On the arithmetic difference of middle Cantor sets

Suppose that $\mathcal{C}$ is the space of all middle Cantor sets. We characterize all triples $(\alpha,~\beta,~\lambda)\in \mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$ that satisfy $C_\alpha- \lambda C_\beta=[-\lambda,~1]. $ Also all triples (that are dense in $\mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$) has been determined such that $C_\alpha- \lambda C_\beta$ forms the attractor of an iterated function system. Then we found a new family of the pair of middle Cantor sets $\mathcal{P}$ in a way that for each $(C_\alpha,~ C_\beta)\in\mathcal{P}$, there exists a dense subfield $F\subset \mathbb{R}$ such that for each $\mu \in F$, the set $C_\alpha- \mu C_\beta$ contains an interval or has zero Lebesgue measure. In sequel, conditions on the functions $f, ~g$ and the pair $(C_\alpha,~C_\beta)$ is provided which $f(C_{\alpha})- g(C_{\beta})$ contains an interval. This leads us to denote another type of stability in the intersection of two Cantor sets. We prove the existence of this stability for regular Cantor sets that have stable intersection and its absence for those which the sum of their Hausdorff dimension is less than one. At the end, special middle Cantor sets $C_\alpha$ and $C_\beta$ are introduced. Then the iterated function system corresponding to the attractor $C_{\alpha}-\frac{2\alpha}{\beta}C_\beta$ is characterized. Some specifications of the attractor has been presented that keep our example as an exception. We also show that $\sqrt{C_{\alpha}}$ - $\sqrt{C_{\beta}}$ contains at least one interval.


Introduction
Regular Cantor sets play a fundamental role in dynamical systems and number theory. Intersections of hyperbolic sets with stable and unstable manifolds of its points are often regular Cantor sets. Also, related to diophantine approximations, many Cantor sets given by combinatorial conditions on the continued fraction of real numbers are regular. In studying the homoclinic bifurcations in dynamical systems, also the classical Markov and Lagrange spectra related to diphantine approximations in number theory, we deal with arithmetic difference of regular Cantor sets [2,3,12]. Many papers have been written about metrical and topological properties of sum or difference of regular Cantor sets [6,13,15]. Before stating 1 arXiv:1306.5880v5 [math.DS] 22 Apr 2014 the results of this paper, we establish some notations.
A Cantor set K is regular or dynamically defined if: i) there are disjoint compact intervals K 1 , K 2 , · · · , K r such that K ⊆ K 1 ∪ K 2 ∪ · · · ∪ K r and the boundary of each K i is contained in K, ii) there is a C 1+ expanding map ψ defined in a neighborhood of the set K 1 ∪ K 2 ∪ · · · ∪ K r such that ψ(K i ) is the convex hull of a finite union of some intervals K j satisfying: ii.1 For each 1 ≤ i ≤ r and n sufficiently large, ψ n (K ∩ K i ) = K, The set {K 1 , K 2 , · · · , K r } is, by definition, a Markov partition of K, and the set D := r i=1 K i is the corresponding Markov domain of K.
The Cantor set K is close on the topology C 1+ to a Cantor set K with the Markov partition { K 1 , K 2 , · · · , K r } defined by expanding map ψ if and only if the extremes of each K i are near the corresponding extremes of K i and supposing ψ ∈ C 1+ with Holder constant C, we must have ψ ∈ C 1+ with Holder constant C such that ( C, ) is near (C, ) and ψ is close to ψ in the C 1 topology. Regular Cantor sets K and K have stable intersection if for any pair of regular Cantor sets ( K, K ) near (K, K ), The concept of stable intersection has been introduced by Moreira in [6] for the first time. The Cantor sets K and K that have stable intersection are useful in two points of view. First, in dynamical systems theory, when stable and unstable Cantor sets associated to a homoclinic bifurcation have a stable intersection, they present open sets in the parameter line with positive density at the initial bifurcating value, for which the corresponding diffeomorphisms are not hyperbolic. Second, in number theory, they guarantee the existence of an open set U including (K, K ) such that for each ( K, K ) ∈ U, the set K − K contains an interval.
