Science in China Series A: Mathematics

, Volume 48, Issue 3, pp 388–412 | Cite as

Julia sets and complex singularities in diamond-like hierarchical Potts models

  • Jianyong Qiao


We study the phase transition of the Potts model on diamond-like hierarchical lattices. It is shown that the set of the complex singularities is the Julia set of a rational mapping. An interesting problem is how are these singularities continued to the complex plane. In this paper, by the method of complex dynamics, we give a complete description about the connectivity of the set of the complex singularities.


Julia set phase transition iteration, rational mapping 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Yang, C. N., Lee, T. D., Statistical theory of equations of state and phase transitions.I.Theory of condensation; II. Lattice gas and Ising model, Phys. Rev., 1952, 87: 404–409, 410–419.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Biskup, M., Kleinwaks, L. J., Chayes, J. et al., General theory of Lee-Yang zeros in models with first-order phase transitions, Physical Review Letters, 2000, 84: 4794–4797.CrossRefGoogle Scholar
  3. 3.
    Bleher, P. M., Lyubich, M. Yu., Julia sets and complex singularities in hierarchical Ising models, Comm. Math. Phys., 1991, 141: 453–474.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Derrida, B., De Seze, L., Itzykson, C., Fractal structure of zeros in hierarchical models, J. Stat. Phys., 1983, 33: 559–569.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Erzan, A., Hierarchical q-state Potts models with periodic and aperiodic renormalization group trajectories, Phys. Lett., 1983, A93A: 237–140.Google Scholar
  6. 6.
    Kaufman, K., Griffiths, R. B., Infinite susceptibility at high temperature in the Migdal-Kadanoff scheme, J. Phys., 1982, A15: L239-L242.MathSciNetGoogle Scholar
  7. 7.
    Mckay, S. R., Berker, A. N., Kirkpatrick S., Spin-glass behavior in frustrated Ising models with chaotic renormali- zation-group trajectories, Phys. Rev. Lett., 1982, 48: 767–770.CrossRefMathSciNetGoogle Scholar
  8. 8.
    Peitgen, H. O., Richter, P. H., The Beauty of Fractals (Images of complex dynamical systems), Berlin: Springer-Verlag, 1986.MATHGoogle Scholar
  9. 9.
    Derrida, B., Itzykson, C., Luck, J. K., Oscillatory critcal amplitudes in hierarchical models, Commun. Math. Phys., 1984, 94: 115–132.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Qiao, J., Li, Y., On connectivity of Julia sets of Yang-Lee zeros, Commun. Math. Phys., 2001, 222: 319–326.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Beardon, A. F., Iteration of Rational Functions, Berlin: Springer, 1991.MATHGoogle Scholar
  12. 12.
    Steinmetz, N., Jordan and Julia, Math. Ann., 1997, 307: 531–541.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Yakobson, M. V., On the boundaries of certain domains of mormality for rational mappings, Uspekhi Mat. Nauk, 1984, 39: 211–212.MathSciNetGoogle Scholar
  14. 14.
    Sullivan, D., Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains, Ann. Math., 1985, 122: 401–418.CrossRefMathSciNetGoogle Scholar
  15. 15.
    Sullivan, D., Thurston, W., Extending holomorphic motions, Acta Math., 1986, 157: 243–257.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Collet, P., Eckmann, P. J., Iterated Maps on the Interval as Dynamical Systems, Boston: Birkhäuser, 1980.MATHGoogle Scholar
  17. 17.
    Singer, D., Stable orbits and bifurcations of maps of the interval, SIAM J. Applied Math., 1978, 35: 260–267.MATHCrossRefGoogle Scholar

Copyright information

© Science in China Press 2005

Authors and Affiliations

  1. 1.Center of MathematicsChina University of Mining & TechnologyBeijingChina

Personalised recommendations