ( p x ) , Dg ′ m ′ y ( p y )) ∈ C ∗ × C ∗ form an open set. Choose a word d ∈ Σ( α ) such that p ∈ G ( d ), α is small enough such that TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 15 there are not more points of the periodic orbit of p contained in G ( d ) and g is holomorphicin V α ( K ) (by lemma 2.19 we may assume this). For 0 < c < K ∩ V cα ( G ( d )) = K ∩ G ( d ). We choose a C ∞ function ψ x such that ψ x ( z ) = ( x · ( z − p ) + p if z ∈ V cα/ ( G ( d )) ,z if z / ∈ V cα/ ( G ( d )) . Where x is in the ball of center 1 and radius δ in C . Define g x = g ◦ ψ x , notice that we can take g x as close as we want to g in the C ∞ topology by choosing δ small enough. If we choose δ smallenough then we can guarantee that the Cantor set K x , associated to g x , and the set ψ x ( K x ) arecontained in V cα/ ( K ) and therefore K x ∩ V cα/ ( G ( d )) \ V cα/ ( G ( d )) = ∅ , which implies that Dg x ( z ) = Dg ( ψ x ( z )) · Dψ x ( z )is conformal for all z ∈ K x . This proves that K x is a conformal Cantor set. Moreover, notice that p x = p is still a periodic point of g x with the same period m , and Dg mx ( p x ) = x · Dg m ( p ). Doing thesame construction for g ′ one sees that for some value of x , y the pair ( Dg mx ( p x ) , Dg ′ m ′ y ( p ′ y )) satisfythe hypothesis of lemma 2.25.Using equation (2), the denseness of the set { a n,m } n,m ∈ Z + and the fact that L has non-emptyinterior, one concludes that for any ( θ, θ ′ , s, t ) there is m, n ∈ Z + , b , b ′ in Σ fin , Σ ′ fin , respectively,and λ ∈ C such that T a n b T ′ a ′ m b ′ ( θ, θ ′ , s, t + λ ) ∈ L . Therefore ( K, K ′ ) ∈ U ∩ Ω ∞ Σ × Ω ∞ Σ .For the final part, let λ ∈ C be such that ( h + λ, h ′ ) is intersecting and take any ε >
0. Thenthere is at least one pair of words a = ( a , a , . . . , a n ) ∈ Σ fin and a ′ = ( a ′ , , a ′ , . . . , a ′ m ) ∈ Σ ′ fin sufficiently large such that (( h + λ ) ◦ f a , h ′ ◦ f a ′ ) are still intersecting and the diameters of thesets G ( a ) and G ( a ′ ) are smaller than ε . Indeed, if it was not the case, the sets ( h + λ )( G ( a )) and h ′ ( G ( a )) would be disjoint for all a and a ′ sufficiently large, a contradiction with the intersectinghypothesis. We will prove that if ε is small there exists ˜ λ ∈ C such that | ˜ λ − λ | . ε and ( h + ˜ λ, h ′ )has stable intersections.Now, following definition 3.10 of [6], we can scale these pairs of configurations by normalizingin the second coordinate, obtaining another pair with intersection. More precisely, we choose A ′ ∈ Af f ( C ) such that( A ′ ◦ h ′ ◦ f a ′ )( c a ′ m ) = 0 and D ( A ′ ◦ h ′ ◦ f a ′ )( c a ′ m ) = Id and consider now the pair of configurations ( A ′ ◦ ( h + λ ) ◦ f a , A ′ ◦ h ′ ◦ f a ′ ). This pair of configurationsis intersecting, because this property is clearly preserved under composition on the left by affinetransformations.Let r ∈ (1 ,
2) be such that h and h ′ are both C r . Reasoning as in the proof of lemma 3.11 of [6](see claim 3.12), we observe that this pair of configurations can be written as( A ′ ◦ ( h + λ ) ◦ f a , A ′ ◦ h ′ ◦ f a ′ ) = (˜ h ◦ B ◦ k θ , ˜ h ′ ◦ k θ ′ ) , where: θ ∈ Σ − ends with a ; θ ′ ∈ Σ ′− ends with a ′ ; the maps ˜ h and ˜ h ′ are close to the identity inthe C r topology, and B is a bounded affine transformation in Af f ( C ). More precisely, if we set DB = DA ′ · Dh ( c a ) · Df a ( c a n ) = s ∈ C and B (0) = A ′ ◦ ( h + λ ) ◦ f a ( c a n ) = t ∈ C , and choose a and a ′ with appropriate lengths, then e − R ≤ | s | ≤ e R and there is some constant c > ε ) such that | B (0) | < c e R and the distance of the maps ˜ h and ˜ h ′ to theidentity is bounded by c diam( G ( a )) r − and c diam( G ( a ′ )) r − Consider now the relative configuration ( θ, θ ′ , s, t ). By the previous part, there is some ˜ t ∈ C suchthat, writing ˜ B ( z ) = sz + ˜ t , the pair of configurations (ˆ h ◦ ˜ B ◦ k θ , ˆ h ′ ◦ k θ ′ ) has stable intersections forevery pair of maps ˆ h, ˆ h ′ δ -close to the identity in the C r metric. Notice that, since (˜ h ◦ B ◦ k θ , ˜ h ′ ◦ k θ ′ ) Notice that if Dh ( x ) , Dh ′ ( y ) were not conformal for all x ∈ K, y ∈ K ′ we could not guarantee that B ∈ Aff ( C ). is intersecting, (cid:12)(cid:12) t − ˜ t (cid:12)(cid:12) is bounded by e R (cid:16) diam( k θ ) + diam( k θ ′ ) (cid:17) . Therefore, by compactness of theset of all pairs of limit geometries ( k θ , k θ ′ ), we may enlarge c , still independently of ε , so that (cid:12)(cid:12) t − ˜ t (cid:12)(cid:12) is bounded by c e R .Now we choose ˜ λ such that A ′ ( h ( c a ) + ˜ λ ) = ˜ t ; it follows that( A ′ ◦ ( h + ˜ λ ) ◦ f a , A ′ ◦ h ′ ◦ f a ′ ) = (ˆ h ◦ ˜ B ◦ k θ , ˜ h ′ ◦ k θ ′ ) , where D ˜ B = s , ˜ B (0) = ˜ t , and the distances of ˆ h and ˜ h ′ to the identity are bounded from aboveby c diam( G ( a )) r − and c diam( G ( a ′ )) r − . Choosing ε sufficiently small, and so a and a ′ very big,these distances to the identity are less than δ . This implies that ( h + ˜ λ, h ′ ) have stable intersections.Notice, finally, that (cid:12)(cid:12)(cid:12) ˜ λ − λ (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) DA ′ (cid:12)(cid:12) − · (cid:12)(cid:12) ˜ t − t (cid:12)(cid:12) . diam( G ( a ′ )) c e R . εc e R . Hence, making ε very small, we approximate ( h + λ, h ′ ) by ( h + ˜ λ, h ′ ) that has stable intersections,completing the proof. (cid:3) Remark . Notice that thanks to remark 2.21, the set U ⊂ Ω Σ × Ω Σ ′ is dense, with respect tothe topology of Ω Σ × Ω Σ ′ , inside the set { ( K, K ′ ) : HD ( K ) + HD ( K ′ ) > } . Lemma 2.25.
Let ( t, v ) , ( t ′ , v ′ ) ∈ ( R \ { } ) × ( R / (2 π Z )) , and consider the subgroup E = { m ( t, v ) + n ( t ′ , v ′ ) : m, n ∈ Z } . Let w, w ′ ∈ R be representatives of v, v ′ , respectively. Then E ⊂ R × T isdense if and only there is not ( β , β ) ∈ Z \ { } such that β · tt ′ + β (cid:18) w − w ′ tt ′ (cid:19) ∈ Z . Moreover, if E is dense and t/t ′ > then { m ( t, v ) − n ( t ′ , v ′ ) : m, n ∈ Z + } is dense. The set ofpairs (( t, v ) , ( t ′ , v ′ )) ∈ ( R × T ) such that E is dense is a countable intersection of open and densesets.Proof. The lemma is proved using Kronecker’s theorem. It states that a vector ( w , ..., w k ) ∈ T k generates a dense subgroup if and only if there is not ( a , ..., a k ) ∈ Z k \ { } such that a w + ... + a k w k = 0.Let p : R → T be the canonical projection and choose w, w ′ ∈ R such that p ( w ) = v , p ( w ′ ) = v ′ .Note that E is dense if and only if the set { ( t, w ) , ( t ′ , w ′ ) , (0 , } generates a dense subgroup in R .Moreover, this last property is invariant under invertible linear transformations in R . Let A : R → R be the linear map such that A ( t ′ , w ′ ) = (1 ,
0) and A (0 ,
1) = (0 , E is denseif and only if the set { A ( t, w ) , (1 , , (0 , } generates a dense subgroup in R . It is clear that this happens if and only if the projection of A ( t, w ) to T generates a dense subgroup in T . Thus, using Kronecker’s theorem we see that E is dense if and only if there is not ( β , β ) ∈ Z \ { } such that h ( β , β ) , A ( t, w ) i ∈ Z . Moreover, it is not difficult to see that A ( t, w ) = ( t/t ′ , w − ( w ′ t/t ′ )). This proves the first part ofthe lemma.Notice that the set of pairs (( t, v ) , ( t ′ , v ′ )) such that E is dense corresponds to the intersection,varying a ∈ Z \ { } , of the sets { (( t, p ( w )) , ( t ′ , p ( w ′ ))) : h a, A ( t, w ) i / ∈ Z } , TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 17 and each one of these sets is open and dense.Finally, { m ( t, v ) − n ( t ′ , v ′ ) : m, n ∈ Z + } will be dense if and only if { mA ( t, w ) − n (1 ,
0) + r (0 ,
1) : m, n ∈ Z + , r ∈ Z } is dense in R . If E is dense, then the projection of { mA ( t, w ) : m ∈ Z + } to T is dense. If wealso have t/t ′ >
0, then from the expression mA ( t, w ) − n (1 ,
0) + r (0 ,
1) = ( m ( t/t ′ ) − n, m [ w − w ′ ( t/t ′ )] + r ) , it is not difficult to see that { m ( t, v ) − n ( t ′ , v ′ ) : m, n ∈ Z + } is dense in this case. (cid:3) Scale Recurrence Lemma.
In this section we will see how to adapt the scale recurrencelemma from [7] to our context. First we note that limit geometries in [7] were defined slightlydifferent, they were defined as the limit of the function˜ k θn = ˜Φ θ n ◦ f θ n , where ˜Φ θ n is the unique affine function satisfying diam ( ˜Φ θ n ( G ( θ n ))) = 1, D ˜Φ θ n · Df θ n ( c θ ) ∈ R + and ˜Φ θ n ( f θ n ( c θ )) = 0. Denote those limit geometries by ˜ k θ . It is clear that there is a complexnumber R ( θ n ) such that ˜ k θn = R ( θ n ) · k θn . It is not difficult to prove that one can go to the limitand find a complex number R ( θ ) such that ˜ k θ = R ( θ ) · k θ . Moreover, R ( θ ) is uniformly boundedfrom above and below, i.e there is c > c − ≤ | R ( θ ) |≤ c . One can also show that R ( θ )depends Lipschitz in θ in the sense that there is a constant C such that (cid:12)(cid:12)(cid:12)(cid:12) R ( θ ) R ( θ ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cd ( θ , θ ) . We will denote by ˜ F θa the affine function defined by˜ k θ ◦ f a = ˜ F θa ◦ ˜ k θa . This affine function can be written in terms of the numbers ˜ r θa ∈ R + , ˜ v θa ∈ T and ˜ c θa ∈ C by theformula ˜ F θa ( z ) = ˜ r θa exp( i ˜ v θa ) z + ˜ c θa . The maps ˜ F θa and F θa are related by the equations DF θa = R ( θa ) R ( θ ) D ˜ F θa , F θa (0) = 1 R ( θ ) ˜ F θa (0) . Now we assume we have two Cantor sets, K and K ′ , and discuss how to go from the renormalizationoperators in [7] to ours. Define φ θ,θ ′ : J → J and L : R × T → J by φ θ,θ ′ ( s ) = R ( θ ) R ′ ( θ ′ ) · s, L ( t, v, v ′ ) = ( t, v − v ′ ) . Remember that we identify J = C ∗ with R × T through ( t, v ) → e t + iv , and define φ : Σ − × Σ ′− × R × T → Σ − × Σ ′− × J given by φ ( θ, θ ′ , t, v, v ′ ) = ( θ, θ ′ , φ θ,θ ′ ( L ( t, v, v ′ ))) . From the previous equations, one easily proves that the renormalization operators of [7], which aregiven by T a,a ′ ( θ, θ ′ , t, v, v ′ ) = ( θa, θ ′ a ′ , t + log ˜ r θa ˜ r θ ′ a ′ , v + ˜ v θa , v ′ + ˜ v θ ′ a ′ ) and act on Σ − × Σ ′− × R × T , are related to our renormalization operators T a T ′ a ′ by the “semicon-jugancy” φ , i.e. φ ◦ T a,a ′ = T a T ′ a ′ ◦ φ . Using φ , it is not difficult to transport the scale recurrencelemma from [7] to one for T a T ′ a ′ : Lemma 2.26.
Suppose that K , K ′ are non-essentially affine and non-essentially real. If R, c areconveniently large, there exist c , c , c , c , ρ > with the following properties: given < ρ < ρ ,and a family E ( a, a ′ ) of subsets of J R , ( a, a ′ ) ∈ Σ( ρ ) × Σ ′ ( ρ ) , such that m ( J R \ E ( a, a ′ )) ≤ c , ∀ ( a, a ′ ) , there is another family E ∗ ( a, a ′ ) of subsets of J R satisfying:(i) For any ( a, a ′ ) , E ∗ ( a, a ′ ) is contained in the c ρ -neighborhood of E ( a, a ′ ) .(ii) Let ( a, a ′ ) ∈ Σ( ρ ) × Σ ′ ( ρ ) , s ∈ E ∗ ( a, a ′ ) ; there exist at least c ρ − ( d + d ′ ) pairs ( b, b ′ ) ∈ Σ( ρ ) × Σ ′ ( ρ ) (with b , b ′ starting with the last letter of a , a ′ ) such that, if θ ∈ Σ − , θ ′ ∈ Σ ′− end respectively with a , a ′ and T b T ′ b ′ ( θ, θ ′ , s ) = ( θb, θ ′ b ′ , ˜ s ) , the c ρ -neighborhood of ˜ s ∈ J is contained in E ∗ ( b, b ′ ) .(iii) m ( E ∗ ( a, a ′ )) ≥ m ( J R ) /c for at least half of the ( a, a ′ ) ∈ Σ( ρ ) × Σ ′ ( ρ ) .Remark . Here m is the unique measure in R × T giving measure 2 π to J / and invariantby translations. We notice that the lemma remains true if we change m by Lebesgue measure in C ∗ ⊂ C . Since all sets are in J R , one just would need to redefine the constants c and c . Thesame happens for the metric on J , we prove the lemma with the metric d (( t, v ) , ( t ′ , v ′ )) = max {| t − t ′ | , k v − v ′ k} and k [ x ] k = min n ∈ Z | x − πn | ([ x ] ∈ T = R / (2 π Z ) is the class generated by x ∈ R ). In J R we canchange this metric for the usual metric in C .We remark that it can be assumed the sets E ∗ ( a, a ′ ) are closed. To do this we just have toredefine E ∗ ( a, a ′ ) by taking their closure and increase the parameter c . Proof.
