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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.12962 |
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| _version_ | 1866912075083677696 |
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| author | Moreira, Carlos Gustavo Xi, Jinghua Zhang, Yiwei |
| author_facet | Moreira, Carlos Gustavo Xi, Jinghua Zhang, Yiwei |
| contents | Bandt and Kravchenko \cite{BandtKravchenko2010} proved that if a self-similar set spans $\R^m$, then there is no tangent hyperplane at any point of the set. In particular, this indicates that a smooth planar curve is self-similar if and only if it is a straight line. When restricting curves to graphs of continuous functions, we can show that the graph of a continuous function is self-similar if and only if the graph is a straight line, i.e., the underlying function is affine. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12962 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Graphs of continuous but non-affine functions are never self-similar Moreira, Carlos Gustavo Xi, Jinghua Zhang, Yiwei Dynamical Systems 28A80 Bandt and Kravchenko \cite{BandtKravchenko2010} proved that if a self-similar set spans $\R^m$, then there is no tangent hyperplane at any point of the set. In particular, this indicates that a smooth planar curve is self-similar if and only if it is a straight line. When restricting curves to graphs of continuous functions, we can show that the graph of a continuous function is self-similar if and only if the graph is a straight line, i.e., the underlying function is affine. |
| title | Graphs of continuous but non-affine functions are never self-similar |
| topic | Dynamical Systems 28A80 |
| url | https://arxiv.org/abs/2410.12962 |