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Thermodynamical formalism and holomorphic dynamical systems. (Formalisme thermodynamique et systèmes dynamiques holomorphes.) (French) Zbl 0879.58042

Panoramas et Synthèses. 4. Paris: Société Mathématique de France. vi, 96 p. (1996).
The purpose of this book is to present some pure mathematical results which are obtained as corollaries of physical laws, stating thus the symmetry between mathematics and physics. For that, the author chooses the example of two disciplines apparently unlinked: thermodynamics (a part of statistical physics which studies the equilibrium of a gas or of different states of matter) and holomorphical dynamical systems (a theory which analyzes, among other things, the strange fractal sets which appear when iterating a quadratic polynomial).
The first part of the work is dedicated to the physical aspect of the problem: the so-called “thermodynamical formalism” is introduced (that is, the ergodic theory, the different notions of entropy, as well as a thermodynamical model). Next, the main tool of this book, the Perron-Frobenius-Ruelle theorem and its thermodynamical consequences are developed. This theorem makes the link with the mathematical problem of iterating quadratic polynomials \((z \to z^2+c)\), more precisely it allows the author to refind some important results concerning the Hausdorff-dimension \(d(c)\) of the Julia set of \(z\to z^2+c\). Finally, the problem of phase transition is considered and the thermodynamical formalism is generalized to the case of polynomials whose Julia set has no critical point.

MSC:

37A99 Ergodic theory
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37F99 Dynamical systems over complex numbers
82B30 Statistical thermodynamics
37B99 Topological dynamics
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