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

For any complex number , the *filled Julia set* is the set
of points which do not escape to infinity under iteration of the map
. It is a fractal set which depends on . The
*Mandelbrot set* is the set of values of for which
is connected.

The correspondence
is not continuous. A big discontinuity
occurs for , the cusp of . The set for is
known as the *cauliflower* ; when is changed to
,
it undergoes asudden change called *implosion*.

There is an infinite number of copies of in , and there are whole sequences of them. For instance, if is a copy of in , there is a sequence of smaller copies tending to the cusp of . For this sequence a special phenomenon occurs : each 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

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

2001-05-11