eBook ISBN:  9781470438500 
Product Code:  CRMM/4.E 
List Price:  $45.00 
MAA Member Price:  $40.50 
AMS Member Price:  $36.00 
eBook ISBN:  9781470438500 
Product Code:  CRMM/4.E 
List Price:  $45.00 
MAA Member Price:  $40.50 
AMS Member Price:  $36.00 

Book DetailsCRM Monograph SeriesVolume: 4; 1994; 62 ppMSC: Primary 58
Consider a space \(M\), a map \(f:M\to M\), and a function \(g:M \to {\mathbb C}\). The formal power series \(\zeta (z) = \exp \sum ^\infty _{m=1} \frac {z^m}{m} \sum _{x \in \mathrm {Fix}\,f^m} \prod ^{m1}_{k=0} g (f^kx)\) yields an example of a dynamical zeta function. Such functions have unexpected analytic properties and interesting relations to the theory of dynamical systems, statistical mechanics, and the spectral theory of certain operators (transfer operators). The first part of this monograph presents a general introduction to this subject. The second part is a detailed study of the zeta functions associated with piecewise monotone maps of the interval \([0,1]\). In particular, Ruelle gives a proof of a generalized form of the BaladiKeller theorem relating the poles of \(\zeta (z)\) and the eigenvalues of the transfer operator. He also proves a theorem expressing the largest eigenvalue of the transfer operator in terms of the ergodic properties of \((M,f,g)\).
Titles in this series are copublished with the Centre de recherches mathématiques.
ReadershipResearchers in mathematics and mathematical physics.

Table of Contents

Chapters

Chapter 1. An introduction to dynamical zeta functions

Chapter 2. Piecewise monotone maps


Reviews

David Ruelle always has something interesting to say ... and this ... book is no exception.
The Bulletin of Mathematics Books 
This is a welcome guide to the problems and methods of this area. The bibliography should help to take the reader further and along alternative but related directions.
Bulletin of the London Mathematical Society


RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Reviews
 Requests
Consider a space \(M\), a map \(f:M\to M\), and a function \(g:M \to {\mathbb C}\). The formal power series \(\zeta (z) = \exp \sum ^\infty _{m=1} \frac {z^m}{m} \sum _{x \in \mathrm {Fix}\,f^m} \prod ^{m1}_{k=0} g (f^kx)\) yields an example of a dynamical zeta function. Such functions have unexpected analytic properties and interesting relations to the theory of dynamical systems, statistical mechanics, and the spectral theory of certain operators (transfer operators). The first part of this monograph presents a general introduction to this subject. The second part is a detailed study of the zeta functions associated with piecewise monotone maps of the interval \([0,1]\). In particular, Ruelle gives a proof of a generalized form of the BaladiKeller theorem relating the poles of \(\zeta (z)\) and the eigenvalues of the transfer operator. He also proves a theorem expressing the largest eigenvalue of the transfer operator in terms of the ergodic properties of \((M,f,g)\).
Titles in this series are copublished with the Centre de recherches mathématiques.
Researchers in mathematics and mathematical physics.

Chapters

Chapter 1. An introduction to dynamical zeta functions

Chapter 2. Piecewise monotone maps

David Ruelle always has something interesting to say ... and this ... book is no exception.
The Bulletin of Mathematics Books 
This is a welcome guide to the problems and methods of this area. The bibliography should help to take the reader further and along alternative but related directions.
Bulletin of the London Mathematical Society