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Julia Sets and Complex Singularities of Free Energies

Jianyong Qiao School of Science, Beijing University of Posts and Telecommunications, Beijing, People’s Republic of China
Available Formats:
Electronic ISBN: 978-1-4704-2029-1
Product Code: MEMO/234/1102.E
List Price: $76.00 MAA Member Price:$68.40
AMS Member Price: $45.60 Click above image for expanded view Julia Sets and Complex Singularities of Free Energies Jianyong Qiao School of Science, Beijing University of Posts and Telecommunications, Beijing, People’s Republic of China Available Formats:  Electronic ISBN: 978-1-4704-2029-1 Product Code: MEMO/234/1102.E  List Price:$76.00 MAA Member Price: $68.40 AMS Member Price:$45.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 2342014; 89 pp
MSC: Primary 37; Secondary 82;

The author studies a family of renormalization transformations of generalized diamond hierarchical Potts models through complex dynamical systems. He proves that the Julia set (unstable set) of a renormalization transformation, when it is treated as a complex dynamical system, is the set of complex singularities of the free energy in statistical mechanics. He gives a sufficient and necessary condition for the Julia sets to be disconnected. Furthermore, he proves that all Fatou components (components of the stable sets) of this family of renormalization transformations are Jordan domains with at most one exception which is completely invariant.

In view of the problem in physics about the distribution of these complex singularities, the author proves here a new type of distribution: the set of these complex singularities in the real temperature domain could contain an interval. Finally, the author studies the boundary behavior of the first derivative and second derivative of the free energy on the Fatou component containing the infinity. He also gives an explicit value of the second order critical exponent of the free energy for almost every boundary point.

• Chapters
• Introduction
• 1. Complex dynamics and Potts models
• 2. Dynamical complexity of renormalization transformations
• 3. Connectivity of Julia sets
• 4. Jordan domains and Fatou components
• 5. Critical exponent of free energy
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Volume: 2342014; 89 pp
MSC: Primary 37; Secondary 82;

The author studies a family of renormalization transformations of generalized diamond hierarchical Potts models through complex dynamical systems. He proves that the Julia set (unstable set) of a renormalization transformation, when it is treated as a complex dynamical system, is the set of complex singularities of the free energy in statistical mechanics. He gives a sufficient and necessary condition for the Julia sets to be disconnected. Furthermore, he proves that all Fatou components (components of the stable sets) of this family of renormalization transformations are Jordan domains with at most one exception which is completely invariant.

In view of the problem in physics about the distribution of these complex singularities, the author proves here a new type of distribution: the set of these complex singularities in the real temperature domain could contain an interval. Finally, the author studies the boundary behavior of the first derivative and second derivative of the free energy on the Fatou component containing the infinity. He also gives an explicit value of the second order critical exponent of the free energy for almost every boundary point.

• Chapters
• Introduction
• 1. Complex dynamics and Potts models
• 2. Dynamical complexity of renormalization transformations
• 3. Connectivity of Julia sets
• 4. Jordan domains and Fatou components
• 5. Critical exponent of free energy
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