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

### 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 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 $(M,f,g)$.

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

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ISBN 10:  0821869914 ISBN 13:  9780821869918
Publisher: Amer Mathematical Society, 1994
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## 1.Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval (Paperback)

Published by American Mathematical Society, United States (2004)
ISBN 10: 0821836013 ISBN 13: 9780821836019
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Book Description American Mathematical Society, United States, 2004. Paperback. Condition: New. Language: English . Brand New Book. 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 $(M,f,g)$. Seller Inventory # AAN9780821836019

US$27.80 Convert currency Shipping: FREE From United Kingdom to U.S.A. Destination, rates & speeds ## 2.Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval (Paperback) Published by American Mathematical Society, United States (2004) ISBN 10: 0821836013 ISBN 13: 9780821836019 New Paperback Quantity Available: 1 Seller: Book Depository International (London, United Kingdom) Rating Book Description American Mathematical Society, United States, 2004. Paperback. Condition: New. Language: English . Brand New Book. Consider a space$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)$. Seller Inventory # AAN9780821836019 Buy New US$ 32.84
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## 3.Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval (Crm Monograph Series)

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