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

TL;DR

This paper investigates the likelihood that randomly translated self-similar iterated function systems satisfy the strong separation condition, showing that small similarity dimension favors this property.

Contribution

It establishes that for systems with sufficiently small similarity dimension, a Lebesgue typical random translation results in the strong separation condition being satisfied.

Findings

Lebesgue typical systems satisfy the strong separation condition when similarity dimension is small.

The probability of satisfying the strong separation condition increases with decreasing similarity dimension.

The results connect geometric properties of IFS with probabilistic translation models.

Abstract

Given a self-similar iterated function system $\Phi=\{ \phi_i(x)=\rho_i O_i x+t_i \}_{i=1}^m$ 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 $\Phi$ is sufficiently small, then a Lebesgue typical member of this family will satisfy the strong separation condition.