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Dynamical zeta functions for piecewise monotone maps of the interval

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 pro...

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Detalles Bibliográficos
Autor principal: Ruelle, David
Lenguaje:eng
Publicado: American Mathematical Society 2004
Materias:
Acceso en línea:http://cds.cern.ch/record/2279783
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author Ruelle, David
author_facet Ruelle, David
author_sort Ruelle, David
collection CERN
description 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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spelling cern-22797832021-04-21T19:05:36Zhttp://cds.cern.ch/record/2279783engRuelle, DavidDynamical zeta functions for piecewise monotone maps of the intervalMathematical Physics and MathematicsConsider 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).American Mathematical Societyoai:cds.cern.ch:22797832004
spellingShingle Mathematical Physics and Mathematics
Ruelle, David
Dynamical zeta functions for piecewise monotone maps of the interval
title Dynamical zeta functions for piecewise monotone maps of the interval
title_full Dynamical zeta functions for piecewise monotone maps of the interval
title_fullStr Dynamical zeta functions for piecewise monotone maps of the interval
title_full_unstemmed Dynamical zeta functions for piecewise monotone maps of the interval
title_short Dynamical zeta functions for piecewise monotone maps of the interval
title_sort dynamical zeta functions for piecewise monotone maps of the interval
topic Mathematical Physics and Mathematics
url http://cds.cern.ch/record/2279783
work_keys_str_mv AT ruelledavid dynamicalzetafunctionsforpiecewisemonotonemapsoftheinterval