Affine embeddings and intersections of Cantor sets

TL;DR

This paper investigates conditions under which self-similar sets, including Cantor sets, can be embedded into each other via affine or smooth maps, and explores the implications for their intersections and Hausdorff dimensions.

Contribution

It establishes equivalences between affine and smooth embeddings of self-similar sets, and proves new results on the Hausdorff dimension of their intersections, especially for Cantor sets with incommensurable contraction ratios.

Findings

F can be C^1-embedded into E iff F can be affinely embedded into E under mild conditions.

If F cannot be affinely embedded into E, then the Hausdorff dimension of E∩f(F) is less than that of F for any C^1-diffeomorphism f.

When E and F are Cantor sets with incommensurable contraction ratios, their intersection has Hausdorff dimension less than the minimum of their individual dimensions.

Abstract

Let $E, F\subset \R^d$ 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 $\R^d$. 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 $\dim_HE\cap f(F)<\min\{\dim_HE, \dim_HF\}$ when $E$ is any Cantor-$p$ set and $F$ any Cantor-$q$ set, where $p,q\geq 2$ 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.