ON THE TOPOLOGY OF ARITHMETIC SUMS OF REGULAR CANTOR SETS

Carlos G. Moreira^{1} Eduardo Muñoz Morales^{2} Juan Rivera-Letelier^{3}

Abstract

The study of geometrical and topological properties of arithmetic sums and differences of regular Cantor sets appears naturally in distinct fields as dynamical systems (particularly, in the study of homoclinic bifurcations related to non-trivial hyperbolic sets) and number theory (particularly, in the study of geometrical properties of the Markov and Lagrange spectra, related to diophantine approximations). We study the topological structure of the sum of two regular Cantor sets and we obtain some local results related to this problem; more precisely, we give persistent examples of several different topological types of these sums or differences.

1Instituto de Matemática Pura e Aplicada. Estrada Dona Castorina 110, Rio de Janeiro, Brasil. gugu@impa.br . Partially supported by PRONEX-Dynamical Systems, IMPA CNPq, Brasil.

2Universidad Católica del Norte. Avenida Angamos 0610, Antofagasta, Chile.

emunoz@socompa.ucn.cl . Partially supported by Fondecyt Grant 1981241, IMPA CNPq and UCN

3Universidad Católica del Norte.Avenida Angamos 0610, Antofagasta, Chile.

juanrive@socompa.ucn.cl and Université de Paris-Sud Laboratorie de Topologie et Dinamique Batiment 425, 91405 Orsay Cedex, Orsay,France. rivera@topo.math.u-psud.fr .

Partially supported by IMPA CNPq and UCN.

1 Introduction

The study of geometrical and topological properties of arithmetic sums and differences of regular Cantor sets appears naturally in distinct fields as dynamical systems (e.g. in the study of homoclinic bifurcations associated to non trivial hyperbolic sets) and number theory (e.g. in the study of the Markov and Lagrange spectra associated to Diophantine approximations).

Regular Cantor sets are, roughly speaking, defined by an iterate system of expanding differentiable functions on intervals. It is not difficult to see that if the sum of the Hausdorff dimensions of two regular Cantor sets K_{1} and K_{2} is smaller than one then the arithmetic sum K_{1} + K_{2} = {x + y |x ∈ K_{1}, y ∈ K_{2} } and also the arithmetic difference K_{1} − K_{2} = {x − y |x ∈ K_{1}, y ∈ K_{2} } both have zero Lebesgue measure. Recently, Moreira and Yoccoz proved a conjecture of J. Palis, according to which, generically, if the sum of the Hausdorff dimensions of K_{1} and K_{2} is bigger than one then K_{1} + K_{2} (and K_{1} − K_{2}) contains intervals. More precisely, it was shown that there is an open, dense and in some sense a total measure subset U of

Ω = {(K_{1}, K_{2}) | K_{1} and K_{2} regular Cantor sets, HD (K_{1}) + HD (K_{2}) > 1}

(here HD denotes Hausdorff dimension) which is dense in the C^{∞} topology such that if (K_{1}, K_{2}) ∈ Ω then K_{1} + K_{2} and K_{1} − K_{2} have stably dense interior (see [MY]).

Mendes and Oliveira studied in [MO] the topological structure of the sum of some class of affine Cantor sets in the interval [0, 1]. In that paper they pose the question of what can be said about the topological structure of the sum of two regular Cantor sets. We will give here some local results about this problem. Since K_{1} − K_{2} = K_{1} + (−K_{2}) we will restrict our attention to sums instead of differences.

If HD (K_{1}) + HD (K_{2}) < 1 then K_{1} + K_{2} has zero Lebesgue measure, and so is a Cantor set (recall that any two Cantor sets on the real line are homeomorphic). So we will be concerned about sums of regular Cantor sets such that the sum of their Hausdorff dimensions is bigger than one. As we mentioned before, these sums have generically dense interior. This fact is of course not enough to characterize its topological structure. However, there are some regions in Ω where we can characterize or give partial descriptions of the topological structure of K_{1} + K_{2} for generic pairs (K_{1}, K_{2}) in the considered region. The simplest example is the region of pairs (K_{1}, K_{2}) such that τ (K_{1}) τ (K_{2}) > 1, where τ denotes the thickness defined by Newhouse, see for instance [N] or [P T]. In this open region K_{1} + K_{2} is stably a finite union of closed intervals (see example 1 in section 4). The same happens in a larger open set U^{0} defined in [M].

2 Background and Global Facts

In this section we introduce some preliminary concepts and definitions. For X ⊆ ℝ, X̂ denotes the convex hull of X and |X| = |X̂| the length of

the interval X̂, which can be eventually infinite. When X is compact, a connected component of ℝ\X is called a gap of X, and a non trivial connected component of X is called an interval of X. A Cantor set K ⊆ ℝ is a compact perfect set with empty interior. A regular Cantor set is defined whenever:

(i) there exist pairwise disjoint compact intervals I_{1}, I_{2}, …, I_{r} ⊆ I_{0} = K̂

with K ⊆ ⋃_{i=1}^{r} I_{i},

(ii) there exist expanding functions ψ_{i} : I_{i} ⟶ I_{0} of class C^{1+ε} whose images are convex hull of disjoint unions of the I_{i} such that if ψ :

⋃_{i=1}^{r} I_{i} ⟶ I_{0} is defined by ψ |_{I_{i}} = ψ_{i} then for each i there exists n_{i}

with ψ^{n_{i}} (K ∩ I_{i}) = K ∩ I_{0} and besides, K = ⋂_{n⩾0}ψ^{−n} ( ⋃_{i=1}^{r} I_{i}).

