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

Given a self-similar iterated function system $\mathrm{\Phi }=\{{\phi }_i(x)={\rho }_i{O}_{i}x+t_i{\}}_{i=1}^m$ acting on ${\mathbb{R}}^d$, we can generate a parameterised family of iterated function systems by replacing each $t_i$ with a random vector in ${\mathbb{R}}^d$. 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 $\mathrm{\Phi }$ is sufficiently small, then a Lebesgue typical member of this family will satisfy the strong separation condition.