Dynamics of a family of rational maps concerning renormalization transformation
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Dynamics of a family of rational maps concerning renormalization transformation Yuhan ZHANG1 ,
Junyang GAO1 ,
Jianyong QIAO2 , Qinghua WANG1
1 School of Science, China University of Mining and Technology, Beijing 100083, China 2 School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
c Higher Education Press 2020
Abstract Considering a family of rational maps Tnλ concerning renormalization transformation, we give a perfect description about the dynamical properties of Tnλ and the topological properties of the Fatou components F (Tnλ ). Furthermore, we discuss the continuity of the Hausdorff dimension HD(J(Tnλ )) about real parameter λ. Keywords Completely invariant domain, quasi-circle, Hausdorff dimension, renormalization transformation MSC 37F10, 37F45 1
Introduction
The phase transition is an important problem in statistical mechanics. Lee and Yang [12,28] proved the celebrated Yang-Lee theory in statistical mechanics. The theory deals with the analytic continuation of free energy on the complex plane, where the free energy is the logarithm of the partition function. In fact, they studied the distribution of zeros of the partition function that is considered as a function of complex magnetic field. They proved famous circles theorem in an exact mathematical way for an Ising ferromagnet. This theorem states all zeros (so-called Yang-Lee zeros) of the partition function of this physical model lie on the unit circle and thus the complex singularities of the free energy lie on this unit circle as well. Fisher [9] initiated the investigation of zeros of the partition function in the complex temperature plane (so-called Fisher zeros). Lee and Yang [12,28] proposed an important problem which is to investigate the set of phase transitions, i.e., the limit distribution of zeros of the grand partition function. The free energy can be expressed as a logarithmic potential over this distribution. Wilson [27] initiated to study the properties of partition Received March 3, 2020; accepted July 21, 2020 Corresponding author: Junyang GAO, E-mail: [email protected]
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function by the theory of renormalization. Furthermore, Derrida et al. [7] found that the set of phase transitions is very complicated for some physical models. In fact, they proved the Julia set of a family of rational functions associated with renormalization transformation is exact the set of complex phase transitions of this physical model. After this, many examples show that the set of phase transitions locates on the Julia set and interesting relationships among critical exponents, critical amplitudes, and the shape of a Julia set (see [7,8,15]). Bleher and Lyubich [2] studied Julia sets and complex singularities in diamond-like hierarchical Ising models. For a general model, they reformulated the following problem: how are singularities of the free energy continued to the complex space and what is their global structure in the complex space? Considering a λ-state Potts model on a generalized
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