Generalized Julia sets: from Cantor bouquet to Cantor cheese
Stefanie Hittmeyer, Bernd Krauskopf, and Hinke M. Osinga
Abstract
The complex quadratic family z → z ² + c is well known to give rise to artistic images and renderings of Julia sets — for a given complex number c, the Julia set is the boundary of the set of points z that escape to infinity under iteration. There exists a unique critical value c, which is the image of the critical point z = 0. If iterates of c remain bounded then the Julia set is connected, typically a set of closed (Jordan) curves, and if its iterations diverge to infinity then the Julia set is a Cantor set. Here, we consider a non-analytic perturbation of the complex quadratic family and its generalized Julia sets, which are defined in the same way. There are now infinitely many critical values that lie on a circle. Consequently, some critical values may remain bounded under iteration, while others do not. As the result, the (generalized) Julia set can be more complicated, leading to new intriguing mathematical objects that may provide novel inspirations for art.
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