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Using a semiclassical approach we show that the spectrum of a smooth Anosovvector field V on a compact manifold is discrete (in suitable anisotropicSobolev spaces) and then we provide an upper bound for the density ofeigenvalues of the operator (-i)V, called Ruelle resonances, close to the realaxis and for large real parts.

Upper bound on the density of Ruelle resonances for Anosov flows