Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval

Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval

by David Ruelle
     
 

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Consider a space $M$, a map $f:M\to M$, and a function $g:M \to {\mathbb C}$. The formal power series $\zeta (z) = \exp \sum ^\infty _{m=1} \frac {z^m}{m} \sum _{x \in \mathrm {Fix}\,f^m} \prod ^{m-1}_{k=0} g (f^kx)$ yields an example of a dynamical zeta function. Such functions have unexpected analytic properties and interesting relations to the theory of

Overview

Consider a space $M$, a map $f:M\to M$, and a function $g:M \to {\mathbb C}$. The formal power series $\zeta (z) = \exp \sum ^\infty _{m=1} \frac {z^m}{m} \sum _{x \in \mathrm {Fix}\,f^m} \prod ^{m-1}_{k=0} g (f^kx)$ 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 Baladi-Keller 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 $(M,f,g)$.

Editorial Reviews

Booknews
A monograph based on the Aisenstadt lectures given by the author in October 1993 at the U. of Montreal on "Dynamical Zeta Functions," but with a different emphasis. Hyperbolic systems are not discussed in detail. After a general introduction (chapter 1), the concentration is on piecewise monotone maps of the interval, and a detailed proof is given of a generalized form of the theorem of Baladi and Keller (chapter 2). Annotation c. Book News, Inc., Portland, OR (booknews.com)

Product Details

ISBN-13:
9780821869918
Publisher:
American Mathematical Society
Publication date:
07/14/1997
Series:
Crm Monograph Series, #4
Pages:
62
Product dimensions:
7.48(w) x 10.63(h) x (d)

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