Quasi-compactness of transfer operators for contact Anosov flows

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For any Cʳ contact Anosov flow with r≥ 3, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all Cʳ functions, such that the transfer operators for the flow extend to them boundedly and that the extensions are quasi-compact. Further we give explicit bounds on the essential spectral radii of the extensions in terms of the differentiability r and the hyperbolicity exponents of the flow.