Horocycle averages on closed manifolds and transfer operators
Abstract
We adapt to Cr Anosov flows on compact manifolds a construction for Cr discrete-time hyperbolic dynamics (r>1), obtaining anisotropic Banach or Hilbert spaces on which the resolvent of the generator of weighted transfer operators for the flow is quasi-compact. We apply this to study the ergodic integrals of the horocycle flows h of Cr codimension one mixing Anosov flows. In dimension three, for any suitably bunched C3 contact Anosov flow with orientable strong-stable distribution, we establish power-law convergence of the ergodic average. We thereby implement the program of Giulietti-Liverani in the "real-life setting" of geodesic flows in variable negative curvature, where nontrivial resonances exist.
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