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

Abstract

Given a self-similar iterated function system $\hat{I}¦=\{\ddot{I}{•}_i(x)=\ddot{I}{}_i{O}_{i}x+t_i{\}}_{i=1}^m$ 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 $\hat{I}¦$ is sufficiently small, then a Lebesgue typical member of this family will satisfy the strong separation condition.

The main results are covered by Theorem 1.1 in [M. Rams and J. L. Véhel, Results on the dimension spectrum for self-conformal measures. Nonlinearity 20 (2007), no.4, 965-973. https://iopscience.iop.org/article/10.1088/0951-7715/20/4/009]