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Sharp polynomial bounds on the number of Pollicott–Ruelle resonances

Abstract We give a sharp polynomial bound on the number of Pollicott–Ruelle resonances. These resonances, which are complex numbers in the lower half-plane, appear in expansions of correlations for Anosov contact flows. The bounds follow the tradition of upper bounds on the number of scattering resonances and improve a recent bound of Faure and Sjöstrand. The complex scaling method used in scattering theory is replaced by an approach using exponentially weighted spaces introduced by Helffer and Sjöstrand in scattering theory and by Faure and Sjöstrand in the theory of Anosov flows.