Generalization of the Mandelbrot set: Quaternionic quadratic maps

The iterated map Q → Q2 + C, where Q and C are complex 2 × 2 matrices representing quaternions, provides a natural generalisation of the Mandelbrot set to higher dimensions. Using the well-known expansion of the quaternion in terms of the generators of SU(2), the Pauli matrices, it is shown that the fixed point Q = Q2 + C is stable for C inside a cardioidal surface M3 in 4 and the boundary set ∂M3 sprouts domains of stability of multiple cycles. Stability calculations up to 3-cycle leading to explicit expressions for the associated Mandelbrot domain in 4 are presented here for the first time. These analyses lay down the theoretical frame work for characterizing the stability domain for general k-cycles.