GENERALIZATION OF THE MANDELBROT SET - QUATERNIONIC QUADRATIC MAPS

The iterated map Q --> Q(2) + C, where Q and C are complex 2 x 2 matri ces representing quaternions, provides a natural generalisation of the Mandelbrot set to higher dimensions. Using the well-known expansion o f the quaternion in terms of the generators of SU(2), the Pauli matric es, it is shown that the fixed point Q = Q(2) + C is stable for C insi de a cardioidal surface M(3) in R(4) and the boundary set partial deri vative M(3) sprouts domains of stability of multiple cycles. Stability calculations up to 3-cycle leading to explicit expressions for the as sociated Mandelbrot domain in R(4) are presented here for the first ti me. These analyses lay down the theoretical frame work for characteriz ing the stability domain for general k-cycles.