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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2505.07138 |
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| _version_ | 1866912370392039424 |
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| author | Brockmoeller, Thies Scherz, Oscar Srkalovic, Nedim |
| author_facet | Brockmoeller, Thies Scherz, Oscar Srkalovic, Nedim |
| contents | The numerical phenomenon of $π$ appearing at parameters $c = 1/4$, $c=-3/4$ and $c=-5/4$ in the Mandelbrot set $\mathcal{M}$ has been known for over 30 years. In 2001, the first proof was provided by Aaron Klebanoff for the parameter $c=1/4$. Very recently in 2023, an even sharper result for $c=1/4$ was proved using holomorphic dynamics by Paul Siewert. This new proof also provided a conceptual understanding of the phenomenon. In this paper, we give, for the first time, a proof of the known phenomenon for the parameters $c=-3/4$ and $c=-5/4$, which is also conceptual, and we provide a generalization of the phenomenon and the proof for all bifurcation points of the Mandelbrot set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07138 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pi in the Mandelbrot set everywhere Brockmoeller, Thies Scherz, Oscar Srkalovic, Nedim Dynamical Systems The numerical phenomenon of $π$ appearing at parameters $c = 1/4$, $c=-3/4$ and $c=-5/4$ in the Mandelbrot set $\mathcal{M}$ has been known for over 30 years. In 2001, the first proof was provided by Aaron Klebanoff for the parameter $c=1/4$. Very recently in 2023, an even sharper result for $c=1/4$ was proved using holomorphic dynamics by Paul Siewert. This new proof also provided a conceptual understanding of the phenomenon. In this paper, we give, for the first time, a proof of the known phenomenon for the parameters $c=-3/4$ and $c=-5/4$, which is also conceptual, and we provide a generalization of the phenomenon and the proof for all bifurcation points of the Mandelbrot set. |
| title | Pi in the Mandelbrot set everywhere |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2505.07138 |