Bounds on the number of Ruelle resonances

Ruelle resonances are complex numbers appearing in
expansions of correlations. For contact Anosov flows, we prove a sharp upper bound on the number of Ruelle resonances in strips near the real axis. Our approach follows the work of Faure-Sjöstrand, who explained how the inhomogeneous Sobolev spaces used by Baladi, Tsujii, Blank,
Gouëzel, Keller, and Liverani have a clear microlocal interpretation. The refined bounds are obtained using recent work of Sjöstrand-Zworski and Datchev and the speaker on resonance counting. Joint work with K. Datchev and M. Zworski.