On the strong separation condition for self-similar iterated function systems with random translations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929488899604480 |
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| author | Baker, Simon Kong, Derong Wang, Zhiqiang |
| author_facet | Baker, Simon Kong, Derong Wang, Zhiqiang |
| contents | Given a self-similar iterated function system $Φ=\{ ϕ_i(x)=ρ_i O_i x+t_i \}_{i=1}^m$ 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 $Φ$ is sufficiently small, then a Lebesgue typical member of this family will satisfy the strong separation condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14175 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the strong separation condition for self-similar iterated function systems with random translations Baker, Simon Kong, Derong Wang, Zhiqiang Dynamical Systems Classical Analysis and ODEs 28A78, 28A80 Given a self-similar iterated function system $Φ=\{ ϕ_i(x)=ρ_i O_i x+t_i \}_{i=1}^m$ 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 $Φ$ is sufficiently small, then a Lebesgue typical member of this family will satisfy the strong separation condition. |
| title | On the strong separation condition for self-similar iterated function systems with random translations |
| topic | Dynamical Systems Classical Analysis and ODEs 28A78, 28A80 |
| url | https://arxiv.org/abs/2401.14175 |