On the arithmetic difference of middle Cantor sets

Abstract

Suppose that C is the space of all middle Cantor sets. We characterize all triples (α, β, λ) ∈ C × C × R * that satisfy C α − λC β = [−λ, 1]. Also all triples (that are dense in C × C × R * ) has been determined such that C α − λC β forms the attractor of an iterated function system. Then we found a new family of the pair of middle Cantor sets P in a way that for each (C α , C β ) ∈ P, there exists a dense subfield F ⊂ R such that for each µ ∈ F , the set C α − µC β contains an interval or has zero Lebesgue measure. In sequel, conditions on the functions f, g and the pair (C α , C β ) is provided which f (C α ) − g(C β ) 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 α and C β are introduced. Then the iterated function system corresponding to the attractor C α − 2α β C β is characterized. Some specifications of the attractor has been presented that keep our example as an exception. We also show that √ C α -C β contains at least one interval.

Sums of two homogeneous Cantor sets

“…To create a recurrent set, we first construct a set which is "very likely" to be a recurrent set and make it really a recurrent set by applying a small perturbation. Let us remark that in some papers recurrent sets are explicitly constructed for some affine Cantor sets [10], [28], [29]. We will also show that analogous result holds for sums of homogeneous Cantor sets with itself.…”

“…Moreira and Yoccoz used this idea in [21] to consider stable intersections of dynamically defined (nonlinear) Cantor sets. See also [10], [28], [29].…”

When the algebraic difference of two central Cantor sets is an interval?

“…Even in the case of middle Cantor sets there is still "a mysterious region" introduced by Solomyak in [26]. From Corollaries 3.5 and 3.6 we now exactly know the area, where the difference of middle Cantor sets is an interval (a nice picture can be found in [22]). However, most of the "mysterious region" from [25] is still mysterious.…”

“…Although the characterization of this case is known for the difference C(a) − C(a), there was not such characterization for the sets of the form C(a) − C(b) for arbitrary sequences a, b ∈ (0, 1) N . There are only some partial results like Newhouse gap lemma or theorem of Pourbarat from [22], which gives an equivalent condition for the algebraic difference of middle Cantor sets to be an interval (see also Corollary 3.5). There are also some results for more general Cantor sets.…”

Topological structure of the sum of two homogeneous Cantor sets

“…When the sets A and B have exactly two elements, K + K is always a Cantor set, an M-Cantorval or a finite union of closed intervals [3]. Along the same lines, we have characterized all K and K for which their sum is a finite union of closed intervals [18]. In the special cases A = {0, 1 − λ}, λ < 1 2 , homogeneous Cantor sets are the same as middle Cantor sets, and in this context many results have been written about the topological and measure-theoretic structure of their sum [4, 6, 8, 16-18, 20, 21].…”