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The Mandelbrot set in a model for phase transitions. (English) Zbl 0565.58030

Arbeitstag. Bonn. 1984, Proc. Meet. Max-Planck-Inst. Math., Bonn 1984, Lect. Notes Math. 1111, 111-134 (1985).
[For the entire collection see Zbl 0547.00007.]
The authors investigate the structure of the zero set of the partition function in the classical theory of equilibrium statistical mechanics of Yang and Lee. The Yang-Lee theory describes the physical phase as a domain of analyticity for the free energy, viewed as a function of complex temperature. The boundaries of these domains are given by the zeroes of the partition function. The authors present another interpretation of these boundaries: they are the Julia set of the corresponding renormalization transformation. Using recent results of Sullivan, Douady-Hubbard, and others, the authors are able to relate the fractal properties of these Julia sets to the interaction between the renormalization group approach and the Yang-Lee theory. In particular, the authors show that the renormalization map of the Yang-Lee theory reduces to a rational map of the Riemann sphere which has degree three. Using the classification theorem of Sullivan, the stable regions for this map are determined by the behavior of the orbits of the critical points. The authors investigate the behavior of these orbits in detail.
Reviewer: R.Devaney

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
82B05 Classical equilibrium statistical mechanics (general)

Citations:

Zbl 0547.00007