Graphs of continuous but non-affine functions are never self-similar
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- by Carlos Gustavo Moreira, Jinghua Xi and Yiwei Zhang;
- Proc. Amer. Math. Soc. 153 (2025), 2501-2512
- DOI: https://doi.org/10.1090/proc/17099
- Published electronically: April 9, 2025
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Abstract:
Bandt and Kravchenko [Nonlinearity 24 (2011), pp. 2717–2728] proved that if a self-similar set spans $\mathbb {R}^m$, 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.References
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Bibliographic Information
- Carlos Gustavo Moreira
- Affiliation: Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, People’s Republic of China; SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, People’s Republic of China; and Institute of Pure and Applied Mathematics (IMPA), Rio de Janeiro, RJ 22.460-320, Brazil
- MR Author ID: 366894
- Email: gugu@impa.br
- Jinghua Xi
- Affiliation: Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, People’s Republic of China
- ORCID: 0009-0007-6472-3215
- Email: 12110713@mail.sustech.edu.cn
- Yiwei Zhang
- Affiliation: Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, People’s Republic of China; and SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, People’s Republic of China
- MR Author ID: 1012850
- Email: zhangyw@sustech.edu.cn
- Received by editor(s): February 8, 2024
- Received by editor(s) in revised form: September 8, 2024, and September 19, 2024
- Published electronically: April 9, 2025
- Additional Notes: The first author was partially supported by CNPq and FRPERJ. The third author was partially supported by NSFC Nos. 12161141002, 12271432, and Guangdong Basic and Applied Basic Research Foundation No. 2024A1515010974
- Communicated by: Katrin Gelfert
- © Copyright 2025 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 153 (2025), 2501-2512
- MSC (2020): Primary 28A80
- DOI: https://doi.org/10.1090/proc/17099
- MathSciNet review: 4892623