We deal with the iterations of transcendental entire functions, and establish the distribution relations between the Julia sets of entire functions and their derivatives.

we study the phase transition of the Potts model on diamond-like hierarchical lattices. It is shown that the set of the complex singularities is the Julia set of a rational mapping. An interesting problem is how are these singularities continued to the complex plane.In this paper, by the method of complex dynamics, we give a completely description about the connectivity of the set of the complex singularities.

The sets of the points corresponding to the phase transitions of the Potts model on the diamond hierarchical lattice for antiferromagnetic coupling are studied. These sets are the Julia sets of a family of rational mappings. It is shown that they may be disconnected sets. Furthermore, the topological structures of these sets are described completely.

For transcendental meromorphic functions of finite order, we prove that there exist iterated orbits which tend to the Borel directions. This gives a relation between the value distribution theory and the iteration theory of meromorphic functions.

The sets of the points corresponding to the complex phases of the Potts model on the diamond hierarchcal lattice are studied. These sets are the Fatou sets of a family of rational mappings. The topological structures of these sets are described completely.