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Julia Sets and Complex Singularities of Free Energies

About this Title

Jianyong Qiao

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 234, Number 1102
ISBNs: 978-1-4704-0982-1 (print); 978-1-4704-2029-1 (online)
Published electronically: July 28, 2014
Keywords:Julia set, Fatou set, renormalization transformation, iterate

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Table of Contents


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


We study a family of renormalization transformations of generalized diamond hierarchical Potts models through complex dynamical systems. We prove 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. We give a sufficient and necessary condition for the Julia sets to be disconnected. Furthermore, we prove 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, we prove here a new type of distribution: the set of these complex singularities in the real temperature domain could contain an interval. Finally, we study the boundary behavior of the first derivative and second derivative of the free energy on the Fatou component containing the infinity. We also give an explicit value of the second order critical exponent of the free energy for almost every boundary point.

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