MandelbrotMsatz

MandelbrotMsatz is the set of complex numbers c for which the iterative sequence z0 = 0 and zn+1 = zn^2 + c remains bounded. It is the parameter space for the family of quadratic polynomials z^2 + c on the complex plane and is named after Benoit Mandelbrot, who popularized its study in the context of fractals and complex dynamics.

Mathematically, the MandelbrotMsatz M consists of all c for which the orbit of 0 under repeated application of z → z^2 + c does not diverge to infinity. The set is compact and connected, and it is known to lie entirely within the disc |c| ≤ 2. Its boundary is highly intricate and exhibits fractal structure, with self-similar and infinitely detailed features at all scales. The area of M is not known exactly, but numerical estimates place it around 1.50659; its boundary has infinite length.

The MandelbrotMsatz is closely related to Julia sets. For a fixed c in M, the Julia set Jc is connected, and the corresponding filled Julia set is connected as well. If c lies outside M, Jc is a Cantor set (disconnected). This connection links parameter space (MandelbrotMsatz) to dynamical planes (Julia sets) and underpins many results in complex dynamics and bifurcation theory.

Visualization of the MandelbrotMsatz typically uses the escape-time algorithm: iterating zn+1 = zn^2 + c and coloring points by the rate at which they escape (or by staying bounded). Such renderings are common in mathematics education and computer graphics, illustrating the interplay between simple equations and complex geometry.