# Thermodynamic Formalism and Holomorphic Dynamical Systems

Share this page
*Michel Zinsmeister*

A co-publication of the AMS and Société Mathématique de France

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 & Endorsements

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

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.