On the strong separation condition for self-similar iterated function systems with random translations

Given a self-similar iterated function system Φ={ϕ_i(x)=ρ_i O_i x+t_i }_i=1^m acting on ℝ^d, we can generate a parameterised family of iterated function systems by replacing each t_i with a random vector in ℝ^d. In this paper we study whether a Lebesgue typical member of this family will satisfy the strong separation condition. Our main results show that if the similarity dimension of Φ is sufficiently small, then a Lebesgue typical member of this family will satisfy the strong separation condition.