  Let K be a self-similar or self-affine set in R^d, let \mu be a self-similar
or self-affine measure on it, and let G be the group of affine maps,
similitudes, isometries or translations of R^d. Under various assumptions (such
as separation conditions or we assume that the transformations are small
perturbations or that K is a so called Sierpinski sponge) we prove theorems of
the following types, which are closely related to each other;
  Non-stability: There exists a constant c<1 such that for every g\in G we have
either \mu(K\cap g(K)) <c \mu(K) or K\subset g(K).
  Measure and topology: For every g\in G we have \mu(K\cap g(K)) > 0 \iff int_K
(K\cap g(K)) is nonempty (where int_K is interior relative to K).
  Extension: The measure \mu has a G-invariant extension to R^d.
  Moreover, in many situations we characterize those g's for which \mu(K\cap
g(K) > 0, and we also get results about those $g$'s for which $g(K)\su K$ or
$g(K)\supset K$ holds.
