English: Julia set (in black) of the mapping ${\displaystyle z\leftarrow \frac{2}{z^2}-2}$ with equipotentials and field lines. Critical points ${\displaystyle z_c=\pm \mathrm{\infty }}$, critical value ${\displaystyle {\omega }_c=-2}$, pole ${\displaystyle z_p=0}$ at the center, image width ${\displaystyle [-3,3]}$.

There is a single attracting real five-point limit cycle with linear rate of convergence ${\displaystyle -\mathrm{0.256945...}}$ The repelling fixed point ${\displaystyle z_f=\tau -1}$ with divergence rate ${\displaystyle -2{\tau }^2}$ is at the center of the tiny white circle, complex conjugate fixed points ${\displaystyle (-1-\tau \pm i\sqrt{(\tau +1)(3\tau -5)})/2}$ with absolute divergence rate ${\displaystyle \sqrt{\frac{4}{{\tau }^2}}}$ are at the centers of the largest five-armed spirals in the left half of the image. The first preimages of fixed point ${\displaystyle \tau -1}$ are ${\displaystyle 1-\tau }$ and ${\displaystyle \pm \sqrt{\frac{2}{3-\tau }}}$. Tribonacci ratio ${\displaystyle \tau }$ is the real root of equation ${\displaystyle x^3=x^2+x+1}$, approximately ${\displaystyle 1.8392867552}$.^[1]

1. ↑ García-Garrido, Víctor J. (2020) Unveiling the fractal structure of Julia sets with Lagrangian descriptors, DOI:10.48550/arXiv.2001.08937, pages 1-12

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  SupersilverCartouche.jpg Julia set associated with the supersilver ratio with fixed point ς − 1.
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  SupersilverCartouche rec.jpg Julia set of the corres­ponding map with fixed point 2 / (ς − 1).