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Dynamical zeta functions for piecewise monotone maps of the interval by David Ruelle

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Dynamical zeta functions for piecewise monotone maps of the interval David Ruelle
Language: English
Page: 69
Format: djvu
ISBN: 0821869914, 9780821869918
Publisher: American Mathematical Society

Consider a space $M$, a map $f:Mto M$, 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 $(M,f,g)$.