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Let $\bm p_0,...,\bm p_{m-1}$ be points in ${\mathbb R}^d$, and let$\{f_j\}_{j=0}^{m-1}$ be a one-parameter family of similitudes of ${\mathbbR}^d$: $$ f_j(\bm x) = \lambda\bm x + (1-\lambda)\bm p_j, j=0,...,m-1, $$ where$\lambda\in(0,1)$ is our parameter. Then, as is well known, there exists aunique self-similar attractor $S_\lambda$ satisfying$S_\lambda=\bigcup_{j=0}^{m-1} f_j(S_\lambda)$. Each $\bm x\in S_\lambda$ hasat least one address $(i_1,i_2,...)\in\prod_1^\infty\{0,1,...,m-1\}$, i.e.,$\lim_n f_{i_1}f_{i_2}... f_{i_n}({\bf 0})=\bm x$.  We show that for $\lambda$ sufficiently close to 1, each $\bm x\inS_\lambda\setminus\{\bm p_0,...,\bm p_{m-1}\}$ has $2^{\aleph_0}$ differentaddresses. If $\lambda$ is not too close to 1, then we can still have anoverlap, but there exist $\bm x$'s which have a unique address. However, weprove that almost every $\bm x\in S_\lambda$ has $2^{\aleph_0}$ addresses,provided $S_\lambda$ contains no holes and at least one proper overlap. Weapply these results to the case of expansions with deleted digits.  Furthermore, we give sharp sufficient conditions for the Open Set Conditionto fail and for the attractor to have no holes.  These results are generalisations of the corresponding one-dimensionalresults, however most proofs are different.Comment: Accepted for publication in Nonlinearit

Combinatorics of linear iterated function systems with overlaps