Interiors of continuous images of self-similar sets with overlaps

Abstract

Let K be the attractor of the following iterated function system {S1(x) = λx, S2(x) = λx + c − λ, S3(x) = λx + 1 − λ}, where S1(I) ∩ S2(I) = ∅, (S1(I) ∪ S2(I)) ∩ S3(I) = ∅, and I = [0, 1] is the convex hull of K. Let d1 = 1 − c − λ λ < 1 1 − c − λ = d2. Suppose that f is a continuous function defined on an open set U ⊂ R 2. Denote the image fU (K, K) = {f (x, y) : (x, y) ∈ (K × K) ∩ U }. If ∂xf , ∂yf are continuous on U, and there is a point (x0, y0) ∈ (K × K) ∩ U such that ∂yf | (x 0 ,y 0) ∂xf | (x 0 ,y 0) ∈ (d1, d2) or ∂xf | (x 0 ,y 0) ∂yf | (x 0 ,y 0) ∈ (d1, d2), then fU (K, K) contains an interval. As a result, we let c = λ = 1 3 , and if f (x, y) = x α y β (αβ = 0), x α ± y α (α = 0), sin(x) cos(y), or x sin(xy), then fU (C, C) contains an interval, where C is the middle-third Cantor set.