Interiors of continuous images of the middle-third Cantor set

TL;DR

This paper investigates the conditions under which the continuous images of the middle-third Cantor set in two dimensions have non-empty interiors, focusing on the role of derivatives and specific functions.

Contribution

It establishes new criteria involving partial derivatives for the images of the Cantor set to have non-empty interior, including applications to specific functions like power and trigonometric functions.

Findings

Images contain non-empty interior under certain derivative conditions.

Specific functions like power and trigonometric functions produce images with interior.

Provides criteria for the interior of images of Cantor sets under continuous maps.

Abstract

Let $C$ be the middle-third Cantor set, and $f$ a continuous function defined on an open set $U\subset \mathbb{R}^{2}$. Denote the image \begin{equation*} f_{U}(C,C)=\{f(x,y):(x,y)\in (C\times C)\cap U\}. \end{equation*} If $\partial _{x}f$, $\partial _{y}f$ are continuous on $U,$ and there is a point $(x_{0},y_{0})\in (C\times C)\cap U$ such that \begin{equation*} 1<\left\vert \frac{\partial _{x}f|_{(x_{0},y_{0})}}{\partial _{y}f|_{(x_{0},y_{0})}}\right\vert <3\text{ or }1<\left\vert \frac{\partial _{y}f|_{(x_{0},y_{0})}}{\partial _{x}f|_{(x_{0},y_{0})}}\right\vert <3, \end{equation*} then $f_{U}(C,C)$ has a non-empty interior. As a consequence, if \begin{equation*} f(x,y)=x^{\alpha }y^{\beta }(\alpha \beta \neq 0),\text{ }x^{\alpha }\pm y^{\alpha }(\alpha \neq 0)\text{ or }\sin (x)\cos (y), \end{equation*} then $f_{U}(C,C)$ contains a non-empty interior.