Affine embeddings and intersections of Cantor sets

Abstract

Let $E,F\subset {\mathbb{R}}^d$ be two self-similar sets. Under mild conditions, we show that $F$ can be $C^1$-embedded into $E$ if and only if it can be affinely embedded into $E$; furthermore if $F$ can not be affinely embedded into $E$, then the Hausdorff dimension of the intersection $E\cap f(F)$ is strictly less than that of $F$ for any $C^1$-diffeomorphism $f$ on ${\mathbb{R}}^d$. Under certain circumstances, we prove the logarithmic commensurability between the contraction ratios of $E$ and $F$ if $F$ can be affinely embedded into $E$. As an application, we show that ${\dim }_{H}E\cap f(F)<\min \{{\dim }_{H}E,{\dim }_{H}F\}$ when $E$ is any Cantor-$p$ set and $F$ any Cantor-$q$ set, where $p,q\ge 2$ are two integers with $\log p/\log q\not \in \Q$. This is related to a conjecture of Furtenberg about the intersections of Cantor sets.

The paper will appear in J. Math. Pure. Appl

References in corpus (1)

• Self-similar subsets of the Cantor set