Abstract
We present a novel construction of three-dimensional generalizations of complex quadratic Julia set fractals. While some extensions exist in the literature, using, e.g., quaternions, none have been able to extend the intricate fractal nature to higher dimensions.
Here we present a new approach which is based on the so-called Sullivan's dictionary, which builds analogies between the fields of complex dynamics and Kleinian groups. Here Julia sets correspond to Kleinian limit sets, which have known extensions to 3D.
By taking a special Kleinian group and its 3D extension, we can obtain information about the sought generalization of Julia sets. This leads to two extensions - the simpler 'inflated' Julia sets, and truly fractal 3D Julia sets which we were able to construct in several special cases.