We are in the hypothesis of the scale recurrence lemma in [7]. Let ˜ r , ˜ c , ˜ c , ˜ c , ˜ c and ˜ ρ bethe constants given by the lemma. We choose R big enough and c small enough such that ν ( ˜ J ˜ r \ L − ( φ − θ,θ ′ ( E ))) < ˜ c for any set E ⊂ J R with m ( J R \ E ) < c , where ˜ J ˜ r = { ( t, v, v ′ ) : | t |≤ ˜ r } and ν is the Haar measurein R × T such that ν ( ˜ J / ) = 1. This can be done since φ θ,θ ′ is multiplication by a complexnumber, whose norm is uniformly bounded and away from zero. Indeed, if one chooses R big suchthat J ˜ r ⊂ φ − θ,θ ′ ( J R ) and c small enough such that m ( φ − θ,θ ′ ( J R \ E )) < ˜ c , then using that L ∗ ν = m one gets ν ( ˜ J ˜ r \ L − ( φ − θ,θ ′ ( E ))) = ν ( L − ( J ˜ r \ φ − θ,θ ′ ( E ))) = m ( J ˜ r \ φ − θ,θ ′ ( E )) ≤ m ( φ − θ,θ ′ ( J R \ E )) < ˜ c . We choose c = ˜ c , c = ˜ c and ρ = ˜ ρ , the other constants will be chosen along the proof. Supposewe are given a family of sets E ( a, a ′ ) as in the setting of the lemma, define a new family ˜ E ( a, a ′ )by ˜ E ( a, a ′ ) = [ θ,θ ′ L − ( φ − θ,θ ′ ( E ( a, a ′ ))) ∩ ˜ J ˜ r , where the union is over all θ, θ ′ finishing in a, a ′ , respectively. Notice that thanks to the previousdiscussion one gets that ν ( ˜ J ˜ r \ ˜ E ( a, a ′ )) < ˜ c . Then we can apply the scale recurrence lemma from TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 19 [7] to get a new family ˜ E ∗ ( a, a ′ ), which satisfies the properties given in [7]. Now we go back to thespace J R , define a new family E ∗ ( a, a ′ ) by E ∗ ( a, a ′ ) = [ ˜ θ, ˜ θ ′ φ ˜ θ, ˜ θ ′ ( L ( ˜ E ∗ ( a, a ′ ))) , where the union is over all pairs ending in a , a ′ respectively. We will prove that the family E ∗ ( a, a ′ )satisfies the desired properties:(i) Note that L is Lipschitz with constant 2. We can also choose constants c , C such that φ ˜ θ, ˜ θ ′ , φ − θ, ˜ θ ′ are Lipschitz with constant c and | φ ˜ θ, ˜ θ ′ ◦ φ − θ,θ ′ − |≤ C [ d ( θ, ˜ θ ) + d ( θ ′ , ˜ θ ′ )]. Using thiswe get E ∗ ( a, a ′ ) = [ ˜ θ, ˜ θ ′ φ ˜ θ, ˜ θ ′ ( L ( ˜ E ∗ ( a, a ′ ))) ⊂ [ ˜ θ, ˜ θ ′ φ ˜ θ, ˜ θ ′ ( L ( V ˜ c ρ ( ˜ E ( a, a ′ )))) ⊂ [ ˜ θ, ˜ θ ′ V c ˜ c ρ φ ˜ θ, ˜ θ ′ ◦ L [ θ,θ ′ L − ◦ φ − θ,θ ′ ( E ( a, a ′ )) = [ ˜ θ, ˜ θ ′ V c ˜ c ρ φ ˜ θ, ˜ θ ′ ◦ φ − θ,θ ′ [ θ,θ ′ E ( a, a ′ ) ⊂ [ ˜ θ, ˜ θ ′ V c ˜ c ρ V Rc Cρ [ θ,θ ′ E ( a, a ′ ) = V (2 c ˜ c + Rc C ) ρ ( E ( a, a ′ )) . Taking c > c ˜ c + Rc C gives the desired property.(ii) Let s ∈ E ∗ ( a, a ′ ), then s ∈ φ ˜ θ, ˜ θ ′ ( L ( ˜ E ∗ ( a, a ′ ))) for some (˜ θ, ˜ θ ′ ) ending in ( a, a ′ ). Let ˜ s ∈ ˜ E ∗ ( a, a ′ ) such that s = φ ˜ θ, ˜ θ ′ ( L (˜ s )). Let ( b, b ′ ) be one of the ˜ c ρ − ( d + d ′ ) pairs, associated to˜ s , given by the scale recurrence lemma in [7]. If we write T b T ′ b ′ (˜ θ, ˜ θ ′ , s ) = T b T ′ b ′ ◦ φ (˜ θ, ˜ θ ′ , ˜ s ) = φ ◦ T b,b ′ (˜ θ, ˜ θ ′ , ˜ s ) = (˜ θb, ˜ θ ′ b ′ , φ ˜ θb, ˜ θ ′ b ′ ( L ( s ∗ ))) , we know that the ball B ( s ∗ , ρ ) is contained in ˜ E ∗ ( b, b ′ ). This implies B ( φ ˜ θb, ˜ θ ′ b ′ ( L ( s ∗ )) , c − ρ ) ⊂ φ ˜ θb, ˜ θ ′ b ′ ( B ( L ( s ∗ ) , ρ )) ⊂ φ ˜ θb, ˜ θ ′ b ′ ◦ L ( B ( s ∗ , ρ )) ⊂ φ ˜ θb, ˜ θ ′ b ′ ◦ L ( ˜ E ∗ ( b, b ′ )) ⊂ E ∗ ( b, b ′ ) . Thus it is enough to take c < c − .(iii) Let ( a, a ′ ) such that ν ( ˜ E ∗ ( a, a ′ )) ≥ ν ( ˜ J ˜ r ) / θ, ˜ θ ′ ) ending in ( a, a ′ ), we have m ( E ∗ ( a, a ′ )) ≥ m ( φ ˜ θ, ˜ θ ′ ( L ( ˜ E ∗ ( a, a ′ )))) & m ( L ( ˜ E ∗ ( a, a ′ )))= ν ( L − ( L ( ˜ E ∗ ( a, a ′ )))) ≥ ν ( ˜ E ∗ ( a, a ′ )) ≥ ν ( ˜ J ˜ r ) / . The fact that ν ( ˜ E ∗ ( a, a ′ )) ≥ ν ( ˜ J ˜ r ) / a, a ′ ) implies immediatly thedesired property with c big enough (cid:3) Random Perturbations of conformal Cantor sets
Random perturbations.
For the proof of theorem 2.22, we first perturb, in the C ∞ topology,the pair of Cantor sets ( K, K ′ ) so they satisfy the hypothesis of the Scale Recurrence Lemma andthe map g defining the Cantor set K is holomorphic on a neighborhood V of K . All this can be donethanks to lemma 2.20. Applying now the scale recurrence lemma gives constants R, c , c , c , c , c verifying the conclusions of the lemma. With the aim of reducing the number of constants, we willalso assume, without loss of generality, that the diameters of the sets G θ ( θ ) = k θ ( G ( θ )) are allless than one, this can be achieved by changing the metric. To prove theorem 2.22, we will nowonly perturb the Cantor set K , leaving K ′ unaltered.Notice that a neighbourhood in Ω ∞ Σ contains a neighbourhood in Ω k Σ for some integer k ≥
2. Sofrom now on we fix this integer k . The desired C k perturbation for g will be picked by a probabilisticargument out of a family of random perturbations that we will now construct.The following constructions and arguments are made having a parameter ρ > ρ sufficiently small.We first pick a subset Σ of Σ( ρ /k ) such that K = [ a ∈ Σ K ( a )is a partition of K into disjoint cylinders.We then define Σ as the subset of Σ formed of the words a ∈ Σ such that no word in Σ( ρ / k )appears twice in a . See the claim in the proof of lemma 2.20.Let ˜ c > sufficiently close to 0 to have the following: letˆ G ( a ) := V ˜ c · diam( G ( a )) ( G ( a )) , for a ∈ Σ ; then the ˆ G ( a ), a ∈ Σ , are pairwise disjoint.For each a ∈ Σ we choose a smooth function χ a : C → R satisfying: χ a ( z ) = 1 for z ∈ V ˜ c · diam( G ( a )) ( G ( a )) ,χ a ( z ) = 0 for z ˆ G ( a ) . Notice that, since a ∈ Σ( ρ /k ), we can choose these functions in a way that k D j χ a k≤ ˜ C ρ − j/k ,for all j ∈ N and ˜ C some constant independent of ρ (but not from ˜ c ).The probability space underlying the family of random perturbations is Ω = D Σ , where D isthe unitary disk in C , equipped with the normalized Lebesgue measure.For ω = ( ω ( a )) a ∈ Σ ∈ Ω, we define Φ ω to be the time-one map of the vector field X ω ( z ) = − c ρ (1+1 / k ) X a χ a ( z ) ω ( a ) , where c > g ω to be g ◦ Φ ω .By our previous estimative on k D j χ a k we have that k Φ ω − Id k C k is O ( ρ / k ).Since Φ ω , for any ω ∈ Ω, is close to the identity in the C k -topology, then g ω is close to g . Taking ρ small enough we can suppose that g ω generates a Cantor set (with the same family of sets G ( a ), a ∈ A ), which we denote by K ω . Moreover, taking ρ small it can be proven that this Cantor setis in fact a conformal Cantor set. Indeed, let V be the open set containing K where the function This constant corresponds to c in [1], since we already used this symbol in the scale recurrence lemma then wechanged it to ˜ c . TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 21 g is holomorphic. If ρ is sufficiently small then , for any x ∈ K ω ( a ) and a ∈ Σ , Φ ω ( x ) ∈ V and x ∈ V ˜ c · diam( G ( a )) ( G ( a )). It follows that D ( g ◦ Φ ω )( x ) = Dg (Φ ω ( x )) · D Φ ω ( x ) = Dg (Φ ω ( x ))which is a conformal linear transformation.Our task will be to find ω ∈ Ω such that the pair of Cantor sets determined by ( g ω , g ′ ) have anon empty compact recurrent set of relative configurations. Remark . All the objects introduced in section 2 are well defined for the Cantor sets K ω andwe will denote them by adding a superscript indicating the corresponding value of ω ∈ Ω, such as G ω ( a ), k θ,ω , c θ,ωa and F θ,ωa for example. Notice however that these Cantor sets have the same typeas K , and therefore are close to K in the topology described in definition 2.4. Besides, we considerfor each ω ∈ Ω the natural conjugation between the dynamical systems ( K ω , g ω | K ) and (Σ , σ ) H ω : K ω → Σ , which carries each point x ∈ K ω to the sequence { a n } n ≥ that satisfies ( g ω ) n ( x ) ∈ G ( a n ). For each a ∈ A , we fix a sequence x a ∈ Σ that begins with a . From now on, we consider that the set of basepoints c ωa ∈ G ( a ) for a ∈ A satisfies c ωa = ( H ω ) − ( x a )for every ω ∈ Ω. This is important for the study of limit geometries.3.2.
Some properties of the family g ω . Let a ′ ∈ Σ and a − ∈ A be such that ( a − , a ) ∈ B and a ′ begins with ( a − , a ); let a − a = a ′ . Any perturbed inverse branch f ωa − ,a is well defined inthe neighborhood V ρ ( G ( a )) and for any x ∈ V ρ ( G ( a ))(3) f ωa − ,a ( x ) = ( f a − ,a ( x ) if a ′ ∈ Σ − Σ ,f a − ,a ( x ) + c ρ / k ω ( a ′ ) if a ′ ∈ Σ . Notice that k Φ ω − Id k C = O ( ρ / k ), therefore V ρ ( G ( a ′ )) ⊂ Φ ω ( V ˜ c diam ( G ( a ′ )) ( G ( a ′ ))) ⊂ V ˜ c diam ( G ( a ′ )) ( G ( a ′ )) , for ρ small enough. This implies that V ρ ( G ( a )) ⊂ g ( V ρ ( G ( a ′ ))) ⊂ g ω ( V ˜ c diam ( G ( a ′ )) ( G ( a ′ )))and g ω ( z ) = ( g ( z ) if a ′ ∈ Σ − Σ ,g ( z − c ρ / k ω ( a ′ )) if a ′ ∈ Σ , for all z ∈ V ˜ c diam ( G ( a ′ )) ( G ( a ′ )). This in turn immediately implies the formula for f ωa ,a . Lemma 3.2.
