Sturmian external angles of primitive components in the Mandelbrot set
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916390566363136 |
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| author | Itzá-Ortiz, Benjamín A. Rocha, Mónica Moreno Nopal-Coello, Víctor |
| author_facet | Itzá-Ortiz, Benjamín A. Rocha, Mónica Moreno Nopal-Coello, Víctor |
| contents | In this work we introduce the broken line construction, which is a geometric and combinatorial algorithm that computes periodic Sturmian angles of a given period, yielding the locations of their landing parameters in the Mandelbrot set. An easy to implement method to compute the conjugated angle of a periodic Sturmian angle is also provided. Furthermore, if $θ$ is a periodic Sturmian angle computed by the broken line construction, then we show the existence of a one-to-one correspondence between its binary expansion and its associated kneading sequence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07636 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sturmian external angles of primitive components in the Mandelbrot set Itzá-Ortiz, Benjamín A. Rocha, Mónica Moreno Nopal-Coello, Víctor Dynamical Systems 37F10 (Primary), 37B10 (Secondary) In this work we introduce the broken line construction, which is a geometric and combinatorial algorithm that computes periodic Sturmian angles of a given period, yielding the locations of their landing parameters in the Mandelbrot set. An easy to implement method to compute the conjugated angle of a periodic Sturmian angle is also provided. Furthermore, if $θ$ is a periodic Sturmian angle computed by the broken line construction, then we show the existence of a one-to-one correspondence between its binary expansion and its associated kneading sequence. |
| title | Sturmian external angles of primitive components in the Mandelbrot set |
| topic | Dynamical Systems 37F10 (Primary), 37B10 (Secondary) |
| url | https://arxiv.org/abs/2409.07636 |