Renormalization group method and Julia sets

The renormalization group (RG) method has been used successfully in treating a variety of phase change and critical-point problems (Wilson KG, Kogut J. Phys Rev C 1974;12:75; Wilson KG. Rev Mod Phys 1975;773; Wilson KG. Phys Rev B 1971;3174). A relatively simple system is considered at the smallest scale; the problem is then renormalized in order to utilize the same system at next larger scale. The process is repeated at larger and larger scales. In the following we consider a model for the flow of a fluid through a porous-medium. The RG transformations for the flow of a fluid through a porous-medium in two and three dimensions are derived and generalized to the complex plane, and the types of the corresponding Julia sets are found and generated. Also, the RG transformation for Ising model on a square lattice is derived and the corresponding Julia set is found.