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Meromorphic extension of the zeta function for Axiom A flows. (English) Zbl 0694.58035
We prove the meromorphy of the zeta function on shifts of finite type for Hölder continuous functions assuming that the essential spectrum of the associated Ruelle operator is contained in the open unit disc.
This result further extends previous results by Ruelle and the improvement subsequently achieved by Parry and Pollicott. The main result of the paper is proved in the Banach space of complex functions whose variation decays exponentially fast. By general theory one obtains meromorphic extensions of the zeta function for Axiom A diffeomorphisms to a region which merely depends upon the regularity of the potential considered. Unlike to all the previous work on this subject the rate of decay of the variation is controlled by a positive function (modulus of continuity) rather than by a constant. This allows us to extend the region of meromorphy of the zeta function for Axiom A flows by a strip whose width is determined by the contraction rate of the flow.
Reviewer: N.T.A.Haydn

MSC:
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
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