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Zeta functions and the periodic orbit structure of hyperbolic dynamics. (English) Zbl 0726.58003
Centre National de la Recherche Scientifique, Paris. Astérisque, 187-188. Paris: Société Mathématique de France. 268 p. $ 36.00 (1990).
As the authors state in the introduction: “Axiom A diffeomorphisms and flows, introduced by Smale, are generalizations of Anosov systems which in turn are based on the prototypical hyperbolic toral automorphisms and geodesic flows on surfaces of constant negative curvature”. To study these systems one usually models them by introducing Markov partitions, shifts and suspensions. In this work the emphasis is on problems associated with periodic orbits.
After introducing basic concepts, a.o. the Ruelle operator and entropy, the authors discuss the relation between zeta functions and periodic points. One of the important points is the relation between the spectra of Ruelle operators (real and complex) and the poles of zeta functions. Among the various themes of the book is the proof of temporal, spatial and symmetrical distribution theorems. Five appendices have been added with basic material.

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37A30 Ergodic theorems, spectral theory, Markov operators
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37D99 Dynamical systems with hyperbolic behavior