Self-similar and self-affine sets; measure of the intersection of two copies
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Let K be a self-similar or self-affine set in R^d, let μbe 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 μ(K\cap g(K)) <c μ(K) or K\subset g(K).
Measure and topology: For every g\in G we have μ(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 μhas a G-invariant extension to R^d.
Moreover, in many situations we characterize those g's for which μ(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.