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arXiv · 2401.14175

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

Simon Baker·Derong Kong·Zhiqiang Wang
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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.

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Simon Baker, Derong Kong, Zhiqiang Wang. 2024-01-25. On the strong separation condition for self-similar iterated function systems with random translations. https://arxiv.org/abs/2401.14175

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