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Dynamics of holomorphic transformations; Mandelbrot and Julia sets.
13
votes
Exponential towers of $i$'s
The single-valued case of the exponential tower of $i$ was studied by Peter Lynch. The tower converges along three spirals to the fixed point $Q=i^Q$, given in terms of the Lambert W-function by
$$Q= …
21
votes
Is the area of the Mandelbrot set known?
New Approximations for the Area of the Mandelbrot Set gives the state of the art from 2014, and a related paper from 2015 is On a numerical approximation of the boundary structure and of the area of t …
6
votes
Precise location of the Mandelbrot Bulb Attachment to the main Cardioid
in the parameterization where the main cardioid is a circle, the bulbs are attached at rational angles $\phi=2\pi m/n$: see R.L. Devaney, The Mandelbrot bulbs.
9
votes
Renormalization in physics vs. dynamical systems
The renormalization approach to dynamical systems pioneered by Chen, Goldenfeld and Oono [1] applies the Gell-Mann and Low renormalization group from quantum physics [2] to extract the global behavior …