Let ω ∈ Ω and H ω : K ω → Σ be the homeomorphism defined in remark 3.1. If ρ issufficiently small,(i) for any a ∈ Σ fin , we have (cid:13)(cid:13) f ωa − f a (cid:13)(cid:13) C ≤ c c ρ k ;(ii) for any a ∈ Σ , we have (cid:12)(cid:12) ( H ω ) − ( a ) − H − ( a ) (cid:12)(cid:12) ≤ c c ρ k ;(iii) for θ ∈ Σ − , we have (cid:13)(cid:13)(cid:13) k θ,ω − k θ (cid:13)(cid:13)(cid:13) C ≤ c c ρ − k ; This is consequence of lemma 3.2 part (ii), note that the proof of this part of the lemma does not use theconformality of g at the points in the Cantor set. We can also get this from the fact that K is an hyperbolic set for g and use continuation of the hyperbolic set (see Theorem 7.8 in [10]). (iv) for θ ∈ Σ − and a word a = ( a , a , . . . , a m ) with a = θ such that diam ( G ( a )) > c − ρ ,we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) DF θa DF θ, ωa − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c c ρ − k ; (cid:12)(cid:12)(cid:12) log r θ,ωa − log r θa (cid:12)(cid:12)(cid:12) ≤ c c ρ − k . The constant c is independent of θ , ω , a , ρ , and the size c of the perturbation.Proof. (i): Let x n = max | a | = n (cid:13)(cid:13) f ωa − f a (cid:13)(cid:13) C be the maximum distance between corresponding inversebranches of g n and ( g ω ) n . We will prove that x n ≤ c c ρ k by induction on n . For n = 1, thisis a direct consequence of equation (3).Observe that g ( G ( a )) covers all the pieces G ( b ) it intersects, therefore there exists δ > a, b ) ∈ B , then V δ ( G ( b )) ⊂ g ( G ( a )). Consequently, if x ∈ G ( b ), anypoint x ′ such that | x − x ′ | < δ and the line segment joining x and x ′ are contained in the ex-tended domain V δ ( G ( b )) of f ( a,b ) = ( g | G ( a ) ) − | V δ ( G ( b )) . In this domain, (cid:13)(cid:13) Df ( a,b ) (cid:13)(cid:13) ≤ µ − . Suppose x n ≤ c c ρ k . If ρ is sufficiently small, then x n < δ . Given a word b = ( b , b , . . . , b n , b n +1 ), wewrite b ′ = ( b , b , . . . , b n +1 ). Given a point x ∈ G ( b n +1 ), f ωb ( x ) − f b ( x ) = f ω ( b ,b ) ( f ωb ′ ( x )) − f ( b ,b ) ( f b ′ ( x )) = f ( b ,b ) ( f ωb ′ ( x )) − f ( b ,b ) ( f b ( x )) + f ω ( b ,b ) ( f ωb ′ ( x )) − f ( b ,b ) ( f ωb ′ ( x )) . Of course, when writing this, we are assuming that f ωb ′ ( x ) belongs to the domain V δ ( G ( b )) of f ( b ,b ) . But this is the case when (cid:12)(cid:12)(cid:12) f ωb ′ ( x ) − f b ′ ( x ) (cid:12)(cid:12)(cid:12) ≤ x n < δ , which is true by hypothesis. Morethan that, because the segment joining the two points is inside this domain and (3), | f ωb ( x ) − f b ( x ) |≤ µ − | f ωb ( x ) − f b ( x ) | + c ρ k ≤ µ − x n + c ρ k . In this manner, choosing c ≥ − µ − , we obtain x n +1 ≤ c c ρ k , finishing this part.(ii): Let a = ( a , a , . . . ) ∈ Σ. It follows that H − ( a ) = lim n →∞ f a n ( G ( a n )) and ( H ω ) − ( a ) =lim n →∞ f ωa n ( G ( a n )). As the diameters of these sets converge exponentially to zero, the result followsfrom (i).(iii): We now study the perturbed limit geometries. Notice that the base point used to define k θ,ω is not the same as the one for k θ , but the estimate of (ii) gives us control over this displacement.Fix θ ∈ Σ − and let z ∈ G ( θ ) = G ω ( θ ). Let the base point c θ ∈ K ( G ( θ )) be given by c θ = H − ( x a ) and the base point c ωθ ∈ K ω ( G ( θ )) be given by c ωθ = ( H ω ) − ( x a ) for some fixedsequence x a ∈ Σ. From (ii), (cid:12)(cid:12)(cid:12) c ωθ − c θ (cid:12)(cid:12)(cid:12) ≤ c c ρ k . Write c n = f θ n ( c θ ) and z n = f θ n ( z ); and c ωn = f ωθ n ( c ωθ ) and z ωn = f ωθ n ( z ) for n ≥
1. Notice that | z ωn − z n | = (cid:12)(cid:12)(cid:12) ( f ωθ n − f θ n )( z ) (cid:12)(cid:12)(cid:12) ≤ c c ρ k by (i). Likewise, seeing that c ωθ ∈ V δ ( G ( θ )),(4) | c ωn − c n | ≤ (cid:12)(cid:12)(cid:12) ( f ωθ n − f θ n )( c ωθ ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f θ n ( c ωθ ) − f θ n ( c θ ) (cid:12)(cid:12)(cid:12) . c ρ k , by (i) again and the estimate for (cid:12)(cid:12)(cid:12) c ωθ − c θ (cid:12)(cid:12)(cid:12) above ( f θ n is a contraction). TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 23
Remember that k θn = Φ θ n ◦ f θ n , where Φ θ n is an affine transformation, and k θ = lim n →∞ k θn .Hence k θn ( z ) = Φ θ n ( z n ) − Φ θ n ( c n ) = (cid:0) Df θ n ( c θ ) (cid:1) − ( z n − c n )and similarly k θ,ωn ( z ) = Φ ωθ n ( z ωn ) − Φ ωθ n ( c ωn ) = (cid:16) Df ωθ n ( c ωθ ) (cid:17) − ( z ωn − c ωn ) . The difference (cid:12)(cid:12)(cid:12) k θ,ωn ( z ) − k θn ( z ) (cid:12)(cid:12)(cid:12) is thus equal to(5) (cid:0) Df θ n ( c θ ) (cid:1) − ( z ωn − z n + c n − c ωn ) + (cid:20)(cid:16) Df ωθ n ( c ωθ ) (cid:17) − − (cid:0) Df θ n ( c θ ) (cid:1) − (cid:21) ( z ωn − c ωn ) . Let us analyze this expression for n not very large. Define A n := (cid:0) Df θ n ( c θ ) (cid:1) − = Dg n ( c n ) ,B n := (cid:16) Df ωθ n ( c ωθ ) (cid:17) − = D ( g ω ) n ( c ωn ) . for n ≥
1. If n is such that | A n | ≤ c ρ − k , then, by the previous estimates, the first term of (5) is . ρ − k c ρ k = c ρ − k . On the other hand, for every m ≥ A m +1 = A m · Dg ( c m +1 ) and B m +1 = B m · Dg ω ( c ωm +1 ) , therefore A m +1 − B m +1 = ( A m − B m ) · Dg ( c m +1 ) + B m · ( Dg ( c m +1 ) − Dg ω ( c ωm +1 )) , from which one can deduce, by induction on n , that for any m, n ≥ A m + n − B m + n ) − ( A m − B m ) · Dg n ( c m + n )= n − X j =0 B m + j · (cid:16) Dg ( c m + j +1 ) − Dg ω ( c ωm + j +1 ) (cid:17) · Dg n − − j ( c m + n )= n − X j =0 B m + j · (cid:16) Dg ( c m + j +1 ) − Dg ω ( c ωm + j +1 ) (cid:17) · A m + j +1 − · A m + n . (6)By (4) and the fact that the maps are C ∞ , (cid:12)(cid:12)(cid:12) Dg ( c m + n − j ) − Dg ω ( c ωm + j +1 ) (cid:12)(cid:12)(cid:12) . c ρ k . Now write C n := − ( A n − B n ) · ( A n ) − . It follows that B n = ( C n + Id ) · A n . Then, making m = 0in (6) and dividing it by | A n | , we get that(7) | C n | . n − X j =0 c ρ k · | Id + C j |·| A j |·| A j +1 | − . c ρ k · n − X j =0 | C j | . Let m be the largest value such that | A m | ≤ c ′ ρ − k . Thus m . − k log ρ and, again byinduction on n , | C n | . n ≤ m . Indeed, if it is true for all j ≤ n −
1, then | C n | . c ρ k · n − X j =0 | C j | . c ρ k · n ≤ m c ρ k . − k log ρ · c ρ k . , if ρ is sufficiently small. Plugging this estimate for | C j | with 0 ≤ j ≤ n − | C n | . c ρ − k for n ≤ m if ρ is sufficiently small. We also know that | z n − c n | . | A n | − and hence | z ωn − c ωn | . | A n | − for n ≤ m . Hence the second term in (5) is . | C n | . c ρ − k . We are left with controlling the difference k θ,ωn ( z ) − k θn ( z ) for n > m . Notice that if | A n | > c ′ ρ − k ,then the four points z n , c n , z ωn and c ωn belong to the same piece G ( a ) where a ∈ Σ . Thus, z ωn +1 − c ωn +1 = f ω ( θ − n − ,θ − n ) ( z ωn ) − f ω ( θ − n − ,θ − n ) ( c ωn ) = f ( θ − n − ,θ − n ) ( z ωn ) − f ( θ − n − ,θ − n ) ( c ωn ) . This way, if ρ is small enough so that the segments joining z ωn to c ωn and z n to c n belong to thedomain of f ( θ − n − ,θ − n ) , k θ,ωn +1 ( z ) = B n +1 · Z Df ( θ − n − ,θ − n ) ( c ωn + ( z ωn − c ωn ) t ) dt · ( z ωn − c ωn )= B n +1 · Z Df ( θ − n − ,θ − n ) ( c ωn + ( z ωn − c ωn ) t ) dt · B − n · k θ,ωn ( z ) ,k θn +1 ( z ) = A n +1 · Z Df ( θ − n − ,θ − n ) ( c n + ( z n − c n ) t ) dt · A − n · k θn ( z ) . Write I n := Z Df ( θ − n − ,θ − n ) ( c n + ( z n − c n ) t ) dt,I ωn := Z Df ( θ − n − ,θ − n ) ( c ωn + ( z ωn − c ωn ) t ) dt. Notice that I n and I ωn are both conformal matrices. This happens because c n + ( z n − c n ) t and c ωn + ( z ωn − c ωn ) t belong, for every t ∈ [0 , f ( θ − n − ,θ − n ) is holomorphic,provided ρ is sufficiently small. Besides, the difference between these two integrals is . c ρ k ,because f is C ∞ and | (1 − t )( c ωn − c n ) + t ( z ωn − z ωn ) | . c ρ k . Furthermore, Df ( θ − n − ,θ − n ) ( c ωn ) = Df ω ( θ − n − ,θ − n ) ( c ωn ) by (3) and so (cid:12)(cid:12) I ωn − Df ( θ − n − ,θ − n ) ( c ωn ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) I ωn − Df ω ( θ − n − ,θ − n ) ( c ωn ) (cid:12)(cid:12)(cid:12) . | z ωn − c ωn | . | B n | − and (cid:12)(cid:12) I n − Df ( θ − n − ,θ − n ) ( c n ) (cid:12)(cid:12) . | z n − c n | . | A n | − respectively. This implies that there exists some constant c > c such that (cid:12)(cid:12) A n +1 · I n · A − n − Id (cid:12)(cid:12) ≤ c | A n | − , (cid:12)(cid:12) B n +1 · I ωn · B − n − Id (cid:12)(cid:12) ≤ c | B n | − and (cid:12)(cid:12) A n +1 · I n · A − n − B n +1 · I ωn · B − n (cid:12)(cid:12) ≤ (cid:12)(cid:12) I n ( A − n · A n +1 − B − n · B n +1 ) (cid:12)(cid:12) + (cid:12)(cid:12) ( I n − I ωn ) B − n · B n +1 (cid:12)(cid:12) ≤ | I n | · (cid:12)(cid:12) Dg ( c n +1 ) − Dg ω ( c ωn +1 ) (cid:12)(cid:12) + | I n − I ωn | · (cid:12)(cid:12) Dg ω ( c ωn +1 ) (cid:12)(cid:12) ≤ cc ρ k , since the matrices A − n , B − n , A n +1 , B n +1 and I n commute (they are all conformal). Therefore,defining a n := (cid:12)(cid:12)(cid:12) k θ,ωn ( z ) − k θn ( z ) (cid:12)(cid:12)(cid:12) , for n ≥ m a n +1 = (cid:12)(cid:12)(cid:12) k θ,ωn +1 ( z ) − k θn +1 ( z ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) A n +1 · I n · A − n · (cid:16) k θ,ωn ( z ) − k θn ( z ) (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( A n +1 · I n · A − n − B n +1 · I ωn · B − n ) · k θ,ωn ( z ) (cid:12)(cid:12)(cid:12) ≤ (1 + c | A n | − ) (cid:12)(cid:12)(cid:12) k θ,ωn ( z ) − k θn ( z ) (cid:12)(cid:12)(cid:12) + c min { c ρ k , max {| A n | − , | B n | − }} = (1 + c | A n | − ) a n + c min { c ρ k , max {| A n | − , | B n | − }} . TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 25
For n & − (1 + k ) log ρ the minimum above is equal to max {| A n | − , | B n | − } , which decaysexponentially. Up to such a value, the formula above implies that a n . a m + c ρ k ( − log ρ ).Using the fact shown before that a m . c ρ − k and choosing ρ sufficiently small, it follows thatthe sequence a n is . c ρ − k . Making n → ∞ we conclude (iii).(iv) By lemma 2.12, r θa = diam( k θ ◦ f a ( G ( a m ))) = diam( F θa ◦ k θ ( G ( a m ))) = (cid:12)(cid:12)(cid:12) DF θa (cid:12)(cid:12)(cid:12) diam( k θ ( G ( a m )))and the analogous relation is valid for the perturbed version. To show that (cid:12)(cid:12)(cid:12) log r θ,ωa − log r θa (cid:12)(cid:12)(cid:12) ≤ c c ρ − k it is thus sufficient to prove that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) DF θa (cid:12)(cid:12)(cid:12) diam( k θ ( G ( a m ))) (cid:12)(cid:12)(cid:12) DF θ, ωa (cid:12)(cid:12)(cid:12) diam( k θ,ω ( G ( a m ))) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . c ρ − / k . Notice that by (iii), diam( k θ ( G ( a m ))) − diam( k θ,ω ( G ( a m ))) . c ρ − / k , and so, as these diameters are uniformly bounded away from zero, (cid:12)(cid:12)(cid:12)(cid:12) diam( k θ,ω ( G ( a m )))diam( k θ ( G ( a m ))) − (cid:12)(cid:12)(cid:12)(cid:12) . c ρ − / k . This way, we are left with analysing the derivatives of the affine maps F θa and F θ a,ω . Also fromlemma 2.12, DF θa = lim n →∞ (cid:0) Df θ n ( c θ ) (cid:1) − · Df ( θa ) n + m ( c a m ) DF θ, ωa = lim n →∞ (cid:16) Df ωθ n ( c ωθ ) (cid:17) − · Df ω ( θa ) n + m ( c ωa m ) . To meet our objectives we need only to show that for all n ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) Df θ n ( c θ ) (cid:1) − · Df ( θa ) n + m ( c a m ) (cid:16) Df ωθ n ( c ωθ ) (cid:17) − · Df ω ( θa ) n + m ( c ωa m ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . c ρ − / k . To each θ ′ ∈ Σ − let n ( θ ′ ) be the largest integer n such that (cid:12)(cid:12)(cid:12) Df θ ′ n ( c θ ′ ) (cid:12)(cid:12)(cid:12) ≥ c ′ ρ /k . The analysisof C n in (iii) implies that, uniformly on θ ′ ∈ Σ − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) Df ωθ ′ n ( c ωθ ′ ) (cid:17) − − (cid:16) Df θ ′ n ( c θ ′ ) (cid:17) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Df θ ′ n ( c θ ′ ) (cid:12)(cid:12)(cid:12) − . c ρ − / k . for all n ≤ n ( θ ′ ). This also implies that for all n ≤ n ( θ ′ ) (cid:12)(cid:12)(cid:12) Df ωθ ′ n ( c ωθ ′ ) − Df θ ′ n ( c θ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Df θ ′ n ( c θ ′ ) (cid:12)(cid:12)(cid:12) . c