A map ψ is C^{1+ε} if ψ is C^{1} and there exists k > 0 such that |ψ′ (x) − ψ′ (y)| ⩽ k |x − y|^{ε} for all x, y. C ^{1+} denotes the union of the spaces C^{1+ε}, ε > 0, with the induced topology. In particular, when the ψ_{i} are affine functions we say that the Cantor set is affine. Note that for a regular Cantor set K, f(K) is a regular Cantor set for every C^{1+} diffeomorphism f of ℝ.

To define the thickness of a Cantor set K, denoted by τ (K) , we consider a gap U of K, and let I_{U} be the interval bounded by U and the closest gap V that is larger than U. Then the thickness of K is

τ (K) = inf \{\frac{|I_{U}|}{|U|} ; U is a bounded gap of K\} .

Note that τ (K) = τ (−K). The importance of the thickness is the following lemma:

Gap-lemma: For given K_{1} and K_{2} Cantor sets with τ (K_{1}) τ (K_{2}) > 1 then exactly one of the following three possibilities holds:

• K̂_{1} is contained in a gap of K_{2}.

• K̂_{2} is contained in a gap of K_{1}.

• K_{1} ∩ K_{2} is non empty.

Thus, if τ (K_{1}) τ (K_{2}) > 1, then K_{1} + K_{2} is a finite union of intervals. Finally, we recall that the thickness, as well as the Hausdorff dimension of a regular Cantor set K depends continuously on K (see [P T] ).

Given a Cantor set K ⊆ ℝ and a finite covering of K by open intervals U = {U_{i}}_{i∈A}, put |U| = max {|U_{i}| : i ∈ A} . Given α ∈ ℝ, we set H_{α} (U) = ∑_{i∈A} |U_{i}|^{α}, and we define the Hausdorff α−measure of K,

m_{α} (K) = lim_{ϵ⟶0}(inf {H_{α} (U) | U is a covering of K with |U| < ϵ}).

The Hausdorff dimension of K is defined as being the unique real number HD (K) such that

m_{α} (K) = \{ \begin{matrix}+∞, & for α < HD (K) . \\ 0, & for α > HD (K) . \end{matrix}

3 Local Results

In this section some local results are presented. To start with, it is considered a simple case where the arithmetic sum is stably an interval.

Example 0: For α ∈ (0, \frac{1}{2}) let K_{α} be the affine Cantor set which is the

maximal invariant set contained in [0, 1] of ψ_{1} and ψ_{2} which are the orientation preserving affine functions such that ψ_{1} ([0, α]) = ψ_{2} ([1 − α, 1]) = [0, 1]. Then, for α > \frac{1}{3} and K_{1}, K_{2} close to K_{α}, K_{1} + K_{2} is an interval.

Note that τ (K_{α}) = \frac{α}{1 − 2α} so, if α > \frac{1}{3} , τ (K_{α}) > 1, and since τ is

continuous, there is a neighbourhood U of K_{α} such that τ (K) > 1 for all K ∈ U. Moreover, by reducing U if necessary, assume that for all K_{1}, K_{2} ∈ U, every gap of K_{2} is shorter than |K_{1}|. Then, by the Gap-lemma, K_{1} + K_{2} is an interval, for all K_{1}, K_{2} ∈ U.

Remark: This example can be generalized, so that the arithmetic sum is stably the union of n > 1 disjoint intervals. In fact, let n > 1, ϵ ∈ (0, 1) and K_{ε} be the Cantor set generated by the orientation preserving affine maps ψ_{0}, …, ψ_{n−1} such that ψ_{i} ([4i, 4i + 2 + ϵ]) = [0, 4n − 2 + ϵ]. Thus,

τ (K_{ε}) = \frac{2 + ϵ}{2 − ϵ} > 1 and there is a neighbourhood V of K_{ε}, such that for all

K_{2} ∈ V, τ (K_{2}) > 1. Moreover, by reducing V and U neighbourhoods as in the previous example, if necessary, assume that for all K_{1} ∈ U, K_{2} ∈ V, |K_{1}| < |K_{2}|, and K_{2} has exactly n − 1 gaps longer than |K_{1}|. Then, by the Gap-lemma, K_{1} + K_{2} is a disjoint union of n intervals, for all K_{1} ∈ U and K_{2} ∈ V.

In general, the arithmetic sum of pairs of regular Cantor sets on the line is more complex, and it can be classified according to the nature of its different types of gaps. It can be distinguished four kind of bounded gaps :

• LR − gap is a gap, between intervals,

• R − gap is a gap with an interval only at its left extreme,

• L − gap is a gap with an interval only at its right extreme,

• C − gap is a Cantor-like gap, i.e. it does not contain intervals at its extremes.

Remark: In this description an interval means a non-trivial interval contained in the sum of the Cantor sets.

