Affine embeddings of Cantor sets in the plane

Abstract

Let $F,E\subseteq \mathbb{R}^2$ be two self similar sets. First, assuming $F$ is generated by an IFS $\Phi$ with strong separation, we characterize the affine maps $g:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $g(F)\subseteq F$. Our analysis depends on the cardinality of the group $G_\Phi$ generated by the orthogonal parts of the similarities in $\Phi$. When $|G_\Phi|=\infty$ we show that any such self embedding must be a similarity, and so (by the results of Elekes, Keleti and M\'ath\'{e}) some power of its orthogonal part lies in $G_\Phi$. When $|G_\Phi| < \infty$ and $\Phi$ has a uniform contraction $\lambda$, we show that the linear part of any such embedding is diagonalizable, and the norm of each of its eigenvalues is a rational power of $\lambda$. We also study the existence and properties of affine maps $g$ such that $g(F)\subseteq E$, where $E$ is generated by an IFS $\Psi$. In this direction, we provide more evidence for a Conjecture of Feng, Huang and Rao, that such an embedding exists only if the contraction ratios of the maps in $\Phi$ are algebraically dependent on the contraction ratios of the maps in $\Psi$. Furthermore, we show that, under some conditions, if $|G_\Phi|=\infty$ then $|G_\Psi|=\infty$ and if $|G_\Phi|<\infty$ then $|G_\Psi|<\infty$.