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 $Φ$ 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 $G_Φ$ generated by the orthogonal parts of the similarities in $Φ$. When $|G_Φ|=\infty$ we show that any such self embedding must be a similarity, and so (by the results of Elekes, Keleti and Máthé) some power of its orthogonal part lies in $G_Φ$. When $|G_Φ| < \infty$ and $Φ$ has a uniform contraction $λ$, 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 $λ$. We also study the existence and properties of affine maps $g$ such that $g(F)\subseteq E$, where $E$ is generated by an IFS $Ψ$. 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 $Φ$ are algebraically dependent on the contraction ratios of the maps in $Ψ$. Furthermore, we show that, under some conditions, if $|G_Φ|=\infty$ then $|G_Ψ|=\infty$ and if $|G_Φ|<\infty$ then $|G_Ψ|<\infty$.