[MO] introduced the concept of an M-Cantorval. It is a compact perfect set on the real line with dense interior, whose all bounded gaps are C-gaps. We will give here an example of an open set U_{1} ⊂ Ω such that, for generic pairs (K_{1}, K_{2}) ∈ U_{1}, K_{1} + K_{2} is an M-Cantorval. It is proved in [MO] that all M-Cantorvals are homeomorphic, so it characterizes the topology of K_{1} + K_{2} for generic (K_{1}, K_{2}) in U_{1}. Suppose that K_{1} and K_{2} are regular Cantor sets defined by increasing surjective differentiable expanding functions ψ_{j}^{(i)} : I_{j}^{(i)} ⟶ I_{j} such that if

ψ_{j} : I_{j}^{(1)} ∪ I_{j}^{(2)} ∪ ... ∪ I_{j}^{(n_{j})} ⟶ I_{j} is defined by ψ_{j} |_{I_{j}^{(i)}} = ψ_{j}^{(i)} then

K_{j} = ⋂_{n⩾0}ψ_{j}^{−n} (I_{j}^{(1)} ∪ I_{j}^{(2)} ∪ ... ∪ I_{j}^{(n_{j})})
= {x ∈ I_{j}^{(1)} ∪ ... ∪ I_{j}^{(n_{j})} |ψ_{j}^{n} (x) ∈ I_{j}^{(1)} ∪ ... ∪ I_{j}^{(n_{j})}, ∀ n ⩾ 0} .

We can order the intervals I_{j}^{(i)} such that the left extreme of I_{j} belongs to I_{j}^{(1)} and the right extreme of I_{j} belongs to I_{j}^{(n_{j})}.

Given two such Cantor sets K_{1} and K_{2} there are diffeomorphisms φ_{j} of I_{j}, j = 1, 2 such that φ_{1} ∘ ψ_{1}^{(n_{1})} ∘ φ_{1}^{−1} and φ_{2} ∘ ψ_{2}^{(1)} ∘ φ_{2}^{−1} are affine, with slope λ_{1} and λ_{2} respectively. We define the right linearized Cantor set of K_{1} as K̃_{1}^{R} := φ_{1} (K_{1}), and the left linearized Cantor set of K_{2} as K̃_{2}^{L} := φ_{2} (K_{2}), so that K̃_{1}^{R} and K̃_{2}^{L} are defined by the functions ψ̃_{1}^{R} := φ_{1} ∘ ψ_{1} ∘ φ_{1}^{−1} and ψ̃_{2}^{L} := φ_{2} ∘ ψ_{2} ∘ φ_{2}^{−1}.

An interval J_{1} = [a, b] is left linked with J_{2} = [c, d] if a < c, b < d and c < b. We say that a pair of Cantor sets (K_{1}, K_{2}) as before satisfies the *L property if there are λ > 0, t ∈ ℝ such that the support interval of K̃_{1}^{R} is left-linked with the support interval of λK̃_{2}^{L} + t and K̃_{1}^{R} ∩ (λK̃_{2}^{L} + t) = ∅. The *R property is defined in the symmetric way: (K_{1}, K_{2}) satisfies the *R property if and only if (K_{2}, K_{1}) satisfies the *L property .

It was proved in [M] (Theorem IV. 1) that if \frac{ln λ_{1}}{ln λ_{2}} ∉ Q, then for λ > 0,

K_{1} + (−λK_{2}) does not contains any interval containing the left extreme of K_{1} + (−λK_{2}). As a consequence we can prove the proposition below.

Remark: The results of the all following propositions hold with total measure in generic 1-parameter families.

Proposition 1 In the open set U_{1} = {(K_{1}, K_{2}) pairs of regular Cantor sets defined by increasing functions such that (K_{1}, −K_{2}) and (K_{2}, −K_{1}) satisfy the *L property and HD (K_{1}) + HD (K_{2}) > 1}. Then for pairs (K_{1}, K_{2}) in a residual subset U_{1}, K_{1} + K_{2} is an M-Cantorval.

Proof. Suppose that (K_{1}, −K_{2}) ∈ U, the open and dense subset of Ω = {(K_{1}, K_{2}) |K_{1}, K_{2} regular Cantor sets, HD (K_{1}) + HD (K_{2}) > 1} defined in the introduction. Then K_{1} +K_{2} have dense interior, i.e. K_{1} +K_{2} = \overline{int(K_{1} + K_{2}}). Suppose moreover that for any a, b extremes of gaps of K_{1} and c, d extremes of gaps of K_{2} we have a + c ≠ b + d. This last property is generic, since for any (K_{1}, K_{2}) there is only a countable subset of λ ∈ ℝ such that the property fails for (K_{1}, λK_{2}). Suppose moreover that the ratio of the logarithms of the slopes of ψ̃_{1}^{L} |Ĩ_{1}^{(1)L}

and ψ̃_{2}^{L} |Ĩ_{2}^{(1)L} is irrational and the same for ψ̃_{1}^{R} |Ĩ_{1}^{(n_{1})L} and ψ̃_{2}^{R} |Ĩ_{2}^{(n_{2})L}

(in our previous notation Ĩ_{1}^{(n_{1})R} := φ_{1} (I_{1}^{(n_{1})}) and Ĩ_{2}^{(1)L} := φ_{2} (I_{2}^{(1)})). We claim that under these conditions K_{1} + K_{2} is an M-Cantorval. Indeed, an extreme of a gap of K_{1}+K_{2} is always of the form a+b where

a ∈ K_{1} and b ∈ K_{2} are extremes of gaps. If we consider a left extreme, then a and b are left extremes of gaps in K_{1} and K_{2} respectively, so there are intervals of the construction of K_{1} and K_{2}, say J_{1} and J_{2} of which a and b are its left extremes, and if \overline{K}_{1} = K_{1}∩J_{1} and \overline{K}_{2} = K_{2}∩J_{2} then \overline{K}_{1} and \overline{K}_{2} are regular Cantor sets conjugated to K_{1} and K_{2}, and so their linearized Cantor sets are similar to the linearized of K_{1} and K_{2}.

