Decay of correlations on Abelian covers of isometric extensions of Anosov flows (joint with IP seminar)

I will report on a recent work with M. Cekić and T. Lefeuvre. We prove a complete asymptotic expansion of the correlation function in inverse powers of the time variable, for flows which arise as Abelian extensions, that is, extensions to Z^d-covers, of certain partially hyperbolic flows. This includes the frame flow of an Abelian cover of a negatively curved closed Riemannian manifold (M, g), if the frame flow on (M, g) is ergodic. As a special case, our theorem also applies to Abelian-extensions of Anosov flow. The proof uses Fourier series in Z^d (Floquet theory), (microlocal) anisotropic Sobolev spaces tailored to the dynamics, as well as the semiclassical Borel-Weil calculus on principal bundles.