Growth rate for beta-expansions

Let $\beta>1$ and let m>\be be an integer. Each x\in I_\be:=[0,\frac{m-1}{\beta-1}] can be represented in the form $$x=\sum_{k=1}^\infty \epsilon_k\beta^{-k},$$ where $\epsilon_k\in\{0,1,...,m-1\}$ for all $k$ (a $\beta$-expansion of $x$). It is known that a.e. $x\in I_\beta$ has a continuum of distinct $\beta$-expansions. In this paper we prove that if $\beta$ is a Pisot number, then for a.e. $x$ this continuum has one and the same growth rate. We also link this rate to the Lebesgue-generic local dimension for the Bernoulli convolution parametrized by $\beta$. When $\beta<\frac{1+\sqrt5}2$, we show that the set of $\beta$-expansions grows exponentially for every internal $x$.Comment: 21 pages, 2 figure

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

Typical self-affine sets with non-empty interior

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\| <1/2$ for $1\leq i\leq m$. We provide some sufficient conditions on these matrices such that the self-affine set generated by the iterated function system $\{T_ix+a_i\}_{i=1}^m$ on $\Bbb R^d$ has non-empty interior for almost all $(a_1,\ldots, a_m)\in \Bbb R^{md}$