Thus, by the hypothesis, \overline{K}_{1}+\overline{K}_{2} does not contain any interval adjacent to a+b, and so a+b is accumulated by gaps and intervals. Analogously the same is true for right extremes. Since \overline{K}_{1} + \overline{K}_{2} contains a neighbourhood of a + b in K_{1} + K_{2} (by uniqueness of a and b), K_{1} + K_{2} is an M-Cantorval, which completes the proof.

Example: Consider the Cantor set K which is the maximal invariant set contained in [0, 23] of the orientation preserving affine functions ψ_{i}, 1 ⩽ i ⩽ 4 , such that ψ_{i}(I_{i}) = [0, 23] where

I_{1} = [0, 1] , I_{2} = [3, 11] , I_{3} = [12, 20] and I_{4} = [22, 23] .

Then, there is a neighborhood U of K in the C^{1+} topology and a residual set R ⊆ U ×U such that for all (K_{1}, K_{2}) ∈ R, K_{1} +K_{2} is an M-Cantorval.

In fact, the maximal invariant set K̃ ⊆ K of ψ_{2} and ψ_{3} has thickness

\frac{8}{7} > 1, so there is a neighbourhood U of K, such that for each K_{1} ∈ U

there exists K̃_{1} ⊆ K_{1} a Cantor set with τ \binom{}{K_{1}} > 1. Therefore, by the Gap- lemma, for all K_{1}, K_{2} ∈ U, K_{1} + K_{2} has dense interior. Moreover, since (K, K) satisfies the ∗L and the ∗R properties defined above, there is a residual R ⊆ U × U such that every bounded gap of K_{1} + K_{2} is a C−gap. It follows that for such (K_{1}, K_{2}), K_{1} + K_{2} is an M−Cantorval.

Similarly, as in [MO], an R-( resp. L-) Cantorval is a perfect compact subset of the real line, with dense interior, whose all its bounded gaps are R-( resp. L-) gaps.

As in [M], we define left extremal stable intersection as follows: (K_{1}, K_{2}) has left extremal stable intersection if there are δ > 0 and a neighbourhood V of (K_{1}, K_{2}) such that (K̃_{1}, K̃_{2}) ∈ V ⟹ K̃_{1} + (−K̃_{2}) contains an

interval of lenght δ to which the left extreme of K̃_{1} + (−K̃_{2}) belongs. Then we have the following proposition, whose proof is analogous to the previous one.

Proposition 2 In the open set U_{2}^{R(resp. L)} = {(K_{1}, K_{2}) pairs of regular Cantor sets defined by increasing functions such that (K_{1}, −K_{2}) satisfy the *L property and (K_{1}, −K_{2}) have extremal stable intersection (resp. (−K_{2}, K_{1}) satisfy the *L property and (−K_{2}, K_{1}) have extremal stable intersection) and HD (K_{1}) + HD (K_{2}) > 1}. Then for generic pairs (K_{1}, K_{2}) ∈ U_{2}^{R(resp. L)}, K_{1} + K_{2} is an R(resp. L)-Cantorval .

Example: If K is defined by the increasing affine functions ψ_{i}, i = 1, 2, 3 such that

ψ_{1} (I_{i}) = [0, 14] , where I_{1} = [0, 5] , I_{2} = [6, 11] and I_{3} = [13, 14] .

Then there is a neighbourhood U of K in the C^{1+} topology such that for generic (K_{1}, K_{2}) ∈ U × U then K_{1} + K_{2} is an R−Cantorval and for generic (−K_{1}, −K_{2}) ∈ U × U then K_{1} + K_{2} is an R−Cantorval.

Proposition 3 Let K be the affine Cantor set whose convex hull is [0, 24] defined by the affine functions ψ_{i}, i = 1, 2, 3, such that

ψ_{i} (I_{i}) = [0, 24] , where I_{1} = [0, 10] , I_{2} = [11, 21] and I_{3} = [23, 24] , and only ψ_{2} is decreasing. Then there is a neighbourhood U_{3} of K in the C^{1+} topology such that if K_{1}, K_{2} ∈ U_{3} then K_{1} + K_{2} is an infinite disjoint union of closed intervals converging to a point.

Proof. The maximal invariant set of ψ_{1} and ψ_{2} in [0, 21] is an affine

Cantor set K̃ whose convex hull is [0, 21] with thickness τ \binom{}{K} = \frac{5}{2}, so

τ (K) τ \binom{}{K} = \frac{5}{4} > 1. This implies that K has stable intersection with

λK̃ + t if K and λK̃ + t are linked.

If I = [0, 21] then K ⊂ \{24\} ∪ ⋃_{n=0}^{∞} ψ_{3}^{−n} (I), and ψ_{3}^{−n} (I) ⊃ ψ_{3}^{−n} \binom{}{K}, a

Cantor set whose thickness is \frac{5}{2} and whose convex hull is ψ_{3}^{−n} (I).

