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Thermodynamic Formalism and Holomorphic Dynamical Systems
Michel Zinsmeister
A co-publication of the AMS and Société Mathématique de France.
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SMF/AMS Texts and Monographs
2000; 82 pp; softcover
Volume: 2
ISBN-10: 0-8218-1948-8
ISBN-13: 978-0-8218-1948-7
List Price: US$22
Member Price: US$17.60
Order Code: SMFAMS/2
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The purpose of thermodynamics and statistical physics is to understand the equilibrium of a gas or the different states of matter. To understand the strange fractal sets appearing when one iterates a quadratic polynomial is one of the goals of the theory of holomorphic dynamical systems. These two theories are strongly linked: The laws of thermodynamics happen to be an extremely powerful tool for understanding the objects of holomorphic dynamical systems. A "thermodynamic formalism" has been developed, bringing together notions that are a priori unrelated. While the deep reasons of this parallelism remain unknown, the goal of this book is to describe this formalism both from the physical and mathematical point of view in order to understand how it works and how useful it can be.

This translation is a slightly revised version of the original French edition. The main changes are in Chapters 5 and 6 and consist of clarification of some proofs and a new presentation of the basics in iteration of polynomials.

Titles in this series are co-published with Société Mathématique de France. SMF members are entitled to AMS member discounts.

Readership

Graduate students, research mathematicians and physicists interested in analysis, specifically measure and integration.

Reviews

"Mathematics (holomorphic systems, e.g., fractals) is explained using physics (thermodynamics and statistical physics). Using the thermodynamic formalism, the author establishes interesting mathematical results ... Mostly self-contained; excellent references."

-- American Mathematical Monthly

"Display[s] ... the vitality and diversity of an area of mathematics still in the full flood of development ... Elegant little monograph ... as a concrete illustration of the power of the thermodynamic formalism, Zinsmeister rigorously proves Ruelle's theorem ... and he establishes Ruelle's asymptotic formula for \(d(c)\) for \(c\) close to zero--yet another triumph from the heroic years of modern holomorphic dynamics."

-- Bulletin of the London Mathematical Society

From a review of the original French edition:

"This book is a pleasant short introduction to both the physics of the dynamic formalism (a formalism developed in statistical mechanics to understand the equilibrium of a gas or of the different states of matter) and its mathematical applications to holomorphic dynamics, in particular to describe the strange fractal sets that appear when iterating quadratic polynomials or other rational maps. The text is interspersed with reminiscences of the author, giving it a warm, human touch."

-- Mathematical Reviews.

Table of Contents

  • Introduction
  • The ergodic hypothesis
  • The concept of entropy
  • Entropy in ergodic theory
  • The Perron-Frobenius-Ruelle theorem
  • Conformal repellers
  • Iteration of quadratic polynomials
  • Phase transitions
  • Hausdorff measures and dimension
  • Bibliography
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