On the other hand, topological and metrical structure of the K − λK plays a key role in investigation of having stable intersection of regular Cantor sets K and K [7]. Hence, we can concentrate on the arithmetic difference of regular Cantor sets of this form. Herein, there are several classical results: I) If τ (K) · τ (K ) > 1, then K − λK contains an interval [12], II) If HD(K) + HD(K ) > 1, then K − λK generically contains an interval [7], III) If HD(K) + HD(K ) > 1, then |K − λK | > 0 for almost every λ ∈ R * [12], IV) There exist regular Cantor sets K and K such that K −K has positive Lebesgue measure, but does not contain any interval [14], A regular Cantor set K is affine if Dψ is constant on every interval K i . Meanwhile, the following conjecture due to Palis is still open: Conjecture. The arithmetic difference of two affine Cantor sets generically, if not always, contains an interval or has zero Lebesgue measure.
Many studies have been done on this conjecture [6,7,8]. This conjecture can be written for middle Cantor sets too; Cantor set C α is a middle-(1 − 2α) or in simple words, middle Cantor set, if the convex hull of C α is [0, 1] and the Markov partition of C α has exactly two members with Dψ = 1 α on their intervals. Regard to above discussion, the morphology of the arithmetic difference C α − λC β on the mysterious region is unclear. This is our motivation in writing the present paper that is organized as follows: In Section 2, we will prove a theorem that introduces the iterated function systems with their attractors C α − λC β , where (C α , C β ) ∈ L. We obtain this theorem by the transferred renormalization operators corresponding to a pair (C α , C β ) on the space R * × R that explained in [4]. A reason of proposing the theorem is that, tracing and controlling points in R under suitable compositions of functions which constitute the iterated function system are easier than tracing and controlling points in R * × R under suitable compositions of transferred operators. Although the number of functions which constitute the iterated function system could be so many, the methods and techniques of the theory of iterated function systems could be profitable. Moreira and Yoccoz in [7] introduced that, a way of having stable intersection of Cantor sets C α and C β is to construct a recurrent compact set of relative configurations corresponding to the renormalization operators. To deal with Palis conjecture, these facts may be a step forward. In this direction, we have found an element of Ω that have stable intersection [10], (see Proposition 4.3 too).
The other applications of the theorem will occur throughout the paper.
In Section 3, the first aim is to establish conditions on the function f, g and the pair (C α , C β ) ∈ L to ensure the existence of an interval in the set f (C α ) − g(C β ). For instance, it will be applied to guarantee the existence of an interval in the sets C 2 + C 2 and sin C + cos C, where C is the middle- 1 3 Cantor set. This allows us to introduce the concept of " weak stable intersection" that could have a pair of arbitrary Cantor sets embedded in the real line; the pair (K, K ) has weak (or geometric) C r -stable intersection, if for all f and g in a C r -neighborhood of the identity, we have f (K) g(K ) = ∅. Note that, the diffeomorphisms In Section 4, we introduce λ ∈ R * and (C α , C β ) ∈ Ω such that C α − λC β is not an affine Cantor set and |C α − λC β | = 0. Regarding to Theorem 2.1, we observe that the set C α − λC β is the attractor of the iterated function system namely S: . It is not clear at this point that the Hausdorff dimension of the attractor is smaller than one since we have twenty one different affine maps with slopes p 2 = 17, 94... < 21. Nevertheless, we calculate its exact value in a different manner and then we present some other results about these Cantor sets. In the context of regular Cantor sets K and K that K − K is not an affine Cantor set and HD(K − K ) < 1 < HD(K) + HD(K ), there exist some properties that prove our example as an exceptional one among others: • The set C α − λC β forms the attractor of an iterated function system that is of finite type.
• The similarity dimension of S is bigger than one and also it is not obvious to determine HD(C α − λC β ) < 1 on the lower steps of the construction C α × C β . In fact, if we do twelve steps in the construction of C α and eighteen steps in the construction of C β , then we can select an iterated function system on I × I ⊂ R 2 of Hausdorff dimension smaller than one, such that C α − λC β becomes the projection of its attractor under angle cot −1 λ. While, this method does not apply for the lower steps of the construction C α × C β , (see Corollary 2.1 and the Remark 2 of [15]). Another purpose in presenting this method is to find an upper bound of Hausdorff dimension and it may be useful for the situations that the iterated function system is not of finite type.
• There exists a dense subgroup G ⊂ R such that for each g ∈ G, the set C α − gC β contains an interval or has zero Lebesgue measure.
• We can not put them in a non constant continuous curve from the pair of regular Cantor sets that Hausdorff dimension of their arithmetic difference is less than one.

