# Why does the Mandelbrot set contain (slightly deformed) copies of itself?

The Mandelbrot set is the set of points of the complex plane whos orbits do not diverge. An point $c$’s orbit is defined as the sequence $z_0 = c$, $z_{n+1} = z_n^2 + c$.

The shape of this set is well known, why is it that if you zoom into parts of the filaments you will find slightly deformed copies of the original shape, for example: I measured some points on the Mandelbrot, and the corresponding points from one of these smaller Mu-molecules. Comparing the orbit sequences it was possible to find points on each sequence which were very close – but this experiment did not really help me to understand anything new.

Suppose that $f$ is a quadratic polynomial. Suppose that there is an integer $n$ and a domain $U \subset \mathbb{C}$ so that the $n$-th iterate, $f^n$, restricted to $U$ is a “quadratic-like map.” Then we’ll call $f$ renormalizable. (See Chapter 7 of McMullen’s book “Complex dynamics and renormalization” for more precise definitions.) Now renormalization preserves the property of having a connected Julia set. Also the parameter space of “quadratic-like maps” is basically a copy of $\mathbb{C}$.
So, fix a quadratic polynomial $f$ and suppose that it renormalizes. Then, in the generic situation, all $g$ close to $f$ also renormalize using the same $n$ and almost the same $U$. This gives a map from a small region about $f$ to the space of quadratic-like maps. This gives a partial map from the small region to the Mandelbrot set and so explains the “local” self-similarity.