涉及相变问题Julia集的Hausdorff维数

Abstract

Abstract:

Considering the sets of the points corresponding to the phase transitions of the Potts model on the diamond hierarchical lattice for antiferromagnetic coupling, it is shown that these sets are the Julia sets of a family of rational mappings. In this paper, we prove that they may be buried points of J(T_λ(z)) for some λ∈R. Further, the asymptotic formula of the Hausdorff dimension of the Julia set is given as λ→∞, which gives a lower bound the Hausdorff dimension of the Julia set of J(T_λ(z)). Finally, other topological structures of Julia set are discussed completely.

Key words: Julia set, Hausdorff dimension, Buried points, Sierpinski curve

CLC Number:

O174.5

Wang Youming. The Hausdorff Dimnension of the Julia Set Concerning Phase Transitions[J].Acta mathematica scientia,Series A, 2016, 36(6): 1048-1056.

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References

[1] Bleher P, Lyubich M. Julia sets and complex singularities in hierarchical Ising models. Commun Math Phys, 1991, 141:453-474
[2] Bleher P, Lyubich M, Roeder R. Lee-Yang-Fisher zeros for DHL and 2d rational dynamics I:Foliation of the physical cylinder. 2010, arxiv:1009.4691v1
[3] Bleher P, Lyubich M. Julia sets and complex singularities in hierarchical Ising models. Comm Math Phys, 1991, 141:453-474
[4] Derrida B, Itzykson C, Luck J K. Oscillatory critical amplitudes in hierarchical models. Comm Math Phys, 1984, 94:115-132
[5] Derrida B, De Seze L. Fractal structure of zeros in hierarchical models. J Stat Phys, 1983, 33:559-569
[6] Falconer K. Fractal Geometry:Mathematical Foundations and Applications (Second Edition). England:John Wiley Sons, 2003
[7] Lee T D, Yang C N. Statistical theory of equations of state and phase transitions II:Lattice gas and Ising model. Phys Rev, 1952, 87:410-419
[8] Mane R, Sad P, Sullivan D. On the dynamics of rational maps. Ann Sci Ecole Norm Sup, 1983, 16(4):193-217
[9] McMullen C T. Complex Dynamics and Renormalization. Princeton:Princeton Univ Press, 1994
[10] McMullen C T. Hausdorff dimension and conformal dynamics II:Geometrically finite rational maps. Comment Math Helv, 2000, 75:535-593
[11] Milnor J. Dynamics in One Complex Variable. Vieweg Wiesbaden:Braunschweig, 1999
[12] Peitgen H O, Richter P H. The Beauty of Fractals (Images of Complex Dynamical Systems). Berlin:Springer-Verlag, 1986
[13] Qiao J. On Julia sets concerning phase transitions. Sci China, 2003, 46(3):415-431
[14] Qiao J. Complex Dynamics on Renormalization Transformations (in Chinese). Beijing:Science Press, 2010
[15] Ruelle D. Repellers for real analytic maps. Ergod Th Dynam Sys, 1982, 2:99-107
[16] Yang C N, Lee T D. Statistical theory of equations of state and phase transitions I:Theory of condensation. Phys Rev, 1952, 87:404-409
[17] Yang F, Zeng J S. On the dynamics of a family of generated renoralization transformations. Journal of Mathematical Analysis and Applications, 2014, 413:361-377
[18] Whyburn G T. Topological characterization of the Siepinski curve. Fund Math, 1958, 45:320-324

The Hausdorff Dimnension of the Julia Set Concerning Phase Transitions

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