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 t...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.展开更多
We develop a Thurston-like theory to characterize geometrically finite rational maps, and then apply it to study pinching and plumbing deformations of rational maps. We show that under certain conditions the pinching ...We develop a Thurston-like theory to characterize geometrically finite rational maps, and then apply it to study pinching and plumbing deformations of rational maps. We show that under certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.展开更多
The sets of the points corresponding to the phase transitions of the Potts model on the diamondhierarchical lattice for antiferromagnetic coupling are studied. These sets are the Julia sets of a family ofrational mapp...The sets of the points corresponding to the phase transitions of the Potts model on the diamondhierarchical lattice for antiferromagnetic coupling are studied. These sets are the Julia sets of a family ofrational mappings. It is shown that they may be disconnected sets. Furthermore, the topological structures ofthese sets are described completely.展开更多
The sets of the points corresponding to the complexphases of the Potts model on the diamond hierarchcal lattice arestudied. These sets are the Fatou sets of a family of rationalmappings. The topological structures of ...The sets of the points corresponding to the complexphases of the Potts model on the diamond hierarchcal lattice arestudied. These sets are the Fatou sets of a family of rationalmappings. The topological structures of these sets are described completely.展开更多
Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that i...Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two.展开更多
An oblique edge crack problem in a semi-infinite plane is discussed. Re concentrated forces are applied on the edge crack face, or on the line boundary of the cracked semi-infinite plane. The rational mapping function...An oblique edge crack problem in a semi-infinite plane is discussed. Re concentrated forces are applied on the edge crack face, or on the line boundary of the cracked semi-infinite plane. The rational mapping function approach is suggested to solve the boundary value problem and a solution in a closed form is obtained. Finally, several numerical examples with the calculated results are given.展开更多
基金This work was supported by the“973”Project Foundation of China.
文摘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.
基金supported by National Natural Science Foundation of China (Grant No. 11125106)
文摘We develop a Thurston-like theory to characterize geometrically finite rational maps, and then apply it to study pinching and plumbing deformations of rational maps. We show that under certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.
基金This work was supported by the 973 Project Foundation of China.
文摘The sets of the points corresponding to the phase transitions of the Potts model on the diamondhierarchical lattice for antiferromagnetic coupling are studied. These sets are the Julia sets of a family ofrational mappings. It is shown that they may be disconnected sets. Furthermore, the topological structures ofthese sets are described completely.
基金This work was supported by the 973 Project Foundation of China.
文摘The sets of the points corresponding to the complexphases of the Potts model on the diamond hierarchcal lattice arestudied. These sets are the Fatou sets of a family of rationalmappings. The topological structures of these sets are described completely.
基金Project supported by the National Natural Science Foundation of China (Grant No. 19531050)the Special Foundation of the Chinese Academy of Sciences, Hong Kong Qiu-Shi Foundationthe Education Foundation of Tsinghua University.
文摘Some homotopy classes of the unit spheres are explicitly represented by entire rational maps.
基金Supported by National Natural Science Foundation of China(Grant Nos.10871089 and 11271179)
文摘Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two.
文摘An oblique edge crack problem in a semi-infinite plane is discussed. Re concentrated forces are applied on the edge crack face, or on the line boundary of the cracked semi-infinite plane. The rational mapping function approach is suggested to solve the boundary value problem and a solution in a closed form is obtained. Finally, several numerical examples with the calculated results are given.