Let K_{1}, K_{2} be Cantor sets close to K defined by (ψ_{1}^{(1)}, ψ_{2}^{(1)}, ψ_{3}^{(1)})

and (ψ_{1}^{(2)}, ψ_{2}^{(2)}, ψ_{3}^{(2)}) respectively. Let K̃_{1} and K̃_{2} be constructed as K̃.

We may suppose τ \binom{}{K_{1}} τ (K_{2}) > 1 and τ \binom{}{K_{2}} τ (K_{1}) > 1, since they are

close to \frac{5}{4}. It is not difficult to check ( by self similarity, as in the Theorem IV.1 of [M] ) that the rightmost point of K_{1} + K_{2} is accumulated by a sequence of gaps of K_{1} + K_{2}. Let b_{1} and b_{2} be the rightmost points of K_{1} and K_{2}, respectively. The results follows from

K_{1} + K_{2} ⊂ \{b_{1} + b_{2}\} ∪ ⋃_{n=0}^{∞} ((ψ_{3}^{(1)})^{−n} (I^{(1)}) + (ψ_{3}^{(2)})^{−[ \frac{n}{2} ]} (K_{2}))

∪ ⋃_{n=0}^{∞} ((ψ_{3}^{(1)})^{−[ \frac{n}{2} ]} (K_{1}) + (ψ_{3}^{(2)})^{−n} (I^{(2)}))

= \{b_{1} + b_{2}\} ∪ ⋃_{n=0}^{∞} ((ψ_{3}^{(1)})^{−n} \binom{}{K_{1}} + (ψ_{3}^{(2)})^{−[ \frac{n}{2} ]} (K_{2}))

∪ ⋃_{n=0}^{∞} ((ψ_{3}^{(1)})^{−[ \frac{n}{2} ]} (K_{1}) + (ψ_{3}^{(2)})^{−n} \binom{}{K_{2}}) ⊂ K_{1} + K_{2},

where I^{(1)} and I^{(2)} are the convex hull of K̃_{1} and K̃_{2}. The equality holds by the Gap-lemma since the sums in the right hand side are sums of Cantor

sets whose product of thicknesses is close to \frac{5}{4} , and (ψ_{3}^{(i)})^{−[ \frac{n}{2} ]} (K_{i}) is never contained in a gap of (ψ_{3}^{(j)})^{−n} \binom{}{K_{j}} , i, j = 0, 1 . Moreover, these sums are finite unions of intervals that converge to {b_{1} + b_{2}}, which shows that b_{1} + b_{2} is the only accumulation point of the boundary of K_{1} + K_{2}. Thus, the proof has been completed.

Remark: If K_{1} and K_{2} are near K and λ > 0 is small (for instance, if

λ < \frac{1}{30}) then the topological type of K_{1} + λK_{2} is a finite union of (2 or more) copies homeomorphic to K + K, and the number of copies increases as λ tends to 0.

Proposition 4 Given ε > 0 consider the affine Cantor set K_{ε} which is the maximal invariant set, contained in [0, 11] of the affine functions ψ_{i}, such that ψ_{i} (I_{i}) = [0, 11], 1 ⩽ i ⩽ 4, where

I_{1} = [0, 4] , I_{2} = [6, 10] , I_{3} = [11 − 2ε, 11 − ε] , I_{4} = [11 − ε^{2}, 11] ,

and only ψ_{1} preserves orientation. If ε > 0 is small enough (for example

ε ∈ (0, \frac{1}{300})) there is a neighbourhood U_{4} of K_{ε} in the C^{1+} topology

such that for all K_{1}, K_{2} ∈ U_{4}, every gap of K_{1} + K_{2} is an RL-gap and the accumulation points of the boundary of K_{1} + K_{2} are not countable. (In particular K_{1} + K_{2} is not a finite union of intervals).

Proof. 1.- Note that the maximal invariant set K̃_{ε} of ψ_{i}, i = 1, 2 is

contained in K_{ε} and has thickness \frac{4}{3}. Moreover, since ψ_{4} reverses orientation, K_{ε} contains another subset at the right extreme which has

thickness \frac{4}{3}. Therefore, by the Gap-lemma all K_{1} and K_{2} close to K_{ε} in the C^{1+} topology are such that every gap of K_{1} + K_{2} is an RL-gap for any λ > 0.

2.- Given a neighbourhood V of K_{ε} there is a neighbourhood U_{4} of K_{ε} such that for K ∈ U_{4} and for each interval I of the construction of K ∩ I is the image by an affine map of an element of V.

We shall prove that if V is small enough and U_{4} is as above, then, for any λ > 0 and K_{1}, K_{2} ∈ U_{4}, we can find two disjoint open intervals I′, I′′ such that (K_{1} + λK_{2}) ∩ I′ = μ′K′_{1} + λ′K′_{2} and (K_{1} + λK_{2}) ∩ I′′ = μ′′K′′_{1} + λ′′K′′_{2} where K′_{1}, K′_{2}, K′′_{1} and K′′_{2} belong to U_{4}, and both μ′λ′ and μ′′λ′′ are positive.

