Multiplication on self-similar sets with overlaps

2018/07/14 by Tian, Li, Gu, Jiangwen, Ye, Qianqian +2 · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1807.05368

Abstract

Let A,B⊂ℝ. Define A⋅ B=\x⋅ y:x∈ A, y∈ B\. In this paper, we consider the following class of self-similar sets with overlaps. Let K be the attractor of the IFS \f₁(x)=λx, f₂(x)=λx+c-λ,f₃(x)=λx+1-λ\, where f₁(I)∩ f₂(I)≠ ∅, (f₁(I)∪ f₂(I))∩ f₃(I)=∅, and I=[0,1] is the convex hull of K. The main result of this paper is K⋅ K=[0,1] if and only if (1-λ)²≤ c. Equivalently, we give a necessary and sufficient condition such that for any u∈[0,1], u=x⋅ y, where x,y∈ K.

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