Fatou, Julia and Mandelbrot sets for functions with non -integer exponent

This thesis examines the dynamics of the function F( z) = zalpha + c, where alpha is not an integer. We run into many new situations compared to the classical case: z zd + c where d is an integer, d ≥ 2. The major new complication is a lack of continuity in considering where this function is defined. The natural space to consider this function is the Riemann surface W alpha, but we show that this does not suffice; i.e., the function still remains discontinuous because of the branch cut on the surface. It is this branch cut and its iterates that cause the "Julia sets" for this function to have "shears". This had been noticed by various people using computer programs to generate the images but without a satisfactory explanation of the mathematics involved. We redefine the Fatou and Julia sets for these functions and prove theorems regarding invariant properties of these sets. We also prove a result similar to one by A. Douady regarding the continuity of the set-valued function cKc . We examine the parameter spaces for these functions and define analogs to the Mandelbrot and Multibrot sets for the integer valued functions. We prove a number of conjectures regarding the properties of these sets and clarify the discontinuities.