On the quaternion Julia sets via Picard–Mann iteration

AbstractIn recent years, an extensive study on the use of various iteration schemes from fixed point theory for the generation of Mandelbrot and Julia sets in complex space has been carried out. In this work, inspired by these progresses, we study the use of the Picard–Mann iteration scheme for the Julia sets in the quaternion space. Specifically, in our study, we prove the escape criterion of the Picard–Mann orbit and examine the symmetry of the Julia set for the quadratic function. Moreover, we present and discuss some 2D and 3D graphical examples of the sets generated using the Picard–Mann iteration scheme. We further analyse the influence of a parameter of interest used in the Picard–Mann iteration scheme on the average number of iterations for 2D cross sections of quaternion Julia sets of different degrees.