Interiors of continuous images of the middle-third Cantor set

Abstract

Let C be the middle-third Cantor set, and f a continuous function defined on an open set U ⊂ R 2 . Denote the imageIf ∂xf , ∂yf are continuous on U, and there is a point (x 0 , y 0 )) has a non-empty interior. As a consequence, if f (x, y) = x α y β (αβ = 0), x α ± y α (α = 0) or sin(x) cos(y), then f U (C, C) contains a non-empty interior.

On the measure of products from the middle-third Cantor set

“…Other arithmetic operations with x, y ∈ K are considered in [2], which describes the structure of quotients y/x with x = 0 and proves that [0, 1] is covered by products x 2 y, so that in particular any element of [0, 1] is the product of 3 factors in K. In [5] it is proved that sums x 2 1 + x 2 2 + x 2 3 + x 2 4 with x i ∈ K for i = 1, 2, 3, 4 cover [0, 4], which was conjectured in [2]. In [6] is described a general condition on maps f : R 2 → R such that f (K × K) has non-empty interior, where such condition is obviously satisfied by the arithmetic operation mentioned above. For the image under affine maps, and in particular for S(x, y) := x + y, a much larger class of Cantor sets and other fractals have been studied.…”

On the sum of squares of middle-third Cantor set

“…is a classical object in fractal geometry. The arithmetic on middle-third Cantor set has been studied in [1,2,3,4,5,6,8]. The first classical result is that the set .…”

Multiplication on uniform $λ$-Cantor sets

“…Jiang and Xi [14] proved that if ∂ x f , ∂ y f are continuous on U, and there is a point (x 0 , y 0 ) ∈ (C × C) ∩ U such that one of the following conditions is satisfied,…”