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 \((M,f,g)\).

Titles in this series are co-published with the Centre de Recherches Mathématiques.

Readership

Researchers in mathematics and mathematical physics.

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

Table of Contents

• An introduction to dynamical zeta functions
• Piecewise monotone maps
• Bibliography