Equicontractive weak separation property on the line does not imply convex finite type condition
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909125320900608 |
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| author | Hare, Kevin G. |
| author_facet | Hare, Kevin G. |
| contents | Let $\{S_1, S_2, \dots, S_n\}$ be an iterated function system on $\mathbb{R}$ with attractor $K$. It is known that if the iterated function system satisfies the weak separation property and $K = [0,1]$ then the iterated function system also satisfies the convex finite type condition. We show that the condition $K = [0,1]$ is necessary. That is, we give two examples of iterated function systems on $\mathbb{R}$ satisfying weak separation condition, and $0< \dim_H(K) < 1$ such that the IFS does not satisfy the convex finite type condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00693 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equicontractive weak separation property on the line does not imply convex finite type condition Hare, Kevin G. Dynamical Systems 28A80 Let $\{S_1, S_2, \dots, S_n\}$ be an iterated function system on $\mathbb{R}$ with attractor $K$. It is known that if the iterated function system satisfies the weak separation property and $K = [0,1]$ then the iterated function system also satisfies the convex finite type condition. We show that the condition $K = [0,1]$ is necessary. That is, we give two examples of iterated function systems on $\mathbb{R}$ satisfying weak separation condition, and $0< \dim_H(K) < 1$ such that the IFS does not satisfy the convex finite type condition. |
| title | Equicontractive weak separation property on the line does not imply convex finite type condition |
| topic | Dynamical Systems 28A80 |
| url | https://arxiv.org/abs/2403.00693 |