涉及相变问题Julia集的Hausdorff维数

The Hausdorff Dimnension of the Julia Set Concerning Phase Transitions

考虑反铁磁链对应的金刚石型等级晶格上的λ-态Potts模型的配分函数零点的极限点集,这极限点集被证明是一族有理函数T_λ(z)的Julia集J(T_λ(z)).该文得到当λ→∞时,其Julia集J(T_λ(z))的Hausdorff维数的渐近估计,即J(T_λ(z))的Hausdorff维数的一个下界估计,另外研究这族有理函数的Julia集的其他拓扑性质.

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（Tλ（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（Tλ（z））. Finally, other topological structures of Julia set are discussed completely.