Repeating the process for the sums μ′K′_{1} + λ′K′_{2} and μ′′K′′_{1} + λ′′K′′_{2} and successively, we get our result.

Our previous statement follows from the following analysis:

(i) If \frac{1}{100} ≤ λ ≤ 100, K_{1}+λK_{2} ⊃ (ψ̃_{4}^{−1} (K_{1}) + λ\overline{ψ}_{4}^{−1} (K_{2}))∪(ψ̃_{3}^{−1}ψ̃_{4}^{−1} (K_{1}) +

λ\overline{ψ}_{3}^{−1}\overline{ψ}_{4}^{−1} (K_{2})) we denote the functions that define K_{1} and K_{2} by ψ̃

and \overline{ψ} respectively). Since support intervals of the two previous sums are disjoint, we get the statement because ψ̃_{j}, \overline{ψ}_{j}, j = 3, 4 reverse orientation.

(ii) If λ ≤ \frac{1}{100}, K_{1} + λK_{2} ⊃ (ψ̃_{1}^{−1} (K_{1}) + λK_{2}) ∪ (ψ̃_{2}^{−2} (K_{1}) + λK_{2}) . As in the previous case, the support intervals of the sums of the right side are disjoint, and both ψ̃_{1} and ψ̃_{2}^{2} preserve orientation, so again we get the desired result. The case λ ≥ 100 is proved in the same way, changing the role of K_{1} and K_{2}. Therefore, the proof has been completed.

Proposition 5 Given ε > 0 small, let K_{ε} be the Cantor set defined by the affine increasing functions ψ_{i} such that ψ_{i}, 1 ⩽ i ⩽ 6, ψ_{i} (I_{i}) = [0, 25], where

I_{1} = [0, 10] , I_{2} = [11, 11 + \frac{ε^{2}}{2} ] , I_{3} = [12, 12 + 2ε^{2}] ,
I_{4} = [13, 13 + \frac{ε^{2}}{2} ] , I_{5} = [14, 14 + 2ε^{2}] and I_{6} = [15, 25] .

If ε small enough (for instance if 0 < ε < \frac{1}{100}) then there is a neighbourhood U_{5} of K_{ε} in the C^{1+} topology such that if (K_{1}, K_{2}) ∈ U_{5} × U_{5} and λ ∈ (\frac{ε}{3}, 3ε) then K_{1} +λK_{2} has non-countably many connected components and all of its gaps are of the type RL.

Proof. Let U_{5} be a small neighbourhood of K_{ε} and K̃ the maximal invariant set of ψ_{1} and ψ_{6}, then τ \binom{}{K} = 2 and the convex hull of K̃

is [0, 25]. In a similar way, given K_{1} ∈ U_{5} one defines K̃_{1}, which has thickness close to 2, and whose convex hull is equal to that of K_{1}.Thus if (K_{1}, K_{2}) ∈ U_{5} × U_{5} \} then τ \binom{}{K_{1}} τ \binom{}{K_{2}} > 1 since it is close to 4, and therefore all gaps of K_{1} + K_{2} are of type RL.

Let Ũ be a small neighbourhood of K_{ε}, thus if U_{5} is small enough, then every sub-Cantor set of K_{1} ∈ U_{5} belongs to Ũ after been properly reescaled.

Let (K_{1}, K_{2}) ∈ U_{5} × U_{5}, and ψ_{i}^{(j)}, 1 ⩽ i ⩽ 6 ,1 ⩽ j ⩽ 2 be the functions that define K_{1} and K_{2}. For λ ∈ (\frac{ε}{3}, 3ε), we have

K_{1} + λK_{2} = ⋃_{i=1}^{6} ((ψ_{i}^{(1)})^{−1} (K_{1}) + λK_{2}) .

Since λ ∈ (\frac{ε}{3}, 3ε), the diameter of λK_{2} is smaller than the lengh of the gaps of K_{1} in the first step of its construction, so the convex hulls of

(ψ_{i}^{(1)})^{−1} (K_{1}) + λK_{2}, 1 ⩽ i ⩽ 6 are disjoint.

Since λ belongs to (\frac{ε}{3}, 3ε) and if U_{5} is small enough, the ratio

of the diameters of the Cantor sets (ψ_{i}^{(1)})^{−1} (K_{1}) and λK_{2} belongs to

(\frac{2ε}{5}, \frac{5ε}{2}) ⊂ (\frac{ε}{3}, 3ε) for at least two values of i between 2 and 5. For

these values of j, the sets (ψ_{j}^{(1)})^{−1} (K_{1}) + λK_{2} are open and closed in

K_{1} + λK_{2}, and (\frac{25}{|K_{2}|} · K_{2}, \frac{25}{λ |K_{2}|} · ψ_{j}^{(1)} (K_{1})) ∈ U_{5} × U_{5}, so we can repeat

the argument to these pairs and conclude that K_{1} + K_{2} has non-countably many connected components. Therefore the proof has been completed.

Remark: If U_{5} is small enough, λ is close to 1 and (K_{1}, K_{2}) ∈ U_{5} × U_{5} then (K_{1}) + λK_{2} is an interval, so the topological structure of K_{1} + λK_{2} for K_{1} and K_{2} close to K depends strongly on the size of λ, that is not the case of the Proposition 4.

