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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
##### Keywords:
Axiom A flow; basic set; Gibbs state; correlation function
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