Baby Mandelbrot sets are born in Cauliflowers. Adrien Douady (Université de Paris-Sud)

For any complex number $c$, the filled Julia set $K_c$ is the set of points which do not escape to infinity under iteration of the map $z \mapsto z^2+c$ . It is a fractal set which depends on $c$ . The Mandelbrot set $M$ is the set of values of $c$ for which $K_c$ is connected.

The correspondence $c \mapsto K_c$ is not continuous. A big discontinuity occurs for $c = 1/4$ , the cusp of $M$ . The set $K_c$ for $c = 1/4$ is known as the cauliflower ; when $c$ is changed to $1/4 + \epsilon$ , it undergoes asudden change called implosion.

There is an infinite number of copies of $M$ in $M$ , and there are whole sequences of them. For instance, if $M'$ is a copy of $M$ in $M$ , there is a sequence $(M_n)$ of smaller copies tending to the cusp of $M'$ . For this sequence a special phenomenon occurs : each $M_n$ is encaged in a nest of decorations, the first one being a copy of an imploded cauliflower, the other ones being the same object duplicated, quadruplated, etc, and wrapped around $M_n$

We shall show and describe this phenomenon,and try to explain how it is produced.