ρ − / k . Let us now show that these estimates remain valid for a much larger value of n , that is, n = ⌈ ρ − /k ⌉ when ρ is sufficiently small. Define for each θ ′ ∈ Σ − and n ≥ x n ( θ ′ ) := Df ωθ ′ n ( c ωθ ′ ) Df θ ′ n ( c θ ′ ) , f ωθ ′ n ( c ωθ ′ ) := c ωn ( θ ′ ) , f θ ′ n ( c θ ′ ) := c n ( θ ′ ) . Notice that for n ≥ n ( θ ′ ), the points c ωn ( θ ′ ) and c n ( θ ′ ) are always on the same piece G ( b ), with b ∈ Σ( ρ /k ), because of the definition of this number. Thus, in a neighborhood of c ωn ( θ ′ ), f ω ( θ − n − ,θ − n ) is just f ( θ − n − ,θ − n ) composed with a translation (see (3)), and therefore (cid:12)(cid:12)(cid:12)(cid:12) x n +1 ( θ ′ ) x n ( θ ′ ) − (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Df ω ( θ − n − ,θ − n ) ( c ωn ( θ ′ )) Df ( θ − n − ,θ − n ) ( c n ( θ ′ )) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Df ( θ − n − ,θ − n ) ( c ωn ( θ ′ )) − Df ( θ − n − ,θ − n ) ( c n ( θ ′ )) Df ( θ − n − ,θ − n ) ( c n ( θ ′ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . c ρ / k , because of (i) and the fact that the f ( θ − n − ,θ − n ) are C ∞ with uniformly (on θ ′ ∈ Σ − ) boundedderivatives. It follows that for every n such that n ( θ ′ ) ≤ n ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x n ( θ ′ ) x n ( θ ′ ) ( θ ′ ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . c ρ / k ρ − /k . c ρ − / k . Hence (cid:12)(cid:12) x n ( θ ′ ) − (cid:12)(cid:12) . c ρ − / k for all n ≤ n , because (cid:12)(cid:12) x n ( θ ′ ) − (cid:12)(cid:12) . c ρ − / k for all n ≤ n ( θ ′ )by the discussion above. Moreover, since m . log ρ , the same estimations imply that (cid:12)(cid:12) x n ( θ ′ ) − (cid:12)(cid:12) . c ρ − / k for all n ≤ n + m (this is the only part we use diam( G ( a )) ≥ c − ρ ).Observe that n ≫ max { n ( θ ) , n ( θa ) } ≈ log ρ if ρ is sufficiently small. The estimates aboveimply that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) Df θ n ( c θ ) (cid:1) − · Df ( θa ) n + m ( c a m ) (cid:16) Df ωθ n ( c ωθ ) (cid:17) − · Df ω ( θa ) n + m ( c ωa m ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) x n ( θ ) x n + m ( θa ) − (cid:12)(cid:12)(cid:12)(cid:12) . c ρ − / k , and so (8) is true for n ≤ n .For n ≥ n , let y n = (cid:0) Df θ n ( c θ ) (cid:1) − · Df ( θa ) n + m ( c a m ) and y ωn = (cid:16) Df ωθ n ( c ωθ ) (cid:17) − · Df ω ( θa ) n + m ( c ωa m ) . Following this notation, (cid:12)(cid:12)(cid:12)(cid:12) y n +1 y n − (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Df ( θ − n − ,θ − n ) ( c n + m ( θa )) Df ( θ − n − ,θ − n ) ( c n ( θ )) − (cid:12)(cid:12)(cid:12)(cid:12) . | ( c n + m ( θa ) − c n ( θ )) | ≤ diam( G ( θ n ))and the analogous relation is valid for the perturbed versions. However, remember that there is C > G ( θ n )) , diam( G ω ( θ n )) ≤ Cµ − n . This geometric control implies that there isa positive constant C ′ such that (cid:12)(cid:12)(cid:12)(cid:12) y n y ωn − (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ′ µ − n , for every n ≥ n . Since n ≥ ρ − /k , if ρ is sufficiently small, then µ − n ≪ c ρ − / k , and so theestimate (3) is valid for all n ≥
0, concluding the first part of the proof. (cid:3)
Remark . Some remarks relating the perturbation: • Note that since k f ωa − f a k C . c ρ / k then, supposing ρ is small enough, we have G ω ( a ) ⊂ V ρ ( G ( a )). • From now on we assume that the reference points c a , a ∈ A , are pre-periodic. From this,it is easy to prove that the reference points do not depend on ω , i.e. c ωa = c a . Indeed, let αββ...β... be the symbolic sequence associated to the points c ωa . Then we can write c ωa = lim n →∞ f ωαβ n ( x ) , TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 27 where x is any element in G ( β ), and β is the last letter of β . Notice that if n is big enoughthen β n contains a word of Σ( ρ / k ) repeated twice, thus any γ ∈ Σ( ρ /k ) containing β n can not be in Σ . This implies that f ωαβ n ( x ) = f αβ n − n ( f ωβ n ( x )) , for all n > n . Making n go to infinity we conclude that c ωa does not depend on ω . • If diam ( G ( a )) ≥ c − ρ then | DF θa |≈ | DF θ,ωa | and using DF θ,ωa = Dk θ,ω ( c a ) · Df ωa ( c a )one sees that | Df a ( c a ) |≈ | Df ωa ( c a ) | and then diam ( G ( a )) ≈ dim ( G ω ( a )). On can arrive to asimilar estimate if diam ( G ( a )) ≥ c − ρ , in this case we can decompose a as a concatenationof at most 4 words in Σ( ρ ) and use the fact that diam ( G ( a a )) ≈ diam ( G ( a )) diam ( G ( a )).Notice that with this approach the constants get worse if we increase the power of ρ in whichwe are interested, ρ will be enough for us. • Let ω , ω ∈ Ω and θ ∈ Σ − , suppose that f ω θ n ( z ) = f ω θ n ( z )for all n ≤ N and all z in a neighborhood of z , and in a neighborhood of c θ . Rememberthat limit geometries are defined by k θ,ω = lim n →∞ k θ,ωn , where k θ,ωn = Df ωθ n ( c θ ) − ( f ωθ n ( z ) − f ωθ n ( c θ )) . By our assumption we have k θ,ω n ( z ) = k θ,ω n ( z ), for all n ≤ N and z in a neighborhoodof z . From the proof of the existence of limit geometries (see [6]) one has that there is aconstant C such that k k θ,ω − k θ,ωn k≤ Cdiam ( G ω ( θ n ))and k D ( k θ,ω ◦ ( k θ,ωn ) − ) − I k≤ Cdiam ( G ω ( θ n )) , the same constant C works for all Cantor sets K ω , since they depend continuously on ω . Itfollows easily that there is a constant C ′ such that | k θ,ω ( z ) − k θ,ω ( z ) |≤ C ′ diam ( G ω ( θ N ))and | Dk θ,ω ( z ) − Dk θ,ω ( z ) |≤ C ′ diam ( G ω ( θ N )) , for all z in a neighborhood of z . Notice that since diam ( G ω j ( θ n )) ≈ Df ω j θ n ( c θ ), j = 1 , Df ω θ N ( c θ ) = Df ω θ N ( c θ ) then diam ( G ω ( θ N )) ≈ diam ( G ω ( θ N )).4. Proof of theorem 2.22
In this section we will define the set of relative configurations L = L ω , which will be a recurrentcompact set for at least one of the Cantor sets in the family of random perturbations. We first givea primary description of L and prove that assuming a probabilistic estimate, proposition 4.1, thenwe can prove theorem 2.22. The proof of the probabilistic estimate will be given in later sections. The recurrent compact set.
The set L = L ω will depend on ω , but only the translationcoordinate t . The image of L ω under the projection map: C → S will be a subset ˜ L of S independentof ω .We will choose a subset of Σ − with good combinatorial properties, this will be crucial to provethe estimate of lemma 4.1. First, let Σ nr ( ρ ) be the subset of Σ( ρ ) formed by words a such that:(1) no word b ∈ Σ( ρ / k ) appears twice in a ;(2) if c ∈ Σ( ρ / k ) appears at the end of a , then it does not appear elsewhere in a .We next define Σ − nr as the subset of Σ − formed by θ which end with a word in Σ nr ( ρ ). This isan open and closed subset in Σ − .A family of subsets E ( a, a ′ ) of J R , for ( a, a ′ ) ∈ Σ( ρ / ) × Σ ′ ( ρ / ) will be carefully constructedin section 4.2, in relation to Marstrand’s theorem, and it will satisfy the hypothesis Leb ( J R \ E ( a, a ′ )) ≤ c , ∀ ( a, a ′ ) . Then, the Scale recurrence Lemma gives us another family E ∗ ( a, a ′ ), ( a, a ′ ) ∈ Σ( ρ / ) × Σ ′ ( ρ / ),with the properties indicated in the statement of the lemma.The set e L is defined to be the subset of S R formed by the ( θ, θ ′ , s ) such that θ ∈ Σ − nr , and thereexists a ∈ Σ( ρ / ), a ′ ∈ Σ ′ ( ρ / ) with s ∈ E ∗ ( a, a ′ ) and θ , θ ′ ending with a , a ′ respectively.For every ( θ, θ ′ , s ) in e L , we will define in section 4.3, in relation to the conclusions of Marstrand’stheorem, a non empty subset L ω ( θ, θ ′ , s ), depending on ω ∈ Ω, of the fiber of C over ( θ, θ ′ , s ).Let L ω = { ( θ, θ ′ , s, t ) : ( θ, θ ′ , s ) ∈ e L , t ∈ L ω ( θ, θ ′ , s ) } ;consider next the ρ -neighbourhood L ω of L ω in e L × C : L ω = { ( θ, θ ′ , s, t ) : ( θ, θ ′ , s ) ∈ e L and ∃ ( θ , θ ′ , s , t ) ∈ L ω with d ( θ, θ ) < ρ / , d ( θ ′ , θ ′ ) < ρ / , | s − s | < ρ, | t − t | < ρ } . Fix u = ( θ, θ ′ , s, t ) ∈ e L × C . We define two subsets Ω ( u ), Ω ( u ) of Ω. First,Ω ( u ) = { ω ∈ Ω , ( θ, θ ′ , s, t ) ∈ L ω } . Second, Ω ( u ) is the set of ω ∈ Ω such that there exists b ∈ Σ( ρ ), b ′ ∈ Σ ′ ( ρ ), with b = θ , b ′ = θ ′ and the image T ωb T ′ b ′ ( u ) = (˜ θ, ˜ θ ′ , ˜ s, ˜ t ) satisfies:(i) for any ˜ s with | ˜ s − ˜ s | < c ρ / , we have (˜ θ, ˜ θ ′ , ˜ s ) ∈ e L ;(ii) ˜ t ∈ L ω (˜ θ, ˜ θ ′ , ˜ s ).The following crucial estimate will be proven in section 6. Proposition 4.1.
Assume that c is chosen conveniently large. Then there exists c > , suchthat, for any u ∈ e L × C , one has P (Ω ( u ) − Ω ( u )) ≤ exp ( − c ρ − k ( d + d ′ − ) . Using the previous estimate, we will prove that for some ω the pair K ω , K has a recurrentcompact set, thus obtaining theorem 2.22. We proceed as in [1]. We discretize the set ˜ L × { t : | t | < e R ) } , i.e. we choose subsets ∆ i , i ∈ I , such that˜ L × { t : | t | < e R ) } = [ i ∈ I ∆ i , and for all ( θ , θ ′ , s , t ) , ( θ , θ ′ , s , t ) ∈ ∆ i one has d ( θ , θ ) < ρ / , d ( θ ′ , θ ′ ) < ρ / , | s − s | < ρ , | t − t | < ρ . TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 29
It is not difficult to see that this can be done in such a way that I is polynomial in ρ − . For each i ∈ I , choose u i ∈ ∆ i . If ρ is small enough, we have P ( ∪ i ∈ I Ω ( u i ) − Ω ( u i )) ≤ I · exp (cid:16) − c ρ − k ( d + d ′ − (cid:17) < . Therefore, the set ∪ i ∈ I Ω ( u i ) − Ω ( u i ) is not the whole Ω and we can choose ω outside of it.Observe that for all i ∈ I we have that ω ∈ Ω ( u i ) implies ω ∈ Ω ( u i ). Now define L ω = { ( θ, θ ′ , s, t ) ∈ ˜ L × C : ∃ (˜ θ, ˜ θ ′ , ˜ s, ˜ t ) ∈ L ω with d ( θ, ˜ θ ) ≤ ρ / , d ( θ ′ , ˜ θ ′ ) ≤ ρ / , | s − ˜ s |≤ ρ/ , | t − ˜ t |≤ ρ/ } . We will prove that L ω is a recurrent compact set for K ω , K ′ . First, notice that L ω ⊂ L ω ⊂ L ω .Then L ω is not empty. Now, let u = ( θ, θ ′ , s, t ) ∈ L ω , we have that there is (˜ θ, ˜ θ ′ , ˜ s, ˜ t ) ∈ L ω with the properties in the definition of L ω . Since ˜ t ∈ L ω (˜ θ, ˜ θ ′ , ˜ s ) then | ˜ t |≤ e R ) (this is clearfrom the definition of L in subsection 4.3). Thus | t | < e R ) and u ∈ ∆ i for some i ∈ I . For u i = ( θ i , θ ′ i , s i , t i ) we have that d ( θ, θ i ) < ρ / , d ( θ ′ , θ ′ i ) < ρ / , | s − s i | < ρ , | t − t i | < ρ . Therefore u i ∈ L ω and ω ∈ Ω ( u i ), this implies that ω ∈ Ω ( u i ) and there is some pair ( b, b ′ ) ∈ Σ( ρ ) × Σ ′ ( ρ ) such that T ω b T ′ b ′ ( u i ) = (˜ θ i , ˜ θ ′ i , ˜ s i , ˜ t i ) satisfies the properties (i) and (ii) described above.We will prove that T ω b T ′ b ′ ( u ) is in the interior of L ω . Write T ω b T ′ b ′ ( u ) = ( θb, θ ′ b ′ , ˆ s, ˆ t ), usingequation (2) we have˜ t i = (cid:16) DF θ ′ i b ′ (cid:17) − · ( t i + s i c θ i ,ω b − c θ ′ i b ′ ) , and ˆ t = (cid:16) DF θ ′ b ′ (cid:17) − · ( t + sc θ,ω b − c θ ′ b ′ ) . Therefore | ˜ t i − ˆ t | . ρ / . Analogously one hasˆ s = DF θ,ω b DF θ ′ b ′ · s, and ˜ s i = DF θ i ,ω b DF θ ′ i b ′ · s i . In this case we get | ˜ s i − ˆ s | . ρ / . One also has d ( θb, ˜ θ i ) . ρ / and d ( θ ′ b ′ , ˜ θ ′ i ) . ρ / .Thanks to property (ii), we know that (˜ θ i , ˜ θ ′ i , ˜ s i , ˜ t i ) ∈ L ω . Moreover, for any ( η, η ′ , r, x ) suchthat d ( η, θb ) < ρ / , d ( η ′ , θ ′ b ′ ) < ρ / , | r − ˆ s | < ρ/ , | x − ˆ t | < ρ/ , we have d ( η, ˜ θ i ) ≤ ρ / , d ( η ′ , ˜ θ ′ i ) ≤ ρ / , | r − ˜ s i |≤ ρ/ , | x − ˜ t i |≤ ρ/ . To conclude that ( η, η ′ , r, x ) ∈ L ω we only need to show that ( η, η ′ , r ) ∈ ˜ L . Property (i) aboveimplies that (˜ θ i , ˜ θ ′ i , r ) ∈ ˜ L , by the definition of the set ˜ L this means that ˜ θ i ∈ Σ − nr and r ∈ E ∗ ( a, a ′ )for a pair ( a, a ′ ) in Σ( ρ / ) × Σ ′ ( ρ / ) such that (˜ θ i , ˜ θ ′ i ) ends in it. However, since d ( η, ˜ θ i ) . ρ / , d ( η ′ , ˜ θ ′ i ) . ρ / then η ∈ Σ − nr and ( η, η ′ ) also ends in ( a, a ′ ). Therefore ( η, η ′ , r ) ∈ ˜ L and( η, η ′ , r, x ) ∈ L ω , which shows that T ω b T ′ b ′ ( u ) is in the interior of L ω . From the fact that thesets E ∗ ( a, a ′ ) are closed and the definition of the sets L ω , it is not difficult to prove that L ω is acompact set. Therefore, L ω is a recurrent compact set for the pair K ω , K ′ . Set of good scales.
