  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 ${\mathbb
R}^d$: $$ f_j(\bm x) = \lambda\bm x + (1-\lambda)\bm p_j, j=0,...,m-1, $$ where
$\lambda\in(0,1)$ is our parameter. Then, as is well known, there exists a
unique self-similar attractor $S_\lambda$ satisfying
$S_\lambda=\bigcup_{j=0}^{m-1} f_j(S_\lambda)$. Each $\bm x\in S_\lambda$ has
at 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 $\bm x\in
S_\lambda\setminus\{\bm p_0,...,\bm p_{m-1}\}$ has $2^{\aleph_0}$ different
addresses. If $\lambda$ is not too close to 1, then we can still have an
overlap, but there exist $\bm x$'s which have a unique address. However, we
prove 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. We
apply these results to the case of expansions with deleted digits.
  Furthermore, we give sharp sufficient conditions for the Open Set Condition
to fail and for the attractor to have no holes.
  These results are generalisations of the corresponding one-dimensional
results, however most proofs are different.