Proposition 6 Let K be the Cantor set which is the maximal invariant set of the maps ψ_{i}, 1 ⩽ i ⩽ 4, such that ψ_{i} (I_{i}) = [0, 52], where

I_{1} = [0, 21] , I_{2} = [22, 43] , I_{3} = [48, 49] , I_{4} = [51, 52] ,

and only ψ_{2} reverses orientation.Then, there is a neighbourhood U_{6} of K and a residual set R ⊆ U_{6} × U_{6} such that for all (K_{1}, K_{2}) ∈ R and λ > 0, (i) the bounded gaps of K_{1} + λ K_{2} are of type RL or L .

(ii) K_{1} + λ K_{2} has infinitely many gaps of type RL and of type L.

Proof. 1.- Fix η > 1 close to 1 and let V = V (η) be a neighbourhood of K such that all the ratios in the first step in the construction of a Cantor set in V are like the ones in K with a factor in (η^{−1}, η).

Then, by bounded distortion, there is a neighbourhood U_{6} = U_{6} (η) of K such that every sub-Cantor set of K ∈ U_{6} belongs to V after been

properly reescaled. Note that τ (K) = \frac{1}{2} and, if K̃ denotes the maximal

invariant set contained [0, 52] of ψ_{1} and ψ_{2}, then τ \binom{}{K} = \frac{21}{10}, so

τ \binom{}{K} · τ (K) > 1.

Therefore, if η > 1 is close to 1 for all K_{1}, K_{2} ∈ U_{6}, τ \binom{}{K_{1}} · τ (K_{2}) > 1,

where K̃_{1} is the analogous set of K̃ for K_{1}. Thus, by the Gap-lemma, K̃_{1} + K_{2} is a finite union of intervals. In particular, the leftmost point of K_{1} + K_{2} belongs to an interval of K_{1} + K_{2}, since it belongs to K̃_{1} + K_{2}.

2.- Let t ∈ K_{1} + K_{2} be the right boundary point of a gap of K_{1} + K_{2}, so t = t_{1} + t_{2} where t_{i} is the left extreme of a sub-Cantor set K′_{i} of K_{i}, i = 1, 2. By the observation in 1 and considering that τ (K) = τ (−K), it follows that if K′_{1} or K′_{2} is an oriented copy of K_{1} or K_{2} , respectively, then t belongs to an interval of K_{1} + K_{2}.

On the other hand, if K′_{1} and K′_{2} are reversed copies of K_{1} and K_{2} let n_{i} be the least integer such that t_{i} is an extreme of an interval in the n_{i}^{th} step of the construction of K_{i}, i = 1, 2. Note that n_{i} > 1 and let I_{i} be the interval in the (n_{i} − 1)^{th} step in the construction of K that contains t_{i}, i = 1, 2.

Thus I_{1} ∩ K_{1} is an oriented copy of K_{1} and t_{1} ∈ intJ ⊆ I_{1}, where J is the support of two thick Cantor set, i.e. J ∩ K_{1} is a Cantor set such that

{ \atop J ∩ K_{1}} = J and τ (J ∩ K_{1}) τ (K_{2}) > 1.

Reducing U_{6}, if necessary, one may suppose that for t′ close to t, −K_{2}+t′ is not contained in a gap of J ∩ K_{1}. Since t_{1} = t − t_{2} ∈ int (J) ∩ (−K_{2} + t), for t′ close to t, J ∩ K_{1} is not contained in a gap of −K_{2} + t′. Then, by the Gap-lemma (J ∩ K_{1}) ∩ (−K_{2} + t′) ≠ ϕ for all t′ close to t. Thus, t belongs to the interior of an interval of K_{1} + K_{2}, contradicting its definition.

This proves that all gaps of K_{1} + K_{2} are of type RL or L, since they all have an interval to their right, so part (i) has been proved.

3.- If K_{1}, K_{2} ∈ V and λ > 0, there are m, n ∈ ℕ (that can be choose n

arbitrarily large) such that \frac{|ψ̃_{4}^{−m}(K_{1})|}{λ|\overline{ψ}_{4}^{−n}(K_{2})|} is close to 1 (where ψ̃, \overline{ψ} denote the

functions that define K_{1} and K_{2}), provided that the ratio of the logarithms of the eigenvalues of ψ̃_{4} and \overline{ψ}_{4} at their fixed points is irrational (which is a generic condition, called irrational condition).

Now, assuming the generic condition that no point can be written in two or more different ways as the sum of extremes points of gaps of K_{1} and K_{2} ( this is called the separation condition ), there is a gap between the convex hulls of ψ̃_{4}^{−m−1} (K_{1}) + λ\overline{ψ}_{4}^{−n−1} (K_{2}) and (ψ̃_{4}^{−m}ψ̃_{3}^{−1} (K_{1}) + λ\overline{ψ}_{4}^{−n−1} (K_{2})) ∪ (ψ̃_{4}^{−m−1} (K_{1}) + λ\overline{ψ}_{4}^{−n}\overline{ψ}_{3}^{−1} (K_{2})) of type L

and there is a gap between the convex hulls of ψ̃_{4}^{−m}ψ̃_{3}^{−1} (K_{1})+λ\overline{ψ}_{4}^{−n}\overline{ψ}_{3}^{−1} (K_{2}) and (ψ̃_{4}^{−m}ψ̃_{3}^{−1} (K_{1}) + λ\overline{ψ}_{4}^{−n}\overline{ψ}_{2}^{−1} (K_{2}))+(ψ̃_{4}^{−m}ψ̃_{2}^{−1} (K_{1}) + λ\overline{ψ}_{4}^{−n}\overline{ψ}_{3}^{−1} (K_{2})) of type RL. Since m, n can be taken arbitrarily large, this gives our result.

