Contents
Introduction 1
Chapter 1. Complex dynamics and Potts models 5
1.1. Iterations of a rational map 5
1.2. Julia sets related to Potts models 6
Chapter 2. Dynamical complexity of renormalization transformations 11
2.1. Factorization of renormalization transformations 11
2.2. Classification of dynamical systems 12
2.3. Iteration of real maps 14
2.4. Iteration of the real map Umnλ 16
2.5. Complex singularities associated with Feigenbaum phenomenon 25
Chapter 3. Connectivity of Julia sets 29
3.1. J(Umnλ) with variant parameters 29
3.2. Connectivity numbers of periodic domains of Umnλ 35
3.3. The proof of Theorem 3.3 44
3.4. Disconnected Julia set J(U2nλ) 45
Chapter 4. Jordan domains and Fatou components 51
4.1. Local connectedness of
J(Umnλ) 51
4.2. Jordan domains in F (Umnλ) 58
4.3. Jordan curve and J(Umnλ) 65
Chapter 5. Critical exponent of free energy 71
5.1. Free energy on Fatou set 71
5.2. Boundary behavior of free energy 76
5.3. Thermodynamical formalism 78
5.4. Critical exponent 82
Acknowledgements 85
Bibliography 87
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