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Generalisation of the Mandelbrot set to integral functions of quaternions
The rich diversity of patterns and concepts intrinsic to the Julia and
the Mandelbrot sets of the quadratic map in the complex plane invite a search
for higher dimensional generalisations. Quaternions provide a natural framework
for such an endeavour. The objective of this investigation is to provide explicit
formulae for the domain of stability of multiple cycles of classes of quaternionic
maps $F(Q)+C$ or $CF(Q)$ where $C$ is a quaternion and $F(Q)$ is an integral function
of $Q$. We introduce the concept of quaternionic differentials and employ this in the
linear stability analysis of multiple cycles.