Remark: In fact is possible to adapt this argument to show that there is an uncountable set C ⊂ K_{1} + K_{2} such that every neighbourhood of any point of C contains infinitely many gaps of both types R and RL.

Proposition 7 : Consider the Cantor set K defined by the affine functions ψ_{i}, 1 ⩽ i ⩽ 4, such that ψ_{i} (I_{i}) = [0, 52], where

I_{1} = [0, 21] , I_{2} = [22, 43] , I_{3} = [48, 49] , I_{4} = [51, 52] ,

and only ψ_{3} reverses orientation. Then, there is a neighbourhood U of K and a residual subset R of U × U such that for all (K_{1}, K_{2}) ∈ R, K_{1} +K_{2} has infinitely many gaps of each four types as described at the beginning of this section.

Proof. Let we first remark that if we denote by K̃ the maximal invariant

set of ψ_{1} and ψ_{2} contained in [0, 52]. Note that K̃ ⊆ K and τ \binom{}{K} = \frac{21}{10}

and so τ (K) τ \binom{}{K} = \frac{21}{20} > 1.

Let K_{1}, K_{2} be regular Cantor sets close to K that satisfy the irrational condition.

As in proposition 6, we have, arbitrarily close to the right extreme of K_{1} + K_{2}, sums of regular Cantor sets which are close to K after reescaling, that are open and closed in K_{1} +K_{2} (and indeed are intersections of K_{1} +K_{2} with intervals). The result follows from noting that if K_{1} and K_{2} are close to K and satisfy the separation condition, then denoting by ψ̃ and \overline{ψ} the functions that define K_{1} and K_{2}, we have:

(i) The gap between the convex hulls of ψ̃_{4}^{−1}(K_{1})+\overline{ψ}_{4}^{−1}(K_{2}) and (ψ̃_{3}^{−1}(K_{1}) + \overline{ψ}_{4}^{−1}(K_{2})) ∪ (ψ̃_{4}^{−1}(K_{1}) + \overline{ψ}_{3}^{−1}(K_{2})) is of type RL.

(ii) The gap between the convex hulls of (ψ̃_{3}^{−1}(K_{1})+\overline{ψ}_{4}^{−1}(K_{2}))∪(ψ̃_{4}^{−1}(K_{1})+ \overline{ψ}_{3}^{−1}(K_{2}) and ψ̃_{3}^{−1}(K_{1}) + \overline{ψ}_{3}^{−1}(K_{2}) is of type R.

(iii) The gap between the convex hulls of ψ̃_{3}^{−1}(K_{1})+\overline{ψ}_{3}^{−1}(K_{2}) and (ψ̃_{2}^{−1}(K_{1})+ \overline{ψ}_{4}^{−1}(K_{2})) ∪ (ψ̃_{4}^{−1}(K_{1}) + \overline{ψ}_{2}^{−1}(K_{2})) is of type C.

(iv) We have, arbitrarily close to the right end point of the gap in C mentioned in last remark, sums of small Cantor sets which are close to −K after rescaling, that are open and closed in K_{1} +K_{2} (this gives, by item (ii) gaps of type L).

Thus, the proof has been completed.

Remark: In fact one can prove, by slightly reducing R, that every end point of a gap is the boundary of an interval, or it is accumulated by infinitely many gaps of each one of the four types.

We can use this fact to show that there is an uncountable set C ⊂ K_{1}+K_{2} such that every neighbourhood of any point of C contains infinitely many gaps of each of the four types (C, L, R and RL).

4 Open Questions

The following is a natural question.

Problem 1: Does there exists a generic set R ⊂ {(K_{1}, K_{2}) | K_{1}, K_{2} are C^{1+} regular Cantor sets} such that each (K_{1}, K_{2}) ∈ R has a neighbourhood U such that (K̃_{1}, K̃_{2}) ∈ R ∩ U ⟹ K̃_{1} + K̃_{2} is homeomorphic to K_{1} + K_{2} ?

In particular, if generic is taken in the Baire sense, it would imply that there are countably many robust topologies of a sum of Cantor sets. The remarks after Example 0 and Proposition 3 show that there are infinitely many robust topologies of sums of Cantor sets.

An interesting particular instance of Problem 1 is the problem of determining the typical topology (or topologies) for sums of pairs of Cantor sets in the open sets U_{2}^{R(resp. L)} of Proposition 2. For generic pairs (K_{1}, K_{2}) ∈ U^{L}, K_{1} + K_{2} is an L-Cantorval, which that not determine its topology. There are two natural possibilities for the topology of K_{1} + K_{2} in this case: the L_{0} topology, obtained from a Cantor set by blowing up each right extreme of a gap creating intervals; and the L_{∞} topology, obtained by blowing up each right extreme of a gap of an M-Cantorval. We do not know robust examples of sums of Cantor sets with the L_{0} topology, (resp. with the L_{∞} topology), but it is reasonable to think that generically in U^{L}, K_{1} + K_{2} is either L_{0} or L_{∞}.

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