Julia sets and complex singularities of free energies / Jianyong Qiao.

Format:

Book

Author/Creator:

Qiao, Jianyong, 1962- author.

Series:

Memoirs of the American Mathematical Society ; Volume 234, Number 1102 (second of 5 numbers)

Memoirs of the American Mathematical Society, 1947-6221 ; Volume 234, Number 1102 (second of 5 numbers)

Language:

English

Subjects (All):

Julia sets.

Fractals.

Physical Description:

1 online resource (89 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2014.

Language Note:

English

Summary:

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.

Contents:

""Cover""; ""Title page""; ""Introduction""; ""Chapter 1. Complex dynamics and Potts models""; ""1.1. Iterations of a rational map""; ""1.2. Julia sets related to Potts models""; ""Chapter 2. Dynamical complexity of renormalization transformations""; ""2.1. Factorization of renormalization transformations""; ""2.2. Classification of dynamical systems""; ""2.3. Iteration of real maps""; ""2.4. Iteration of the real map _{ }""; ""2.5. Complex singularities associated with Feigenbaum phenomenon""; ""Chapter 3. Connectivity of Julia sets""; ""3.1. ( _{ }) with variant parameters""

""3.2. Connectivity numbers of periodic domains of _{ }""""3.3. The proof of Theorem 3.3""; ""3.4. Disconnected Julia set ( _{2 })""; ""Chapter 4. Jordan domains and Fatou components""; ""4.1. Local connectedness of ( _{ })""; ""4.2. Jordan domains in ( _{ })""; ""4.3. Jordan curve and ( _{ })""; ""Chapter 5. Critical exponent of free energy""; ""5.1. Free energy on Fatou set""; ""5.2. Boundary behavior of free energy""; ""5.3. Thermodynamical formalism""; ""5.4. Critical exponent""; ""Acknowledgements""; ""Bibliography""; ""Back Cover""

Notes:

Description based upon print version of record.

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-2029-5