On continuous images of self-similar sets

Abstract

Let $(\mathcal{M},c_k,n_k,\kappa )$ be a class of homogeneous Moran sets. Suppose $f(x,y)\in C^3$ is a function defined on ${\mathbb{R}}^2$. Given $E_1,E_2\in (\mathcal{M},c_k,n_k,\kappa )$, in this paper, we prove, under some checkable conditions on the partial derivatives of $f(x,y)$, that $$f(E_1,E_2)=\{f(x,y):x\in E_1,y\in E_2\}$$ is exactly a closed interval or a union of finitely many closed intervals. Similar results for the homogeneous self-similar sets with arbitrary overlaps can be obtained. Further generalization is available for some inhomogeneous self-similar sets if we utilize the approximation theorem.

To appear in JMAA

References in corpus (1)

• Multiplication on uniform $\lambda$-Cantor sets