Let ( θ, θ ′ , s ) in the space of relative scales, and points x ∈ K ( θ ), x ′ ∈ K ′ ( θ ′ ). Consider λ = π θ,θ ′ ,s ( x, x ′ ) := k θ ′ ( x ′ ) − sk θ ( x ) . Then ( θ, θ ′ , s, λ ) is the unique relative configuration above ( θ, θ ′ , s ) such that A ( k θ ( x )) = A ′ ( k θ ′ ( x ′ )) , (where ( θ, A ) , ( θ ′ , A ′ ) represents this relative configuration).Remember that, for some previously fixed R > J R = { s ∈ J : e − R < | s | < e R } and S R = Σ − × Σ ′− × J R .We equip each set K ( θ ) (resp. K ′ ( θ ′ )) with the d -dimensional (resp. d ′ -dimensional) Hausdorffmeasure µ d (resp. µ d ′ ).Then, for ( θ, θ ′ , s ) ∈ S , we denote by µ ( θ, θ ′ , s ) the image under π θ,θ ′ ,s of µ d × µ d ′ on K ( θ ) × K ′ ( θ ′ ).As in the theory of Cantor sets in the real line, there are constants c > c > θ ∈ A , θ ′ ∈ A ′ : c < µ d × µ d ′ ( K ( θ ) × K ′ ( θ ′ )) < c . This can be proven using the results appearing in Zamudio’s thesis ([8]). Indeed, for a givenconformal Cantor set K of dimension d , by lemma 1.2.2 of that manuscript, one can find a sequenceof coverings with size converging to zero and d -volume bounded by C c d , namely the covering bythe pieces of Σ ( ρ ), showing that µ d ( K ) < ∞ .On the other hand, given a finite cover of K by balls U i of radii r i > i = 1 , . . . , n , by lemma1.2.3, each U i intersects at most Cρ − d r di pieces G ( a ) of Σ ( ρ ) if ρ and r i are sufficiently small. Since U i is a cover and by lemma 1.2.2 ( ρ ) > C − ρ − d , summing for all i yields: Cρ − d (cid:16)X r di (cid:17) ≥ C − ρ − d and so P r di (and µ d ( K )) is always bounded from zero. To obtain the statement just restrict thearguments to K ( θ ) and K ′ ( θ ′ ) and take their product.Notice that the same lemma 1.2.3 implies that there is a constant c > µ := µ d × µ d ′ ,the product measure in C , µ ( B ( x, r )) < cr d + d ′ for any ball of radius r >
0. If d + d ′ > I ( µ ) := Z Z | u − v | − dµ ( u ) dµ ( v ) < ∞ This way, the proof of Marstrand’s theorem in Mattila’s book [11] can be adapted to our contextto show that for fixed ( θ, θ ′ ) the measure µ ( θ, θ ′ , s ) is absolutely continuous with respect to theLebesgue measure for Lebesgue almost every s , with L -density χ θ,θ ′ ,s satisfying Z J R k χ θ,θ ′ ,s k L dLeb ( s ) ≤ c ( R ) , where c ( R ) is independent on θ , θ ′ . all one needs to verify is that for any points u, v ∈ C , Leb ( { s : s ∈ J R , | π θ,θ ′ ,s ( u ) − π θ,θ ′ ,s ( v ) | < δ } ) < cδ | u − v | − ,where c > R . Notice that π θ,θ ′ ,s = π s ◦ F θ,θ ′ , where π s ( u ) = u − s · u for u = ( u , u ) and F θ,θ ′ = ( k θ , k θ ′ ) are diffeomorphisms that distort area in a uniformly bounded way. A simplemanipulation shows that the measure is bounded above by cδ | u − v | − , but since s is in J R a bounded set, | u − v | − is bounded by c R | u − v | − , for some constant c R depending on R . TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 31
When one does control k χ θ,θ ′ ,s k L , this gives, by Cauchy-Schwarz inequality, a lower bound forthe Lebesgue measure of π θ,θ ′ ,s ( X ), X being a subset of K × K ′ with positive ( d + d ′ )-dimensionalHausdorff measure; indeed we have: µ d × µ d ′ ( X ) ≤ Z π θ,θ ′ ,s ( X ) χ θ,θ ′ ,s ( t ) dt ≤ Leb ( π θ,θ ′ ,s ( X )) / k χ θ,θ ′ ,s k L and therefore(9) Leb ( π θ,θ ′ ,s ( X )) ≥ ( µ d × µ d ′ ( X )) k χ θ,θ ′ ,s k − L . Fix ( θ, θ ′ ) in Σ − × Σ ′− . Let a ∈ Σ( ρ / k ), a ′ ∈ Σ ′ ( ρ / k ), with a = θ , a ′ = θ ′ . One has T a T ′ a ′ ( θ, θ ′ , s ) = (cid:16) θa, θ ′ a ′ , s · DF θa /DF θ ′ a ′ (cid:17) and c − ≤ | DF θa || DF θ ′ a ′ | ≤ c . We therefore have Z J R k χ T a T ′ a ′ ( θ,θ ′ ,s ) k L ds ≤ c ′ ( R ) , with c ′ ( R ) independent of θ , θ ′ , a , a ′ . On the other hand, one has ρ / k ) ≤ c ρ − d/ k , ′ ( ρ / k ) ≤ c ρ − d ′ / k . We conclude that Z J R X a,a ′ k χ T a T ′ a ′ ( θ,θ ′ ,s ) k L ds ≤ c ρ − d + d ′ k c ′ ( R ) . We now define, with c > E ( θ, θ ′ ) = (cid:26) s ∈ J R , k χ θ,θ ′ ,s k L ≤ c and X a,a ′ k χ T a T ′ a ′ ( θ,θ ′ ,s ) k L ≤ c ρ − d + d ′ k (cid:27) . For c ∈ Σ( ρ / ), c ′ ∈ Σ ′ ( ρ / ), we define E ( c, c ′ ) as the set of s ∈ J R such that there exists θ , θ ′ ending respectively with c , c ′ such that s ∈ E ( θ, θ ′ ).One has, for any θ ∈ Σ − , θ ′ ∈ Σ ′− : Leb ( J R \ E ( θ, θ ′ )) ≤ c − ( c ( R ) + c c ′ ( R ));therefore, provided that c > c − ( c ( R ) + c c ′ ( R )) , we will have Leb ( J R \ E ( c, c ′ )) ≤ c for all c ∈ Σ( ρ / ), c ′ ∈ Σ ′ ( ρ / ). This means that we can apply the Scale recurrence Lemma withthe family E ( c, c ′ ) of subsets of J R . The sets E ∗ ( c, c ′ ) are then defined using this lemma (see section2.7), we can assume they are closed, this is justified in the remark after the lemma. Construction of L ω . Until now, we have only worked with the unperturbed maps g , g ′ .We now consider the family of random perturbations g ω , and will proceed to construct the sets L ω ( θ, θ ′ , s ), for ( θ, θ ′ , s ) ∈ ˜ L . For a ∈ Σ( ρ / k ), let Σ − ( a ) be the open and closed subset of Σ − formed by the θ ending with a . Choose a subset Σ − of Σ( ρ / k ) such thatΣ − = [ Σ − Σ − ( a )is a partition of Σ − .For a ∈ Σ − , define a subset Σ ( a ) of the subset Σ (recall Σ ⊂ Σ( ρ /k )), as the set of words inΣ starting with a . For θ ∈ Σ − ( a ), we also define Σ ( θ ) = Σ ( a ).Let θ ∈ Σ − . We write Ω = [ − , +1] Σ ( θ ) × [ − , +1] Σ − Σ ( θ ) ,ω = ( ω ′ , ω ′′ )and for such an ω , we set ω ∗ = (0 , ω ′′ ) . This depends on θ , but nearby b θ (with d ( θ, b θ ) < c − ρ / k ) will belong to the same Σ − ( a ) and givethe same projection ω ∗ of ω .For ( θ, θ ′ , s ) ∈ ˜ L , the set L ω ( θ, θ ′ , s ) will actually only depend (as far as ω is concerned) on theprojection ω ∗ of ω associated to θ .We will say that two words b , b ∈ Σ( ρ ) are independent if there is no word b ∈ Σ( ρ / k ) suchthat both b and b start with b .With c > N = (cid:20) c ρ − k ( d + d ′ − (cid:21) . Let ( θ, θ ′ , s ) ∈ ˜ L , and ω ∈ Ω.We define L ω ( θ, θ ′ , s ) to be the set of points ( θ, θ ′ , s, t ) in the fiber for which there exist pairs( b , b ′ ) , . . . , ( b N , b ′ N ) in Σ( ρ ) × Σ ′ ( ρ ), with b i = θ , b ′ i = θ ′ such that, if we set T ω ∗ b i T ′ b ′ i ( θ, θ ′ , s, t ) = ( θ i , θ ′ i , s i , t i ) , the following hold:(i) the words b , . . . , b N are pairwise independent;(ii) for 1 ≤ i ≤ N , θ i ∈ Σ − nr ;(iii) for 1 ≤ i ≤ N , and | ˜ s − s i |≤ c ρ / , ( θ i , θ ′ i , ˜ s ) ∈ ˜ L ;(iv) for 1 ≤ i ≤ N , | t i |≤ e R ).We will use also a slightly smaller set L − ω ( θ, θ ′ , s ); it is defined in the same way than L ω ( θ, θ ′ , s ),but with (iii), (iv) replaced by:(iii)’ for 1 ≤ i ≤ N , and | ˜ s − s i |≤ c ρ / , ( θ i , θ ′ i , ˜ s ) ∈ ˜ L (iv)’ for 1 ≤ i ≤ N , | t i |≤ e R .In the next section, we will prove the following estimate. Proposition 4.2. If c has been chosen sufficiently small, there exists c > such that, for any ( θ, θ ′ , s ) ∈ ˜ L and any ω ∈ Ω , the Lebesgue measure of L − ω ( θ, θ ′ , s ) is > c . TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 33 Proof of proposition 4.2
In this section we will prove proposition 4.2. First we prove some lemmas that are necessary forthe proposition. We follow the same argument as [1] sections 4.8-4.12 with some modifications.Fix ( θ, θ ′ , s ) ∈ ˜ L , we will work with this triple throughout this section, at the end we will provethat Leb ( L − ω ( θ, θ ′ , s )) > c .Choose a subfamily Σ of Σ( ρ / k ) of words starting with θ such that K ( θ ) = [ Σ K ( a )is a partition of K ( θ ). Similarly, choose a subfamily Σ ′ of Σ ′ ( ρ / k ) of words starting with θ ′ suchthat K ′ ( θ ′ ) = [ Σ ′ K ′ ( a ′ ) . There is a constant c > a, a ′ ) ∈ Σ × Σ ′ , we have c − ρ k ( d + d ′ ) ≤ µ d × µ d ′ ( K ( a ) × K ′ ( a ′ )) ≤ c ρ k ( d + d ′ ) . Let J ( a, a ′ ) := π θ,θ ′ ,s ( G ( a ) × G ( a ′ )) and c ( a, a ′ ) := π θ,θ ′ ,s ( c a , c a ′ ) ∈ C for a ∈ Σ and a ′ ∈ Σ ′ .Since s is bounded above and below, we have(10) B ( c ( a, a ′ ) , c − ρ k ) ⊂ J ( a, a ′ ) ⊂ B ( c ( a, a ′ ) , c ρ k ) , if c is sufficiently large. We assume that the previous relations involving c hold for any othertriples in Σ − × Σ ′− × J R and for any value of ω , choosing c large enough this can be easilyguaranteed.Say ( a, a ′ ) is good if there are no more than c − ρ − / k ( d + d ′ − pairs (˜ a, ˜ a ′ ) such that the distancebetween the points c ( a, a ′ ) and c (˜ a, ˜ a ′ ) is less than (2 + 1 / c ρ / k . Otherwise, say it is bad . Lemma 5.1.
The number of bad pairs ( a, a ′ ) is less than · · π c c c ρ − / k ( d + d ′ ) . Before begining the proof, we remember the Vitali covering lemma. Let B , B , . . . , B n ⊂ R d be a finite collection of balls. For i = 1 , . . . , n , denote by 3 B i the ball with same center as B i but having radius three times larger. The lemma states that there exists a subcollection of balls B j , B j , . . . , B j k with the Vitali property, this is • The balls B j , B j , . . . , B j k are pairwise disjoint and • The union B ∪ B ∪ . . . ∪ B n is contained in 3 B j ∪ B j ∪ . . . ∪ B j k .In the case that the balls B i are subsets of the complex plane and have the same radius R , onecan see that every point z ∈ C is covered by no more than 16 of the balls 3 B j , B j , . . . , B j k .Indeed, consider the ball B centered at z with radius 4 R . It contains all the balls B i , . . . , B i l suchthat 3 B i , . . . , B i l cover z . However, the balls B i , . . . , B i l are pairwise disjoint, and so there areno more than (4 R ) /R = 16 of them inside B , otherwise they would overlap. Proof.
By construction of e L , there exists e θ , e θ ′ , ˜ s with d ( θ, e θ ) ≤ c ρ / , d ( θ ′ , e θ ′ ) ≤ c ρ / , | s − ˜ s |≤ c ρ / such that (cid:13)(cid:13)(cid:13) χ e θ, e θ ′ , ˜ s (cid:13)(cid:13)(cid:13) L ≤ c . Because of remark 2.6, the distance between the points k θ ( c a ) and k ˜ θ ( c a ) is of order ρ / forevery ( a, a ′ ) ∈ Σ × Σ ′ and the same is true for their K ′ versions. Thus, if c is sufficiently large,for each bad pair ( a, a ′ ) there are more than c − ρ − / k ( d + d ′ − pairs (˜ a, ˜ a ′ ) satisfying | π ˜ θ, ˜ θ ′ , ˜ s ( c a , c a ′ ) − π ˜ θ, ˜ θ ′ , ˜ s ( c ˜ a , c ˜ a ′ ) |≤ c ρ k . From now on, we denote π ˜ θ, ˜ θ ′ , ˜ s ( c a , c a ′ ) as ˜ c ( a, a ′ ) for any ( a, a ′ ) ∈ Σ × Σ ′ .For each bad pair ( a, a ′ ), consider the disk J ∗ ( a, a ′ ) of radius c ρ k and center at ˜ c ( a, a ′ ).Then the corresponding c − ρ − / k ( d + d ′ − sets π ˜ θ, ˜ θ ′ , ˜ s ( G (˜ a ) × G (˜ a ′ )) are subsets of J ∗ ( a, a ′ ). Thisway, Z J ∗ ( a,a ′ ) χ ˜ θ, ˜ θ ′ , ˜ s ds = ( µ d × µ d ′ )( π − θ, ˜ θ ′ , ˜ s ) ( J ∗ ( a, a ′ )) ≥ X (˜ a, ˜ a ′ ) ( µ d × µ d ′ )( G (˜ a ) × G (˜ a ′ )) ≥ c − ρ − / k ( d + d ′ − · c − ρ d + d ′ k = ( c c ) − ρ /k . Let J ∗ be the union of all the disks J ∗ ( a, a ′ ) corresponding to bad pairs and B be the number ofthese pairs. Choose a subcover of J ∗ as in the Vitali lemma, indexed by the pairs ( a, a ′ ) belongingto a subset V of the set of bad pairs. It follows that B · c − ρ ( d + d ′ )2 k ≤ ( µ d × µ d ′ )( π − θ, ˜ θ ′ , ˜ s ) ( J ∗ )) = Z J ∗ χ ˜ θ, ˜ θ ′ , ˜ s ds ≤ X ( a,a ′ ) ∈ V Z J ∗ ( a,a ′ ) χ ˜ θ, ˜ θ ′ , ˜ s ds. On the other hand, by Cauchy-Schwartz theorem, Z J ∗ ( a,a ′ ) χ ˜ θ, ˜ θ ′ , ˜ s ds ! · ( c c ) − ρ /k ≤ Z J ∗ ( a,a ′ ) χ ˜ θ, ˜ θ ′ , ˜ s ds ! ≤ Z J ∗ ( a,a ′ ) χ θ, ˜ θ ′ , ˜ s ds · Leb(3 J ∗ ( a, a ′ )) = Z J ∗ ( a,a ′ ) χ θ, ˜ θ ′ , ˜ s ds · πc ρ /k , for every bad pair ( a, a ′ ). But the Vitali covering covers each point z ∈ C at most 16 times, so X ( a,a ′ ) ∈ V Z J ∗ ( a,a ′ ) χ ˜ θ, ˜ θ ′ , ˜ s ds ≤ π c c X ( a,a ′ ) ∈ V Z J ∗ ( a,a ′ ) χ θ, ˜ θ ′ , ˜ s ds ≤ · · π c c Z C χ θ, ˜ θ ′ , ˜ s ds. It follows that B ≤ · · π c c c ρ − ( d + d ′ ) / k , concluding the proof. (cid:3) First, we construct the pairs ( b, b ′ ) amongst which the pairs ( b i , b ′ i ) of 4.2 must be looked for.We make the following observation: Lemma 5.2.