Iterated function system
Let us start with introducing the set L. An element (C α , C β ) belongs to L if and only if log α log β ∈ Q. Obviously, L is a dense subset in the space C × C. Let p, q > 2, it has been shown in [4] that the transferred renormalization operators corresponding to pair (C α , C β ) : If log p log q =: n 0 m 0 ∈ Q with (m 0 , n 0 ) = 1, then every vertical line s = λ =: cot θ passes over itself with suitable compositions of the operators (1). Hence, we can transfer the operators (1) on these lines by be two finite sequences of numbers 0 and 1. Then the maps are return maps to the vertical line s = λ and the attractor of iterated function system Proof. Suppose that {b k } ∞ k=0 and {a k } ∞ k=0 are two arbitrary sequences of numbers 0 and 1. For every a k and b k , we can write the operators (1) in form Let m, n ∈ N, then we claim that To prove the claim, we use induction. The case m = n = 1 is true. Assume that formulas are valid for the cases m = i and n = j, then we have Put s = λ, m = m 0 and n = n 0 in the relations (i) and (ii). Then we obtain the maps (2), since give all of the return maps, since the operators T a i and T b j commutate together. Moreover, the maps (2) establish an iterated function system with p m 0 whose attractor is C α − λC β , see page 51 of [7]. This completes the proof. 2 For a given λ ∈ R * , rename the maps (2) to T i λ with 1 ≤ i ≤ 2 m 0 +n 0 , (sometimes they are less than this number) and let S i the iterated function systems corresponding to the pair (C α , C β ) ∈ L, (or the attractors C α − λC β ). Indeed, the set C α − λC β forms a uniformly contracting self-similar set that obeys from the formula If we do m 0 steps in the construction of C α and n 0 steps in the construction of C β , then the squares that obtain from their Cartesian product are called the first step of the construction of C α × C β . The number of the squares are 2 m 0 +n 0 and each of them has length p −m 0 . Let Π θ :=Proj θ be the projection onto the line R × {0} and let C be one of these squares. Therefore the affine map that sends the interval [−λ, 1] to the interval Π θ (C) is one of the maps (2). Basically, the calculation of the maps (2) is easier in this way.
Corollary 2.1. Under the above notations, Proof. I) In this case, tan θ = p(q−1) q(p−1) and it is easy to check that Π θ (1, 1− 1 q ) = ( 1 p , 0). Thus, the number of intervals emerged from the projection of squares in the first step of the construction of C α ×C β is at most 3 4 2 m 0 +n 0 . As above, the iterated function system corresponding to λ = q(p−1) p(q−1) consists of at most 3 4 2 m 0 +n 0 maps, that ensures its similarity dimension is smaller than log p m 0 3 4 2 m 0 +n 0 . Therefore, HD(C α − λC β ) < 1. This completes the proof of (I).
II) The assertion is obtained since operators (1) are affine and the points on the vertical lines s = λ 1 pass over the vertical lines s = λ 2 with suitable compositions of them. 2

Weak stable intersection
In this section, we apply Theorem 2.1 to provide conditions that guarantee the existence of an interval in f (K) − g(K ) and then we bring some results in this direction. First we state a definition. ii) for each 1 ≤ i, j ≤ 2 m 0 +n 0 , we have S i λ (−λ, 1) S j λ (−λ, 1) = ∅. Otherwise there exists d ∈ R such that d ∈ S i λ (−λ, 1) S j λ (−λ, 1) , for every λ ∈ (m 1 , m 2 ).  2 . This process happens for all sub squares C. Putting these facts together, we conclude the existence of an interval in the set Example 1. Suppose that C is a middle-1 3 Cantor set. Then C 2 + C 2 and sin C + cos C contain an interval.
The projection of all the squares in the first step of the construction C × C cover each other, when We can select numbers m 1 and m 2 such that −1 2 ∈ (m 1 , m 2 ) and the elements of the iterated function systems corresponding to the pair (C, C) are regularly linked on λ ∈ (m 1 , m 2 ). let f (x) = −g(x) = x 2 , hence the family of curves x 2 + y 2 = c satisfying the differential equation y = x −y and for the point < m 2 . Considering Proposition 3.1, the set C 2 + C 2 contains an interval.
