On Symmetry Properties of Quaternionic Analogs of Julia Sets

Abstract

By means of theory group analysis, some algebraic and geometrical properties of quaternion analogs of \emph{Julia} sets are investigated. We argue that symmetries, intrinsic to quaternions, give rise to the class of identical \emph{Julia} sets, which does not exist in complex number case. In the case of quadratic quaternionic mapping $X_{k+1} = X_k^2 + C$ these symmetries mean, that the shape of fractal \emph{Julia} set is completely defined by just two numbers, $C_0$ and $|{\bf C}|$. Moreover, for given $C_0$ the vector part of the \emph{Julia} set may be obtained by rotation of a two-dimensional \emph{Julia} subset of arbitrary plane, comprising ${\bf C}$, around the axis ${\bf n} = {\bf C}/|{\bf C}|$.