Horocycle averages on closed manifolds and transfer operators

Abstract

We adapt to $C^r$ Anosov flows on compact manifolds a construction for $C^r$ 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 $h_{\rho }$ of $C^r$ codimension one mixing Anosov flows. In dimension three, for any suitably bunched $C^3$ 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.

Version v4 is the electronic copy of the published version in Tunisian J Math

References in corpus (6)

• Horocycle averages on closed manifolds and transfer operators
• Smooth mixing Anosov flows in dimension three are exponential mixing
• Parabolic Flows Renormalized by Partially Hyperbolic Maps
• FBI Transform in Gevrey Classes and Anosov Flows
• There are no deviations for the ergodic averages of the Giulietti-Liverani horocycle flows on the two-torus
• Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum

Cited by in corpus (6)

• Horocycle averages on closed manifolds and transfer operators
• Parabolic Flows Renormalized by Partially Hyperbolic Maps
• A paradifferential approach for hyperbolic dynamical systems and applications
• There are no deviations for the ergodic averages of the Giulietti-Liverani horocycle flows on the two-torus
• Twisted cohomological equations for translation flows
• Ruelle resonances from cohomological equations