For the second one, when 1 3 < cot θ < 1, the projection of all the squares in the first step of the construction C × C cover each other. Moreover, the family of the curves sin x + cos y = c satisfies y = cos x sin y and for the point ( Although the first condition in Proposition 3.1 seems weak, this defect disappears in the second condition as we have seen in Example 1. We will also see this below and in Example 2 at the end of the Section 4. Definition 3.2. Suppose that K and K are two Cantor sets of the real numbers. We say that the pair (K, K ) has weak stable intersection in the sense of topology C r with r ≥ 1, if f (K) g(K ) = ∅, for all diffeomorphisms of f and g in the C r -neighborhood of the identity.
Thus, when K and K have weak stable intersection, the set f (K) − g(K ) contains an interval, for f and g selected in a C r -neighborhood of the identity. Now, we obtain two important results.
i) If the pair (K, K ) has stable intersection, then we can take neighborhood U of the identity, such that for the f, g ∈ U, the sets f (K) and g(K ) are regular Cantor sets, and f (K), g(K ) has stable intersection. Hence, the pair (K, K ) has weak stable intersection too. Thus, an appropriate way to show that the Cantor sets K and K does not have stable intersection is to introduce a sequence of diffeomorphisms {h n } near h(x) = x, such that the Lebesgue measure of K − h n (K ) is zero.
ii) If HD(K) + HD(K ) < 1 and K and K are the regular Cantor sets, then f (K) − g(K ) does not contain any interval for each f, g ∈ C 1 , since Thus, the pair (K, K ) does not have weak stable intersection.
Note that if we take arbitrary Cantor sets K and K with dim H K = dim B K instead of regular Cantor sets K and K in (ii), then we obtain the same assertion. Before we state a result about the existence of weak stable intersection of Cantor sets C α and λC β , we state following open problem; Open Problem 1. Are there any (C α , C β ) ∈ Ω that has weak stable intersection while does not have stable intersection? what about regular Cantor sets (K, K )?
The pair (C, C) with 1 3 -middle Cantor set does not have weak stable intersection too. In fact, natural variations of Sannami's example [14], which follows from the results of [1], shows that there are central Cantor sets K which are diffeomorphic to C by diffeomorphisms C ∞ very close to the identity such that K − K has empty interior with positive Lebesgue measure.
Note that C − C = [0, 1] and that S i 1). But for the iterated function systems S λ = {S i λ } corresponding to the pair (C α , C β ) that satisfies S i 1 (−1, 1) = (−1, 1), we can select the interval (m −1 , m) such that for every g (y) < m for every x, y ∈ [0, 1] and f, g ∈ U. By planning the arguments similar to what employed in the proof of Proposition 3.1, we observe that f (C α ) − g(C β ) contains an interval. Indeed, it is proved that the pair (C α , C β ) ∈ L has weak stable intersection. The below corollary can be a generalization of this result.

Hausdorff dimension
The results of this section begin by observing the complication of calculating Hausdorff dimension C α −λC β with the simplest non trivial choice of the middle Cantor sets C α and C β ; indeed, log α log β = 3 2 ∈ Q, together with the special number λ. Take α := 1 p := 1 γ 3 and β := 1 q := 1 γ 2 , where γ is the golden number √ 5+1 2 . Also, let C α and C β be two middle Cantor sets with expanding maps φ α and φ β , respectively, as follows: .
on R, that are return maps to the vertical line s = 2 γ . As we mentioned in Section 2, we can easily find a i 's as below: Note that, the numbers in the right side of the notion are the a i 's which are related to the square of the first step of the construction C α × C β appeared in the point presented in the left side. Of course, we can find other a i 's as below: For every 7 < i ≤ 14, we use the relation a i = a i−7 − p(p − 1) = a i−7 − (6γ + 4) and we get a 8 = 2γ + 4, a 9 = 2γ + 2, a 10 = 4, a 11 = 2, Regardless of the indexes, we have Also, it is straightforward to show that Other sets have similar relations. The intervals S i [−λ, 1] overlap each other just on G i or R i . It is important to know that in each G i or R i , the inverse of Gs (also Rs) under the maps T is either the same, or there is not any intersection between them and they situate symmetrical as Figure 3. Henceforth, we show that in each stage of the construction C α × C β , there exists an attractor namely F, with the minimum number of the contractions, that satisfies and then we see that HD(F ) < 1 for the 6th step.
