Self-similar and self-affine sets; measure of the intersection of two copies

Let K ⊂ R be a self-similar or self-affine set, let μ 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. Under various assumptions (such as separation conditions or we assume 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 ⇐⇒ intK(K ∩ g(K)) 6= ∅ (where intK is interior relative to K). • (Extension) The measure μ has a G-invariant extension to R. Moreover, in many situations we characterize those g’s for which μ ( K ∩ g(K) ) > 0 holds, and we also get results about those g’s for which g(K) ⊂ K or g(K) ⊃ K holds. † Supported by Hungarian Scientific Foundation grant no. 37758. † Supported by Hungarian Scientific Foundation grant no. F 43620. ‡ Supported by Hungarian Scientific Foundation grant no. T 49786. Self-similar and self-affine sets 1