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.