Julia Sets and Complex Singularities of Free Energies (Memoirs of the American Mathematical Society) - Softcover

Qiao, Jianyong

Softcover

ISBN 10: 1470409828 ISBN 13: 9781470409821

Publisher: American Mathematical Society, 2015

Synopsis

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.

About this title

About the Author

Jianyong Qiao, School of Science, Beijing University of Posts and Telecommunications, People's Republic of China.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2015
• Language: English
• ISBN 10: 1470409828
• ISBN 13: 9781470409821
• Binding: Paperback
• Number of pages: 89