Horocycle flows on abelian covers of surfaces of negative curvature

TL;DR

This paper studies horocycle flows on infinite Abelian covers of negatively curved surfaces, proving asymptotic ergodic integral results and spectral properties without symbolic dynamics, and deriving power deviation estimates.

Contribution

It introduces a Fourier decomposition approach and spectral analysis of transfer operators for horocycle flows on Abelian covers, avoiding symbolic dynamics.

Findings

Proves asymptotic behavior of ergodic integrals for regular functions.

Recovers known results in constant curvature cases.

Establishes power deviation estimates without pinching conditions.

Abstract

We consider the unit speed parametrization of the horocycle flow on infinite Abelian covers of compact surfaces of negative curvature. We prove an asymptotic result for the ergodic integrals of sufficiently regular functions. In the case of constant curvature, where the unit speed and the uniformly contracting parametrizations of horocycles coincide, we recover a result by Ledrappier and Sarig. Our method, which does not use symbolic dynamics, is based on a general Fourier decomposition for Abelian covers and on the study of spectral theory of weighted (and twisted) transfer operators for the geodesic flow acting on appropriate anisotropic Banach spaces. Finally, as a byproduct result, we obtain a power deviation estimate for the horocycle ergodic averages on compact surfaces, without requiring any pinching condition as in previous results.