Horocycle averages on closed manifolds and transfer operators

TL;DR

This paper develops a framework using transfer operators and anisotropic spaces to analyze horocycle flows on compact manifolds, demonstrating power-law ergodic average convergence in certain Anosov flows.

Contribution

It adapts transfer operator techniques to $C^r$ Anosov flows, establishing quasi-compactness and applying it to ergodic integrals of horocycle flows in variable negative curvature.

Findings

Power-law convergence of ergodic averages in dimension three.

Extension of transfer operator methods to $C^r$ Anosov flows.

Implementation of Giulietti-Liverani program in variable curvature settings.

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.