3. The Mandelbrot set and Julia Sets

3. E. The boundary of the Mandelbrot set

In many ways, the most interesting part of the Mandelbrot set is its boundary. First, what is the boundary of a set S in the plane?.

The boundary of the Mandelbrot set looks very crinkly.

To get an idea of just how complicated the boundary is, we introduce two special types of points in the Mandelbrot set.

Centers of Mandelbrot set components are c values with z = 0 belonging to a cycle.

Misiurewicz points are c values with z = 0 iterating in a finite number of steps to a cycle not containing z = 0.

The boundary of the Mandelbrot set is so fuzzy it is 2-dimensional.

At the Mandelbrot set boundary, the Julia sets and the Mandelbrot set appear to look more and more alike.

Return to the Mandelbrot set and Julia sets.