Let K⊂ℝd be a self-similar or self-affine set and let μ be a self-similar or self-affine measure on it. Let 𝒢 be the group of affine maps, similitudes, isometries or translations of ℝd. Under various assumptions (such as separation conditions, or the assumption that the transformations are small perturbations, or that K is a so-called Sierpiński 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∈𝒢 we have either μ(K∩g(K))<c⋅μ(K) or K⊂g(K).
• (Measure and topology) For every g∈𝒢 we have μ(K∩g(K))>0⟺∫ K(K∩g(K))≠0̸ (where ∫ K is interior relative to K).
• (Extension) The measure μ has a 𝒢-invariant extension to ℝd.
Moreover, in many situations we characterize those
g for which
μ(
K∩
g(
K))>0. We also obtain results about those
g for which
g(
K)⊂
K or
g(
K)⊃
K.