Let θ ∈ Σ − nr . The number of words c ∈ Σ( ρ / ) with c = θ , such that θc . . . c m / ∈ Σ − nr is o ( ρ − d/ ) as ρ → , uniformly in θ . Now let ( θ, θ ′ , s ) ∈ e L . It follows from conclusion (ii) of the Scale recurrence Lemma (lemma 2.26)and the last observation that we can find at least c ρ − / d + d ′ ) pairs ( c i , c ′ i ) ∈ Σ( ρ / ) × Σ ′ ( ρ / )such that, writing T c i T c ′ i ( θ, θ ′ , s ) = ( θ i , θ ′ i , s i ), we have: • θ i ∈ Σ − nr ; • B ( s i , c ρ / ) ⊂ E ∗ ( c i , c ′ i ).As ( θ i , θ ′ i , s i ) again belongs to e L , we can for each i find at least c ρ − / d + d ′ ) pairs ( d ij , d ′ ij ) ∈ Σ( ρ / ) × Σ ′ ( ρ / ) (with the first letter of d ij , d ′ ij being the last one of c i , c ′ i respectively), suchthat writing T d ij T d ′ ij ( θ i , θ ′ i , s i ) = ( θ ij , θ ′ ij , s ij ), we have • θ ij ∈ Σ − nr ; • B ( s ij , c ρ / ) ⊂ E ∗ ( d ij , d ′ ij ).Concatenation of the c i , c ′ i and d ij , d ′ ij gives a family of words ( b ij , b ′ ij ) in Σ( ρ ) × Σ ′ ( ρ ) with atleast c ρ − ( d + d ′ ) elements. TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 35
We now consider the perturbed operators. In this case T ωc i T ′ c ′ i ( θ, θ ′ , s ) = ( θ i , θ ′ i , s i ( ω )) and bylemma 3.2 the distance between s i ( ω ) and s i is of order c ρ − / k . Similarly one has T ωd ij T ′ d ′ ij ( θ i , θ ′ i , s i ( ω )) = ( θ ij , θ ′ ij , s ij ( ω ))and again the distance between s ij ( ω ) and s ij is of order c ρ − / k .Let ( e θ, e θ ′ , ˜ s ) be such that d ( θ, e θ ) ≤ c ρ / , d ( θ ′ , e θ ′ ) ≤ c ρ / , | s − ˜ s |≤ c ρ / and ˜ s ∈ E ( e θ, e θ ′ ). Lemma 5.3. If c has been chosen sufficiently small, there are at least c c − ρ − d + d ′ k pairs ( a, a ′ ) ∈ Σ × Σ ′ which are good and satisfy (cid:13)(cid:13)(cid:13)(cid:13) χ T a T a ′ ( e θ, e θ ′ , ˜ s ) (cid:13)(cid:13)(cid:13)(cid:13) L ≤ c − and such that at least c c − ρ − ( d + d ′ )( − k ) pairs ( c i , c ′ i ) start with ( a, a ′ ) .Proof. The proof is the same as in [1], bearing in mind the different but similar bound in thenumber of bad pairs. (cid:3)
The following general lemma will be used later to estimate the measure of the union of theperturbed version of the sets J ( b ij , b ′ ij ). Lemma 5.4.
Let J α , J ′ α , K α , α ∈ A , be families of sets in C such that for some λ, ǫ, ν, σ ∈ R + ,and c α , c ′ α ∈ C : • B ( c α , ǫ ) ⊂ J α ⊂ B ( c α , λǫ ) , K α ⊂ J α , B ( c ′ α , ǫ ) ⊂ J ′ α . • d ( c α , c ′ α ) ≤ νǫ , Leb ( K α ) ≥ σ − Leb ( J α ) .Then Leb [ α ∈ A J ′ α ! ≥ λ + ν ) Leb [ α ∈ A J α ! , and Leb [ α ∈ A K α ! ≥ σ − λ Leb [ α ∈ A J α ! . Proof.
Notice that J α ⊂ B ( c ′ α , ( λ + ν ) ǫ ). Let ˜ A be a subset of A such that the balls { B ( c ′ α , ( λ + ν ) ǫ ) } α ∈ ˜ A have the Vitali property. Thus Leb [ α ∈ A J α ! ≤ Leb [ α ∈ A B ( c ′ α , ( λ + ν ) ǫ ) ! ≤ Leb [ α ∈ ˜ A B ( c ′ α , λ + ν ) ǫ ) ≤ [3( λ + ν )] Leb [ α ∈ ˜ A B ( c ′ α , ǫ ) ≤ λ + ν ) Leb [ α ∈ A B ( c ′ α , ǫ ) ! ≤ λ + ν ) Leb [ α ∈ A J ′ α ! , where in the passage from the second to the third line we use the fact that the sets B ( c ′ α , ǫ ), α ∈ ˜ A ,are disjoint. This proves the first inequality. For the second, use again Vitali to find a subset A ′ of A such that the balls { B ( c α , λǫ ) } α ∈ A ′ have the Vitali property. Then Leb [ α ∈ A J α ! ≤ Leb [ α ∈ A ′ B ( c α , λǫ ) ! ≤ λ X α ∈ A ′ Leb ( B ( c α , ǫ )) ≤ λ σ X α ∈ A ′ Leb ( K α ) = 9 λ σ · Leb [ α ∈ A ′ K α ! ≤ λ σ · Leb [ α ∈ A K α ! . (cid:3) Lemma 5.5.
Define the sets J ω ( b ij , b ′ ij ) = π ωθ,θ ′ ,s ( G ω ( b ij ) × G ( b ′ ij )) ,J ω ( a, a ′ ) = [ ( c i ,c ′ i ) [ ( b ij ,b ′ ij ) J ω ( b ij , b ′ ij ) . Then
Leb (cid:0) J ω ( a, a ′ ) (cid:1) & c ρ /k . Proof.
First we make the following observation:(11) π η,η ′ ,w ◦ ( f d , f d ′ ) = A ◦ π T d T ′ d ′ ( η,η ′ ,w ) , where A is an affine function with | DA |≈ diam ( G ( d )), and this holds for any ( d, d ′ ) ∈ Σ( α ) × Σ ′ ( α ),for some α ∈ R + , and any ( η, η ′ , w ).For an excellent pair ( a, a ′ ) we consider the associated pairs ( c i , c ′ i ) and the sets J ω ( c i , c ′ i ) = π ωθ,θ ′ ,s ( G ω ( c i ) × G ( c ′ i )) ,J ( c i , c ′ i ) = π θ,θ ′ ,s ( G ( c i ) × G ( c ′ i )) , ˜ J ( c i , c ′ i ) = π ˜ θ, ˜ θ ′ , ˜ s ( G ( c i ) × G ( c ′ i )) ,J ω ( a, a ′ ) = [ ( c i ,c ′ i ) J ω ( c i , c ′ i ) . We will prove that the measure of J ω ( a, a ′ ) is at least of the order c ρ /k . To do this, we uselemma 5.4 to see that the measures of the sets S J ω ( c i , c ′ i ), S J ( c i , c ′ i ) and S ˜ J ( c i , c ′ i ) are of thesame order. It is clear from equation (10) that the sets J ω ( c i , c ′ i ), J ( c i , c ′ i ) and ˜ J ( c i , c ′ i ) are allcontained in, and contain, balls with radius of order ρ / , and centered at the points π ωθ,θ ′ ,s ( c ωc i , c c ′ i ), π θ,θ ′ ,s ( c c i , c c ′ i ) and π ˜ θ, ˜ θ ′ , ˜ s ( c c i , c c ′ i ) respectively. We remark that | π ωθ,θ ′ ,s ( c ωc i , c c ′ i ) − π θ,θ ′ ,s ( c c i , c c ′ i ) | = | s |·| k θ ( c c i ) − k θ,ω ( c ωc i ) |≤ | s |· h | k θ ( c c i ) − k θ,ω ( c c i ) | + | k θ,ω ( c c i ) − k θ,ω ( c ωc i ) | i . c ρ − / k . ρ / , we also have | π ˜ θ, ˜ θ ′ , ˜ s ( c c i , c c ′ i ) − π θ,θ ′ ,s ( c c i , c c ′ i ) | ≤ | k ˜ θ ′ ( c c ′ i ) − k θ ′ ( c c ′ i ) | + | s − ˜ s |·| k θ ( c c i ) | + | ˜ s |·| k θ ( c c i ) − k ˜ θ ( c c i ) | . ρ / , given that d ( θ, ˜ θ ) ≤ c ρ / , d ( θ ′ , ˜ θ ′ ) ≤ c ρ / and | s − ˜ s |≤ c ρ / . All this allows us to concludethat we can use lemma 5.4. TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 37
Now we can estimate the measure of J ω ( a, a ′ ), but first we introduce some notation: given finitewords α , β such that α starts with β , the finite word α/β is defined by α = β ( α/β ). In the followinglines of equations we will be using: lemma 5.4 for the first three lines, observation in equation (11)for the fifth and sixth line, equation (9) in the seventh line, in the last line we use that ( a, a ′ ) is anexcellent pair and the fact that diam ( G ( c i /a )) is of order ρ − k . Leb (cid:0) J ω ( a, a ′ ) (cid:1) = Leb (cid:16)[ J ω ( c i , c ′ i ) (cid:17) & Leb (cid:16)[ J ( c i , c ′ i ) (cid:17) & Leb (cid:16)[ ˜ J ( c i , c ′ i ) (cid:17) = Leb (cid:16) π ˜ θ, ˜ θ ′ , ˜ s (cid:16)[ G ( c i ) × G ( c ′ i ) (cid:17)(cid:17) = Leb (cid:18) A ◦ π T a T ′ a ′ (˜ θ, ˜ θ ′ , ˜ s ) (cid:16)[ G ( c i /a ) × G ( c ′ i /a ′ ) (cid:17)(cid:19) ≈ diam ( G ( a )) · Leb (cid:18) π T a T ′ a ′ (˜ θ, ˜ θ ′ , ˜ s ) (cid:16)[ G ( c i /a ) × G ( c ′ i /a ′ ) (cid:17)(cid:19) & diam ( G ( a )) · µ d × µ d ′ (cid:16)[ G ( c i /a ) × G ( c ′ i /a ′ ) (cid:17) · (cid:13)(cid:13)(cid:13)(cid:13) χ T a T ′ a ′ (˜ θ, ˜ θ ′ , ˜ s ) (cid:13)(cid:13)(cid:13)(cid:13) − L & c ρ /k · (cid:16) ρ ( − k ) ( d + d ′ ) · ρ − ( d + d ′ )( − k ) (cid:17) ≈ c ρ /k We now consider the sets J ω ( b ij , b ′ ij ) = π ωθ,θ ′ ,s ( G ω ( b ij ) × G ( b ′ ij )) ,J ω ( c i , c ′ i ) = [ ( b ij ,b ′ ij ) J ω ( b ij , b ′ ij ) ,J ω ( a, a ′ ) = [ ( c i ,c ′ i ) J ω ( c i , c ′ i ) . We will estimate the measure of J ω ( a, a ′ ). Notice that J ω ( c i , c ′ i ) ⊂ J ω ( c i , c ′ i ) and if we are ableto prove that Leb (cid:0) J ω ( c i , c ′ i ) (cid:1) ≥ σ − · Leb (cid:0) J ω ( c i , c ′ i ) (cid:1) , for some constant σ , then we can use lemma5.4 with K α being J ω ( c i , c ′ i ) and J α being J ω ( c i , c ′ i ) to conclude that Leb (cid:0) J ω ( a, a ′ ) (cid:1) = Leb (cid:16)[ J ω ( c i , c ′ i ) (cid:17) & Leb (cid:16)[ J ω ( c i , c ′ i ) (cid:17) & c ρ /k . To prove that there is such σ we will proceed similarly to what we did when estimating Leb (cid:0) J ω ( a, a ′ ) (cid:1) .Note that T c i T ′ c ′ i ( θ, θ ′ , s ) = ( θ i , θ ′ i , s i ) ∈ ˜ L and then there exists (˜ θ i , ˜ θ ′ i , ˜ s i ) such that | s i − ˜ s i |≤ c ρ / , d (˜ θ i , θ i ) ≤ c ρ / , d (˜ θ ′ i , θ ′ i ) ≤ c ρ / and(12) (cid:13)(cid:13)(cid:13) χ ˜ θ i , ˜ θ ′ i , ˜ s i (cid:13)(cid:13)(cid:13) L ≤ c . The sets π ωθ i ,θ ′ i ,s i ( ω ) (cid:0) G ω ( d ij ) × G ( d ′ ij ) (cid:1) , π θ i ,θ ′ i ,s i (cid:0) G ( d ij ) × G ( d ′ ij ) (cid:1) and π ˜ θ i , ˜ θ ′ i , ˜ s i (cid:0) G ( d ij ) × G ( d ′ ij ) (cid:1) are all contained in, and contain, balls with radius of order ρ / , and centered at the points π ωθ i ,θ ′ i ,s i ( ω ) ( c ωd ij , c d ′ ij ) , π θ i ,θ ′ i ,s i ( c d ij , c d ′ ij ) and π ˜ θ i , ˜ θ ′ i , ˜ s i ( c d ij , c d ′ ij ) respectively. By lemma 3.2 we know that k k θ i ,ω − k θ i k C . c ρ − / k , | c ωd ij − c d ij | . c ρ / k , on the other hand we also have | s i − s i ( ω ) | . ρ / , k k θ i − k ˜ θ i k C . ρ / and k k θ ′ i − k ˜ θ ′ i k C . ρ / , thuswe can conclude that the distance between any two centers is of order less than ρ / . Therefore,we can apply lemma 5.4 taking J α as one of the families π ωθ i ,θ ′ i ,s i ( ω ) (cid:0) G ω ( d ij ) × G ( d ′ ij ) (cid:1) , π θ i ,θ ′ i ,s i (cid:0) G ( d ij ) × G ( d ′ ij ) (cid:1) , π ˜ θ i , ˜ θ ′ i , ˜ s i (cid:0) G ( d ij ) × G ( d ′ ij ) (cid:1) and J ′ α as other of these families.Using the previous analysis together with equations (11), (9), (12) and the fact