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

Abstract

AbstractLetK⊂ℝdbe a self-similar or self-affine set and letμbe 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 thatKis a so-called Sierpiński sponge) we prove theorems of the following types, which are closely related to each other.•(Non-stability)There exists a constantc<1 such that for everyg∈𝒢 we have eitherμ(K∩g(K))<c⋅μ(K) orK⊂g(K).•(Measure and topology)For everyg∈𝒢 we haveμ(K∩g(K))>0⟺∫K(K∩g(K))≠0̸ (where ∫Kis interior relative toK).•(Extension)The measureμhas a 𝒢-invariant extension to ℝd.Moreover, in many situations we characterize thosegfor whichμ(K∩g(K))>0. We also obtain results about thosegfor whichg(K)⊂Korg(K)⊃K.