In the first step, we have the iterated function system which consists of 32 contractions on the square I × I. An equivalence relation on S 1 defines as follows: On our selection of number γ and angle θ, we can take F := {S 1 , ..., S 21 } ⊂ S 1 , in this condition that S i S j for each 1 ≤ i = j ≤ 21. Let F be the attractor of the family F on I × I. Then F satisfies the relation (5) and we have HD(F ) = log 21 γ 6 .
On the n-step, let S n : Again, for elements of S n , we use the equivalence relation (6). If 1 ≤ i ≤ 6, then we define X i := [S] | S(I × I) Π −1 θ (X i ) = ∅ . For 1 ≤ i ≤ 12 the set Y i , and for 1 ≤ i ≤ 3 the set Z i have been defined similarly. Moreover, if 1 ≤ i ≤ 12, then we define G i := [S] | S(I × I) ⊂ Π −1 θ (G i ) . For 1 ≤ i ≤ 6 the set R i has been defined similarly. Regardless of the index i, we take x n := |X i |. Numbers y n , z n , g n and r n are defined similarly. Now we claim that xn yn zn gn rn , n ≥ 2.
The assertion holds for n = 2. Indeed, we know that the elements of S 2 are as follows:   The general case is obtained by using induction method and the relation [x n , y n , z n , g n , r n ] T = A[x n−1 , y n−1 , z n−1 , g n−1 , r n−1 ] T . Now we select the elements of F ⊂ S n , that are just in a class and F is its attractor on the square I × I.
Although, the iterated function system {S i } 21 i=1 corresponding to the attractor C α − 2 γ C β is of finite type and HD(C α − λC β ) could be calculated by characterizing the incidence matrix corresponding to this 21 maps, (we can not do it). But we find an easier way to do this by using the fact that the attractor F of the iterated function system of finite type satisfies 0 < H s (F ) < 1, where s=dim H (F ) [9]. Take A as Proposition 4.1, hence where λ is the largest eigenvalue of the matrix A. Moreover, this number is the Box dimension of C α − 2 γ C β .
Proof. By the same notations used in Proposition 4.1 and s := HD(C α − 2 γ C β ). Noting to the scaling property of the s-dimensional Hausdorff measure H s , we obtain which is equivalent to On the other hand, roots of the characteristic polynomial corresponding to the matrix A are 0, 1, − 0.0943..., 1.8434..., 17.2508.... Due to these facts s has to be log p 2 17.2508 = 0.9863.... The second assertion is a direct result of Theorem 1.1 of [9]. 2 Regarding to our discussion, we convince that C α × C β has a better structure among other members of L. We finish this paper by posing several characteristics of the pair of Cantor sets C α × C β . Take the dense subgroup G := {cγ + d| c, d ∈ Q} from the real numbers. Hence, Proposition 4.3. For all g ∈ G, the set C α − gC β contains an interval or has zero Lebesgue measure.
Proof. Fix g ∈ G. Then we can find the rational number r ∈ Q such that all a g appeared in Theorem 2.1 satisfy b g = − ag p 2 ∈ rZ[γ], since G = Q[γ] is a field (note that 1 cγ+d = c c 2 −cd−d 2 · γ − c+d c 2 −cd−d 2 ). When HD(C α − gC β ) = 1, then by using Theorems 2.9 and 1.3 of [9], we see that (C α − gC β ) • = ∅. Thus, the set C α − gC β contains an interval which proves the corollary. 2 Also, it is easy to find λ's that C α − λC β contains an interval. This can be useful in providing the assumptions of Proposition 3.1, for given f and g. By taking f (x) = g(x) = √ x, we pose the below example.
Example 2. The set √ C α − C β contains an interval.
On the first step of the structure of C α × C β , the projection under the angle θ on all squares overlap each other, when 1 = We can select a basic square in the next structures of C α × C β situated in (x 0 , y 0 ) between lines y = x and y = ( q p−2 ) 2 x (there are plenty of squares close to the point (1, 1)) and so 1 < y 0 x 0 < ( q p−2 ) 2 . On the other hand, we see that the family of curves √ x − √ y = c satisfies y = y x . Now we can select p−2 q < m 1 < x 0 y 0 < m 2 < 1 such that the elements of the iterated function systems corresponding to (C α , C β ) be regularly linked on (m 1 , m 2 ). Regarding to the Proposition 3.1, the set √ C α − C β contains an interval.
Open Problem 2. Does the pair (C α , C β ) have stable intersection, what about weak?