that the numberof ( d ij , d ′ ij ) is a positive proportion of Σ( ρ / ) × Σ ′ ( ρ / ) we obtain Leb (cid:0) J ω ( c i , c ′ i ) (cid:1) = Leb (cid:16)[ J ω ( b ij , b ′ ij ) (cid:17) = Leb (cid:16) π ωθ,θ ′ ,s (cid:16)[ G ω ( b ij ) × G ( b ′ ij ) (cid:17)(cid:17) = Leb (cid:18) A ◦ π ωT ωci T ′ c ′ i ( θ,θ ′ ,s ) (cid:16)[ G ω ( d ij ) × G ( d ′ ij ) (cid:17)(cid:19) ≈ Leb (cid:0) J ( c i , c ′ i ) (cid:1) · Leb (cid:16)[ π ωθ i ,θ ′ i ,s i ( ω ) (cid:0) G ω ( d ij ) × G ( d ′ ij ) (cid:1)(cid:17) ≈ Leb (cid:0) J ( c i , c ′ i ) (cid:1) · Leb (cid:16)[ π θ i ,θ ′ i ,s i (cid:0) G ( d ij ) × G ( d ′ ij ) (cid:1)(cid:17) ≈ Leb (cid:0) J ( c i , c ′ i ) (cid:1) · Leb (cid:16)[ π ˜ θ i , ˜ θ ′ i , ˜ s i (cid:0) G ( d ij ) × G ( d ′ ij ) (cid:1)(cid:17) & Leb (cid:0) J ( c i , c ′ i ) (cid:1) · µ d × µ d ′ (cid:16)[ G ( d ij ) × G ( d ′ ij ) (cid:17) · (cid:13)(cid:13)(cid:13) χ ˜ θ i , ˜ θ ′ i , ˜ s i (cid:13)(cid:13)(cid:13) − L & Leb (cid:0) J ( c i , c ′ i ) (cid:1) · (cid:16) ρ ( d + d ′ ) · ρ − ( d + d ′ ) (cid:17) ≈ Leb (cid:0) J ( c i , c ′ i ) (cid:1) . This guarantees the existence of the desired constant σ and finishes the proof of the lemma. (cid:3) Now we can prove proposition 4.2. Consider the function ϕ = X ( a,a ′ ) J ω ( a,a ′ ) , where 1 B means indicator function of the set B and the sum is over all excellent pairs. We wantto estimate the measure of the set X = { t ∈ C : ϕ ( t ) ≥ c ′′ c ρ − k ( d + d ′ − } , where c ′′ is a constant defined in the following way. Suppose that we have two excellent pairs withthe same first coordinate ( a, a ′ ), ( a, ˜ a ′ ) and such that J ω ( a, a ′ ) ∩ J ω ( a, ˜ a ′ ) = ∅ . Then k θ ′ ( y ) − sk θ,ω ( x ) = k θ ′ (˜ y ) − sk θ,ω (˜ x ) , for some ( x, y ) ∈ G ω ( a ) × G ( a ′ ), (˜ x, ˜ y ) ∈ G ω ( a ) × G (˜ a ′ ). Thus | y − ˜ y |≈ | x − ˜ x | . ρ / k , which shows that d ( G ( a ′ ) , G (˜ a ′ )) . ρ / k . This implies that if we fix ( a, a ′ ), then the number of possible pairs ( a, ˜ a ′ ) such that J ω ( a, a ′ ) ∩ J ω ( a, ˜ a ′ ) = ∅ is bounded by a uniform constant, independent of ρ and ( a, ˜ a ′ ), we denote thisconstant by c ′′ (this last statement is a consequence of lemma 1.2.3 in [8]). TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 39
Notice that since s ∈ J R then ϕ is supported in a ball of radius proportional to 1 + e R centeredat 0, thus there is a constant c such that Z ϕ dt ≤ (sup ϕ ) · Leb ( X ) + cc ′′ c ρ − k ( d + d ′ − . Now we estimate sup ϕ from above and R ϕ dt from below. By lemmas 5.5 and 5.3 there is aconstant c ′ such that(13) Z ϕ dt ≥ c ′ c ρ /k · ρ − d + d ′ k = c ′ c ρ − k ( d + d ′ − . Let x ∈ C and excellent pairs ( a, a ′ ), (˜ a, ˜ a ′ ) such that x ∈ J ω ( a, a ′ ) ∩ J ω (˜ a, ˜ a ′ ). Remember that | π ωθ,θ ′ ,s ( c ωa , c a ′ ) − π θ,θ ′ ,s ( c a , c a ′ ) | . c ρ − / k = o ( ρ / k ) , where the o notation means that, once we have chosen c , we can choose any ǫ > c ρ − / k ≤ ǫρ / k will hold provided ρ is small enough. With this in mind we obtain | π θ,θ ′ ,s ( c a , c a ′ ) − π θ,θ ′ ,s ( c ˜ a , c ˜ a ′ ) | ≤ o ( ρ / k ) + | π ωθ,θ ′ ,s ( c ωa , c a ′ ) − x | + | x − π ωθ,θ ′ ,s ( c ω ˜ a , c ˜ a ′ ) |≤ o ( ρ / k ) + 2 c ρ / k . Given that ( a, a ′ ), (˜ a, ˜ a ′ ) are excellent, we conclude that there can be no more than c − ρ − k ( d + d ′ − excellent pairs intersecting at x . Since ϕ ≤ P ( a,a ′ ) J ω ( a,a ′ ) we get that ϕ ( x ) ≤ c − ρ − k ( d + d ′ − . We are ready to bound
Leb ( X ), using equation (13) and the previous estimates c − ρ − k ( d + d ′ − Leb ( X ) + cc ′′ c ρ − k ( d + d ′ − ≥ c ′ c ρ − k ( d + d ′ − , and from this we get Leb ( X ) ≥ c ( c ′ − cc ′′ c ) . We fix c small enough such that c := c ( c ′ − cc ′′ c ) > { ( θ, θ ′ , s, t ) : t ∈ X } ⊂ L − ω ( θ, θ ′ , s ). Let t ∈ X , then there are at least h c ′′ c ρ − k ( d + d ′ − i excellent pairs ( a, a ′ ), each one with an associatedpair ( b ij , b ′ ij ) which starts with ( a, a ′ ) and such that t ∈ J ω ( b ij , b ′ ij ). By the definition of c ′′ , wecan extract from this family of excellent pairs a subfamily ( a l , a ′ l ), l = 1 , ..., h c ρ − k ( d + d ′ − i ,such that all firsts coordinates are different. For ( a l , a ′ l ) we denote the associated pair ( b ij , b ′ ij ) by( b l , b ′ l ). We will prove the pairs ( b , b ′ ) , ..., ( b N , b ′ N ) have the properties necessary to guaranty that( θ, θ ′ , s, t ) ∈ L − ω ( θ, θ ′ , s ). Write T ωb l T ′ b l ( θ, θ ′ , s ) = ( θ l , θ ′ l , s l , t l )then:(i) Since all firsts coordinates of the excellent pairs are different we conclude that b , ..., b N arepairwise independent.(ii) By the way in which d ij was defined we get that all θ l ∈ Σ − nr .(iii)’ By the scale recurrence lemma we know that B ( s ij , c ρ / ) ⊂ E ∗ ( d ij , d ′ ij ). We also knowthat | s ij − s ij ( ω ) | . c ρ − / k = o ( ρ / ), then, if ρ is small enough, we have { ˜ s : | ˜ s − s ij ( ω ) |≤ c ρ / } ⊂ E ∗ ( d ij , d ′ ij ) . We conclude that ( θ l , θ ′ l , ˜ s ) ∈ ˜ L if | ˜ s − s l |≤ c ρ / (remember that for every l there is i, j such that s l = s ij ( ω )). (iv)’ Given that t ∈ J ω ( b l , b ′ l ), there is ( x, y ) ∈ C such that t = k θ ′ ( f ωb ′ l ( y )) − sk θ,ω ( f ωb l ( x )) F θ ′ b ′ l ( k θ ′ b ′ l ( y )) = t + s · F θ,ωb l ( k θb l ,ω ( x )) k θ ′ b ′ l ( y ) = (cid:16) F θ ′ b ′ l (cid:17) − (cid:16) t + s · F θ,ωb l ( k θb l ,ω ( x )) (cid:17) k θ ′ b ′ l ( y ) = t l + s l · k θb l ,ω ( x ) . Since s l ∈ J R we conclude that | t l |≤ e R .6. Proof of Proposition 4.1
In this section we will prove proposition 4.1. Given u = ( θ, θ ′ , s, t ) ∈ ˜ L × C , remember thedecomposition ω = ( ω ′ , ω ′′ ) where ω ′ ∈ D Σ ( θ ) and ω ′′ ∈ D Σ \ Σ ( θ ) . Recall that the set Σ ( θ ) isgiven by the words in Σ starting with the same word, in Σ( ρ / k ), in which θ finishes. In the sameway as in [1], one uses Fubini’s theorem to reduce the proof of proposition 4.1 to proving(14) P ′ (Ω ′ − Ω ′ ( u )) ≤ exp( − c ρ − k ( d + d ′ − ) , where we have fixed ω ′′ such that u ∈ L ,ω ′′ ) and Ω ′ = D Σ ( θ ) , Ω ′ ( u ) = { ω ′ : ( ω ′ , ω ′′ ) ∈ Ω ( u ) } , P ′ is normalized Lebesgue measure in Ω ′ . u = ( θ, θ ′ , s, t ) ∈ L ,ω ′′ ) means that ( θ, θ ′ , s ) ∈ ˜ L and there is (˜ θ, ˜ θ ′ , ˜ s, ˜ t ) for which d ( θ, ˜ θ ) < ρ / , d ( θ ′ , ˜ θ ′ ) < ρ / , | s − ˜ s | < ρ, | t − ˜ t | < ρ, (˜ θ, ˜ θ ′ , ˜ s ) ∈ ˜ L and ˜ t ∈ L ,ω ′′ ) (˜ θ, ˜ θ ′ , ˜ s ). Notice that Σ ( θ ) = Σ (˜ θ ), moreover u ∈ L ,ω ′′ ) if and onlyif u ∈ L ω ′ ,ω ′′ ) for any ω ′ ∈ Σ ( θ ).Next, ˜ t ∈ L ,ω ′′ ) (˜ θ, ˜ θ ′ , ˜ s ) means that there are pairs ( b , b ′ ) , ..., ( b N , b ′ N ) in Σ( ρ ) × Σ ′ ( ρ ) suchthat if we set T (0 ,ω ′′ ) b i T ′ b ′ i (˜ θ, ˜ θ ′ , ˜ s, ˜ t ) = (˜ θ i , ˜ θ ′ i , ˜ s i , ˜ t i )then:(i) the words b , ..., b N are pairwise independent;(ii) ˜ θ i ∈ Σ − nr ;(iii) (˜ θ i , ˜ θ ′ i , ˜ s ′ i ) ∈ ˜ L if | ˜ s i − ˜ s ′ i | < c ρ / ;(iv) | ˜ t i |≤ e R ).Let a be the word in Σ( ρ / k ) in which θ ends, for each b i define a i in Σ ( θ ) given by theconcatenation of a and a word at the beginning of b i , in such a way that a i ∈ Σ ( θ ). Notice thatthe independence of the words b i imply that the words a i are all different.Now we consider the decomposition of ω ′ ∈ Ω ′ as ω ′ = ( ω , ..., ω N , ˜ ω ′ ), where ˜ ω ′ ∈ D Σ ( θ ) \{ a ,...,a N } and ω i is the component of ω ′ corresponding to a i . We use again Fubini’s theorem to reduce theproof of equation (14) to a similar formula in a smaller space. For ˜ ω ′ fixed, we will prove that theset of ( ω , .., ω N ) such that ω ′ / ∈ Ω ′ ( u ) has measure ≤ exp( − c ρ − k ( d + d ′ − ).To prove the desired inequality, we will prove that for each ω i there is a set with positive measuresuch that whenever ω i is in this set we have ω ′ ∈ Ω ′ ( u ) (no matter the value of ω j , j = i ). Moreprecisely, we will prove that if ω i is in this set then b i , b ′ i verify that if we set T ωb i T ′ b ′ i ( u ) = ( θ i , θ ′ i , s i ( ω ) , t i ( ω ))then:(i) ( θ i , θ ′ i , s ′ i ) ∈ ˜ L if | s ′ i − s i ( ω ) | < c ρ / ; TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 41 (ii) t i ( ω ) ∈ L ω ( θ i , θ ′ i , s i ( ω )).The first property can be easily obtained. In fact, we already know (from (iii) above) that(˜ θ i , ˜ θ ′ i , ˜ s ′ i ) ∈ ˜ L if | ˜ s i − ˜ s ′ i | < c ρ / . Notice that since d ( θ i , ˜ θ i ) . ρ / , d ( θ ′ i , ˜ θ ′ i ) . ρ / then˜ θ i ∈ Σ − nr and the fiber of ˜ L over ( θ i , θ ′ i ) is the same as the one over (˜ θ i , ˜ θ ′ i ), thus we only need toestimate | ˜ s i − s i ( ω ) | . Using that | s − ˜ s | < ρ , d ( θ, ˜ θ ) < ρ / , d ( θ ′ , ˜ θ ′ ) < ρ / and lemma 3.2 onegets that | ˜ s i − s i ( ω ) | = o ( ρ / ), then choosing ρ sufficiently small gives the desired property (for anyvalue of ω i ).For the second property we choose θ ∈ Σ − such that d ( θ, θ ) < ρ / , then θ ends with a , we alsoassume that it does not contain a anywhere else (this is possible since θ ∈ Σ − nr ). Set T ωb i T ′ b ′ i ( θ, θ ′ , s, t ) = ( θ i , θ ′ i , s i ( ω ) , t i ( ω )) . We will prove the following lemmas:
Lemma 6.1.
