Affine embeddings of Cantor sets in the plane

TL;DR

This paper characterizes affine embeddings of Cantor sets in the plane, revealing conditions under which such maps are similarities or have specific eigenvalue properties, and explores implications for a conjecture on contraction ratios.

Contribution

It provides a detailed classification of affine maps embedding self-similar sets, depending on the symmetry group size, and offers evidence for a conjecture relating contraction ratios of different IFSs.

Findings

When the symmetry group is infinite, embeddings are similarities.

Eigenvalues of linear parts are rational powers of contraction ratios.

Embeddings between different self-similar sets depend on algebraic relations of contraction ratios.

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 $|G_\Psi|<\infty$.