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An analytic FBI transform is built on compact manifolds without boundary,that satisfies all the expected properties. It enables the study of microlocalanalytic and Gevrey regularity on such manifolds. This tool is then used tostudy the Ruelle spectrum of Anosov flows with Gevrey coefficients. Inparticular, finite order for the associated dynamical determinant is proved.

FBI Transform in Gevrey Classes and Anosov Flows