Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval (CRM Monograph Series) - Softcover

Ruelle, David

Softcover

ISBN 10: 0821836013 ISBN 13: 9780821836019

Publisher: Amer Mathematical Society, 2004

Synopsis

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 $(M,f,g)$.

About this title

Review

David Ruelle always has something interesting to say ... and this ... book is no exception. --The Bulletin of Mathematics Books

Edition details

• Publisher: Amer Mathematical Society
• Publication date: 2004
• Language: English
• ISBN 10: 0821836013
• ISBN 13: 9780821836019
• Binding: Paperback
• Number of pages: 62