# Julia Sets and Complex Singularities of Free Energies

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*Jianyong Qiao*

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.

#### Table of Contents

# Table of Contents

## Julia Sets and Complex Singularities of Free Energies

- Cover Cover11 free
- Title page i2 free
- Introduction 18 free
- Chapter 1. Complex dynamics and Potts models 512 free
- Chapter 2. Dynamical complexity of renormalization transformations 1118
- Chapter 3. Connectivity of Julia sets 2936
- Chapter 4. Jordan domains and Fatou components 5158
- Chapter 5. Critical exponent of free energy 7178
- Bibliography 8794
- Back Cover Back Cover1102