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Resonances for axiom A flows. (English) Zbl 0658.58026
Let M be a compact \(C^{\infty}\) manifold, \(\{f^ t\}\) an Axiom A flow on M and let \(\Lambda\) be a non-trivial basic set for the flow. Considering the suspension \(\Omega^{\#}\) of a subshift of finite type (\(\Omega\),\(\tau)\) and a suspended flow \((\tau^ t)\) on \(\Omega^{\#}\) as well as a map \(\omega\) : \(\Omega^{\#}\to \Lambda\) connecting the flow \((\tau^ t)\) on \(\Omega^{\#}\) and \((f^ t)\) restricted to \(\Lambda\), the author first defines a Banach space \(C_{\theta}^{\#}\) \((0<\theta <1)\) and then introduces the notion of Gibbs state for \(A\in C(\Omega^{\#},R)\), which is related to the pressure of A. The main purpose of this paper is to study the pair correlation function \(\rho_{BC}\) defined, by the Gibbs state \(\rho\) corresponding to \(A\in C^{\#}_{\theta^ 2}\), for \(B,C\in C_{\theta}^{\#}\) and the meromorphy of its Fourier transform \({\hat \rho}{}_{BC}\). The residues of the poles of \({\hat \rho}{}_{BC}\) are also investigated.
Reviewer: A.Morimoto

MSC:
37D99 Dynamical systems with hyperbolic behavior
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