Pi in the Mandelbrot set everywhere
Thies Brockmoeller, Oscar Scherz, Nedim Srkalovic

TL;DR
This paper proves the occurrence of the mathematical constant pi at specific points in the Mandelbrot set, extending the understanding of this phenomenon to all bifurcation points with a conceptual proof.
Contribution
It provides the first proof of pi's appearance at certain parameters in the Mandelbrot set beyond known points, generalizing to all bifurcation points with a conceptual approach.
Findings
Pi appears at parameters c=-3/4 and c=-5/4 in the Mandelbrot set.
A conceptual proof is established for these phenomena.
The proof is generalized to all bifurcation points.
Abstract
The numerical phenomenon of appearing at parameters , and in the Mandelbrot set has been known for over 30 years. In 2001, the first proof was provided by Aaron Klebanoff for the parameter . Very recently in 2023, an even sharper result for 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 and , which is also conceptual, and we provide a generalization of the phenomenon and the proof for all bifurcation points of the Mandelbrot set.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Quantum chaos and dynamical systems · Chaos control and synchronization
