Convergence of points under the Mandelbrot iterated equation

Question

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 https://www.desmos.com/calculator/2op8thgknv

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

Answer 1

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:

[image from Wikipedia]