Convergence of points under the Mandelbrot iterated equation

Mathematics Asked by Robert Kaman on December 9, 2020

I was looking at $z_n=z_n^2+z_0$ and I realized I didn’t know what convergence actually looked like, even in real numbers. I picked a really easy number, -0.5, to see what it did. It approached a number I don’t know 🙁

A wolfram alpha widget helped me out here. It appears to converge to something like -0.366025403784

enter image description here

In other words, I’m wondering if there’s a closed-form way to write: $$((((x^2+x)^2+x)^2+x)^2+x…)$$ for $x=-.05

One Answer

The behavior and convergence (or not) of $z_{n+1}=z_n^2+z_0$ very much depends on $z_0$. In 1D, see the Logistic map. There is a close relationship with the Mandelbrot set:

enter image description here [image from Wikipedia]

Answered by lhf on December 9, 2020

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