Once ω ′′ has been fixed, the number t i ( ω ) only depends on ω i , not in ˜ ω ′ or ω j for j = i . Moreover, if c is big enough then there is a constant c ′ > such that Leb (cid:16) { ω i : t i ( ω ) ∈ L − ω ( θ i , θ ′ i , s i (ˆ ω )) } (cid:17) ≥ c ′ , where ˆ ω = (0 , ˜ ω ′ , ω ′′ ) . Lemma 6.2. If t i ( ω i ) ∈ L − ω ( θ i , θ ′ i , s i (ˆ ω )) then t i ( ω ) ∈ L ω ( θ i , θ ′ i , s i ( ω )) . These two lemmas imply that
Leb ( { ( ω , ..., ω N ) : ( ω , ..., ω N , ˜ ω ′ , ω ′′ ) / ∈ Ω ( u ) } ) ≤ (1 − c ′ ) N = e N log(1 − c ′ ) , this finishes the proof of proposition 4.1. Proof of lemma 6.1:
From equation (2) we know that t i ( ω ) = (cid:16) DF θ ′ b ′ i (cid:17) − · ( t + sc θ,ωb i − c θ ′ b ′ i ) . The dependency on ω is on the term c θ,ωb i . Let ˆ θ such that θ = ˆ θa , note that ˆ θ does not contain theword a , let b be the last letter of b i , we have(15) c θ,ωb i = k θ,ω ( c ωb i ) = k ˆ θa,ω ( f ωb i ( c ωb )) = (cid:16) F ˆ θ,ωa (cid:17) − ◦ k ˆ θ,ω ( f ωab i ( c ωb )) . We will prove that, assuming ω ′′ is fixed, this expression only depends on ω i and not in ω j for j = i or ˜ ω ′ , and it depends in a very specific way. Remember that the reference points c ωa were chosen tobe pre-periodic points and that in fact they do not depend on ω , i.e. c ωa = c a . Notice that θ i = θb i and d ( θ i , θ i ) . ρ / , then θ i ∈ Σ − nr and a appears only once in θ i . We will study the dependencyon ω i for the different terms in equation (15): • Let α be a finite word at the end of θ i strictly shorter than ab i . It is easy to prove byinduction that f ωα ( c ωb ) does not depend on ω ′ . Indeed, suppose that α and β = ( c, d ) α aretwo such consecutive words. Assume f ωα ( c ωb ) does not depend on ω ′ , we have f ωβ ( c ωb ) = f ω ( c,d ) (cid:0) f ωα ( c ωb ) (cid:1) . If the word β is shorter than some word in Σ( ρ / k ), then f ωα ( c ωb ) ∈ G ω ( γ ) ⊂ V ρ ( G ( γ ))such that ( c, d ) γ ∈ Σ( ρ /k ) and γ contains a word in Σ( ρ / k ) repeated (remember that c ωb is pre-periodic, with a uniform bounded period). Thus ( c, d ) γ can not belong to Σ and f ωβ ( c ωb ) does not depend on ω ′ (in fact, in this case it does not depend on ω ). If the word β is longer than any word in Σ( ρ / k ) then it begins with a word in Σ( ρ / k )which can not be a . This implies that f ωα ( c ωb ) ∈ G ω ( γ ) ⊂ V ρ ( G ( γ )) such that ( c, d ) γ ∈ Σ( ρ /k ), and the word ( c, d ) γ is not in Σ ( θ ). Thus f ωβ ( c ωb ) does not depend on ω ′ . • Let ( a , a ) be the first two letters of a and define α by ab i = ( a , a ) α . Define x = f a ,a ( f ωα ( c ωb )), we already proved that x does not depend on ω ′ . We have f ωab i ( c ωb ) = f ωa ,a ( f ωα ( c ωb )) = x + c ρ / k ω i . • We now study k ˆ θ,ω . By definition k ˆ θ,ω ( z ) = lim n →∞ Df ω ˆ θ n ( c ˆ θ ) − ( f ω ˆ θ n ( z ) − f ω ˆ θ n ( c ˆ θ )) . Using the same arguments as before we see that, since ˆ θ does not contain a and c ˆ θ is pre-periodic, f ω ˆ θ n ( c ˆ θ ) does not depend on ω ′ . This also happens for z in a neighborhood of c ˆ θ ,then Df ω ˆ θ n ( c ˆ θ ) is independent of ω ′ . Again, the same arguments prove that if z ∈ G ω ( a )then f ω ˆ θ n ( z ) = f (0 ,ω ′′ )ˆ θ n ( z ). We conclude that k ˆ θ,ω ( z ) = k ˆ θ, (0 ,ω ′′ ) ( z ) , and Dk ˆ θ,ω ( z ) = Dk ˆ θ, (0 ,ω ′′ ) ( z ) , for all z ∈ G ω ( a ). • We now treat F ˆ θ,ωa . One has that F ˆ θ,ωa (0) = c ˆ θ,ωa = k ˆ θ,ω ( c ωa ) , DF ˆ θ,ωa = Dk ˆ θ,ω ( c ωa ) · Df ωa ( c ωa ) , where a is the last letter in a . Given that a ∈ Σ( ρ / k ) and c ωa = c a is pre-periodic weconclude that c ωa = f ωa ( c ωa ) and Df ωa ( c ωa ) are independent of ω . On the other hand, giventhat c ωa ∈ G ω ( a ) and ˆ θ does not contain a we obtain that k ˆ θ,ω ( c ωa ) does not depend on ω ′ .We conclude that F ˆ θ,ωa is independent of ω ′ .From the previous analysis we get that t i ( ω ) only depends on ω i and not in ω j for j = i or ω ′ .Moreover, we have c θ,ωb i = (cid:16) F ˆ θ, (0 ,ω ′′ ) a (cid:17) − ◦ k ˆ θ, (0 ,ω ′′ ) ( x + c ρ / k ω i ) , taking derivative respect to ω i we get D ω i c θ,ωb i = (cid:16) DF ˆ θ, (0 ,ω ′′ ) a (cid:17) − · Dk ˆ θ, (0 ,ω ′′ ) ( x + c ρ / k ω i ) · c ρ / k . Observe that since x + c ρ / k ω i = f ωab i ( c ωb ) belongs to the Cantor set K ω then the matrix Dk ˆ θ, (0 ,ω ′′ ) ( x + c ρ / k ω i ) is conformal. Moreover D ω i t i ( ω ) = (cid:16) DF θ ′ b ′ i (cid:17) − · sD ω i c θ,ωb i , therefore ω i → t i ( ω ) defines a holomorphic function, which we will denote by t i ( ω i ). From theprevious formulas it is not difficult to see that k D ω i t i ( ω i ) k≈ c .Now we show that t i (0) = t i (0 , ω ′′ ) is uniformly bounded. We already know that | ˜ t i |≤ e R ),using this inequality together with d ( θ, ˜ θ ) < ρ / , d ( θ ′ , ˜ θ ′ ) < ρ / , | t − ˜ t | < ρ , | s − ˜ s | < ρ andequation (2) one gets that | t i (0) | .
1. Therefore, choosing c big enough one guarantees that theimage of t i ( ω i ) contains L − ω ( θ i , θ ′ i , s i (ˆ ω )) (see lemma 6.3). Having chosen c this way, the fact that k D ω i t i ( ω i ) k≈ c and proposition 4.2 gives that there is a constant c ′ such that Leb (cid:16) { ω i : t i ( ω ) ∈ L − ω ( θ i , θ ′ i , s i (ˆ ω )) } (cid:17) ≥ c ′ . TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 43 (cid:3)
In the previous proof we used the following lemma, it is proven using standard arguments inanalysis, for completeness we present the proof.
Lemma 6.3.
Let f : B (0 , ⊂ R n → R n be C on B (0 , and such that m ( Df ( x )) > c for all x ∈ B (0 , . Given r , if c is big enough, depending on r , then B (0 , r ) ⊂ f ( B (0 , .Proof. Redefining f as f − f (0) and taking r as r + f (0), one can assume f (0) = 0. Since f is C and m ( Df ( x )) > c > f is a C local diffeomorphism, hence its image f ( B (0 , h be a point in R n \ f ( B (0 , / λh ∈ f ( B (0 , / ≤ λ <
1. We can cover B (0 , /
2) by a finite number of open sets such that in each one of thesesets f is a diffeomorphism onto its image. We can use this cover to lift curves in f ( B (0 , / f | B (0 , / : B (0 , / → f ( B (0 , / α : [0 , → f ( B (0 , / α ( t ) = th . Denote by β a lifting of α , i.e. f ◦ β = α , such that β (0) = 0. The curve β is C and β (1) / ∈ B (0 , / h .Therefore | h | = length ( α ) = Z (cid:12)(cid:12)(cid:12)(cid:12) ddt ( f ◦ β ) (cid:12)(cid:12)(cid:12)(cid:12) dt ≥ Z m ( Df ( β ( t ))) (cid:12)(cid:12)(cid:12)(cid:12) ddt β (cid:12)(cid:12)(cid:12)(cid:12) dt ≥ c Z (cid:12)(cid:12)(cid:12)(cid:12) ddt β (cid:12)(cid:12)(cid:12)(cid:12) dt ≥ c / . If c > r then | h | > r and the desired result follows. (cid:3) Proof of lemma 6.2:
Let t i ( ω i ) ∈ L − ω ( θ i , θ ′ i , s i (ˆ ω )), then there exists pairs ( d i , d ′ i ), 1 ≤ i ≤ N , suchthat if we write T ˆ ω ( i ) d j T ′ d ′ j ( θ i , θ ′ i , s i (ˆ ω ) , t i ( ω i )) = ( θ i d j , θ ′ i d ′ j , ˆ s ( j ) , ˆ t ( j ) ) , where ˆ ω ( i ) is obtained from ˆ ω by setting the value 0 in the coordinates belonging to Σ ( θ i ), then(i) d ,..., d N are pairwise disjoint.(ii) θ i d j ∈ Σ − nr .(iii)’ | s ∗ − ˆ s ( j ) | < c ρ / implies ( θ i d j , θ ′ i d ′ j , s ∗ ) ∈ ˜ L .(iv)’ | ˆ t ( j ) | < e R .We will prove that t i ( ω ) ∈ L ω ( θ i , θ ′ i , s i ( ω )). To do this, we will prove that if write T ω ( i ) d j T ′ d ′ j ( θ i , θ ′ i , s i ( ω ) , t i ( ω )) = ( θ i d j , θ ′ i d ′ j , s ( j ) , t ( j ) ) , where ω ( i ) is obtained from ω by setting the value 0 in the coordinates belonging to Σ ( θ i ), then(iii) | s ∗ − s ( j ) | < c ρ / implies ( θ i d j , θ ′ i d ′ j , s ∗ ) ∈ ˜ L .(iv) | t ( j ) | < e R ).First, notice that by lemma 3.2 we have that | s i ( ω ) − s i (ˆ ω ) | . ρ − / k . To obtain s ( j ) and ˆ s ( j ) weapplied the same renormalizations, with the same limit geometries but with different values of theperturbation parameter, therefore | s ( j ) − ˆ s ( j ) | . ρ − / k = o ( ρ / ). We conclude that, taking ρ smallenough, item (iii)’ implies item (iii).Now, we will prove that (iv)’ implies (iv). We haveˆ t ( j ) = (cid:16) DF θ ′ i d ′ j (cid:17) − · (ˆ t i ( ω i ) + s i (ˆ ω ) c θ i , ˆ ω ( i ) d j − c θ ′ i d ′ j ) ,t ( j ) = (cid:16) DF θ ′ i d ′ j (cid:17) − · ( t i ( ω ) + s i ( ω ) c θ i ,ω ( i ) d j − c θ ′ i d ′ j ) . We will compare the corresponding terms: • Using that d ( θ, θ ) < ρ / , one has | t i ( ω i ) − t i ( ω ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) DF θ ′ b ′ i (cid:17) − · s h c θ,ωb i − c θ,ωb i i(cid:12)(cid:12)(cid:12)(cid:12) . ρ / . • Notice that ω ( i ) and ˆ ω ( i ) only differ at their values in the coordinates ( ω , ..., ω N ). Usingthat θ i d j ∈ Σ − nr one sees that a is not contained in d j . Then, from the arguments used inlemma 6.1, we see that c ω ( i ) d j = c ˆ ω ( i ) d j . Moreover, since θ i ends in ab i and θ i d j ∈ Σ − nr then f ω ( i ) θ in ( c ω ( i ) d j ) = f ˆ ω ( i ) θ in ( c ˆ ω ( i ) d j )for all n such that θ in is strictly shorter that ab i . For the same reasons we also have f ω ( i ) θ in ( c θ ) = f ˆ ω ( i ) θ in ( c θ )for all n such that θ in is strictly shorter than ab i . Both equalities still hold for z in aneighborhood of either of the points. Thus we can use remark 3.3 and lemma 3.2 to obtainthat | c θ i ,ω ( i ) d j − c θ i , ˆ ω ( i ) d j | = | k θ i ,ω ( i ) ( c ω ( i ) d j ) − k θ i , ˆ ω ( i ) ( c ˆ ω ( i ) d j ) | . diam ( G ω ( i ) ( ab i )) . diam ( G ω ( i ) ( a )) diam ( G ω ( i ) ( b i )) . diam ( G ( a )) diam ( G ( b i )) ≈ ρ / k . • Notice that ω and ˆ ω only differ in the values associated to the coordinates ( ω , ..., ω N ). Wewill use again θ . Consider T ωb i T ′ b ′ i ( θ, θ ′ , s ) = ( θb i , θ ′ b ′ i , s i ( ω ))and T ˆ ωb i T ′ b ′ i ( θ, θ ′ , s ) = ( θb i , θ ′ b ′ i , s i (ˆ ω )) . Since d ( θ, θ ) < ρ / one gets | s i ( ω ) − s i ( ω ) | . ρ / and | s i (ˆ ω ) − s i (ˆ ω ) | . ρ / . Thus we onlyneed to estimate | s i ( ω ) − s i (ˆ ω ) | . Remember ˆ θ which verified θ = ˆ θa , write k θ,ω = ( F ˆ θ,ωa ) − ◦ k ˆ θ,ω ◦ f ωa , then Dk θ,ω = ( DF ˆ θ,ωa ) − · ( Dk ˆ θ,ω ◦ f ωa ) · Df ωa . Using the analysis and notation from lemma 6.1 we see that: F ˆ θ,ωa does not depend on ω ′ , Dk ˆ θ,ω ( z ) = Dk ˆ θ, ˆ ω ( z ) for all z ∈ G ω ( a ) ∪ G ˆ ω ( a ) (in particular c ωab i and c ˆ ωab i ), we also have c ˆ ωab i = x , c ωab i = x + c ρ / k ω i and Df ωa ( c ωb i ) = Df ˆ ωa ( c ˆ ωb i ). Therefore | Dk θ,ω ( c ωb i ) − Dk θ, ˆ ω ( c ˆ ωb i ) | = | Df ωa ( c ωb i ) || DF ˆ θ,ωa | · | Dk ˆ θ,ω ( x + c ρ / k ω i ) − Dk ˆ θ,ω ( x ) | . ρ / k , Here we used that limit geometries are C and the norm of D k ˆ θ,ω can be uniformlybounded. Using that, in general we have DF θ,ωd = Dk θ,ω ( c ωd ) · Df ωd ( c d ), where d is thelast letter in d , one can conclude that | DF θ,ωb i − DF θ, ˆ ωb i | = | Df ωb i ( c b i ) |·| Dk θ,ω ( c ωb i ) − Dk θ, ˆ ω ( c ˆ ωb i ) | . ρ / k . TABLE INTERSECTIONS OF CONFORMAL CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS 45
And from this (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) DF θ,ωb i DF θ ′ b ′ i − DF θ, ˆ ωb i DF θ ′ b ′ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ρ / k , therefore | s i ( ω ) − s i (ˆ ω ) | . ρ / k and | s i ( ω ) − s i (ˆ ω ) | . ρ / k .From the previous estimates we conclude that | ˆ t ( j ) − t ( j ) | . ρ / k , then if ρ is small enough (iv)’implies (iv). (cid:3) References [1] C. Moreira and J.-C. Yoccoz. Stable intersections of regular cantor sets with large hausdorff dimensions.
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