INTRODUCTION 3

dynamics of Umnλ in this article. Furthermore, we give a classification about the

complexity of the dynamics of the renormalization transformation Umnλ (Theorem

2.1). After exploring locations of real fixed points and the post-critical set of Umnλ

deeply, we find a very interesting phenomenon about the distribution of complex

singularities. We prove that the Julia set J(U2nλ) could contain a small Feigenbaum

Julia set which meets the positive real axis at a closed interval. This leads to a

mystical distribution of complex singularities: the set of complex singularities in

the real temperature domain could contain an interval (Theorem 2.2). This is a

very interesting phenomenon.

In Chapter 3, we deal with the connectivity of the Julia set J(Umnλ) of the

renormalization transformation Umnλ. Firstly, in this chapter we prove that J(Umnλ)

is connected when m = n or m and n are both odd (Theorem 3.1 and Theorem

3.2). Furthermore, we give a suﬃcient and necessary condition for the Julia set

J(Umnλ) to be a disconnected set (Theorem 3.3).

In Chapter 4, we deal with topological properties of the Fatou components of

Umnλ. The main result in this chapter is that all components of the Fatou set of

Umnλ are Jordan domains with at most one exception which is a completely invari-

ant domain (Theorem 4.2). In order to prove this result, we need a result about

the local connectivity of the Julia set J(Umnλ) which tells us that all components

of J(Umnλ) are locally connected (Theorem 4.1). When the absolute value of λ is

large enough, we show that the Julia set J(Umnλ) is actually a quasicircle. In this

case the Fatou set F (Umnλ) consists of two Jordan domains (Theorem 4.3).

Chapter 5 is devoted to dealing with the critical exponent of the free energy

of a generalized diamond hierarchical Potts model. Considering the immediate at-

tractive basin Amnλ(∞) which corresponds to the ”high temperature” domain, we

show that the derivative fmnλ of the free energy fmnλ is analytic on Amnλ(∞) and

the boundary ∂Amnλ(∞) is the natural boundary of fmnλ for some parameters m,

n and λ (Theorem 5.2). Noting that the second derivatives fmnλ is not contin-

uous up to the boundary ∂Amnλ(∞) (Theorem 5.3), we give an explicit value of

the second order critical exponent ατ

(2)

of fmnλ for almost every point τ on the

boundary of Amnλ(∞) (Theorem 5.4). The main method used for the proof of this

result is the thermodynamical formalism following Bowen, Ruelle and Sinai (see

[BOW],[RU1],[SINA]).