Distance Ratio Fractal for Complex Mapping zleftarrow z^a

The complex mapping z larr z^a is important in dynamical application, but it is not received as much attention in the literature as the mapping z larr z^a+c, for there is not fractal structure by using escape time algorithm. This paper utilizes a new method named as distance ratio iteration method and discusses the iteration properties of the complex mapping z larr z^a. The distance ratio iteration method can render the convergence region of the mapping, so the image has complex and self-similarity structure. This paper generates fractal image using distance ratio iteration method for various exponents of z larr z^a and discusses their visual properties. There is rich detail fractal structure in the mapping z larr z^a