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Julia sets and complex singularities of free energies. (English) Zbl 1333.37032

Mem. Am. Math. Soc. 1102, v, 89 p. (2015).
The author summarizes a series of his current research [Complex dynamics on renormalization transformations (Chinese). Beijing: Science Press (2010); with Y. Yin and J. Gao, Ergodic Theory Dyn. Syst. 30, No. 5, 1573–1591 (2010; Zbl 1211.37061)] about dynamical properties of the family of rational maps \[ U_{mn\lambda}(z)=\left [\frac{(z+\lambda-1)^n+(\lambda-1)(z-1)^n}{(z+\lambda-1)^n-(z-1)^n} \right ]^m, \quad m, n\geq 2, \quad \lambda\in \mathbb{R}\setminus \{0\}. \] There are two basic problems in iteration theory of rational functions, which are also the topics of this book: 1. The dynamical property of an individual map. Each map \(U_{mm\lambda}\) expands globally since its degree is \(mn>1\), and it contracts locally at critical points. The sphere is divided into two parts: The Fatou set, where the iterative behavior is relatively tame in the sense that points close to each other behave similarly, and the Julia set, where chaotic phenomena take place and which is a fractal in general. In particular, the topological properties of these two sets, such as connectivity, local connectivity, draw a lot of attention. The maps \(U_{mn\lambda}\) are closely related to physical phenomena. The Julia set of \(U_{mn\lambda}\) is the limiting set of zeros of a grant partition function, and the maps \(U_{mn\lambda}\) are renormalization transformations with respect to \(\lambda\)-state Potts models on general diamond hierarchical lattices. 2. The second general problem is to examine the change of dynamics if the function is perturbed. A particular case is a family of functions that depends on one parameter. The main results can be summarized as following:
1. The Julia set \(U_{mn\lambda}\) is connected if \(m=n\) or both \(m\) and \(n\) are odd. Moreover, the Julia set \(U_{mn\lambda}\) is disconnected if the immediate basin of \(1\), \(A_{mn\lambda}(1)\) or the immediate basin of \(\infty\), \(A_{mn\lambda}(\infty)\) contains \(0\) or \((1-\lambda)^n\).
2. Each component of the Julia set is locally connected.
3. Each component of the Fatou set is completely invariant or a Jordan domain.
4. For \(m=2\), \(n\geq 5\) and odd, there exist parameters \(\lambda_1, \lambda_2\in \{1,2\}\) such that \(U_{2n \lambda_ j}\) (\(j=1,2\)) are Feigenbaum-like maps, that is, they are \((2,2,2,\dots)\)-renormalizable.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B28 Renormalization group methods in equilibrium statistical mechanics

Citations:

Zbl 1211.37061
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