Affine embeddings and intersections of Cantor sets

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.Comment: The paper will appear in J. Math. Pure. App

Report of computer experiments on the Rauzy fractals (Algebraic system, Logic, Language and Related Areas in Computer Sciences II)

For a Pisot primitive unimodular substitution over the alphabet A with d letters, a substitution dynamical system consisting of a subset of the full A shift and a shift map is constructed. And we obtain d - 1 dimensional domain, so called the Rauzy fractals as a geometrical realization of the substitution dynamical system. The authors conducted computer experiments to observe geometrical properties of the Rauzy fractals. In this report, examples of the Rauzy fractals are given