# There are no deviations for the ergodic averages of the Giulietti-Liverani horocycle flows on the two-torus

## TL;DR

This paper proves that ergodic averages for Giulietti-Liverani horocycle flows on the two-torus either grow linearly or stay bounded, with no deviations, using zeta functions and transfer operators.

## Contribution

It establishes the absence of deviations in ergodic averages for these flows and analyzes correlation decay rates for Anosov diffeomorphisms on the two-torus.

## Key findings

- Ergodic averages either grow linearly or are bounded.
- Correlations decay at a rate slower than exp(-h_top(F)).
- Topological invariance of Artin-Mazur zeta function is crucial.

## Abstract

We show that the ergodic averages for the horocycle flow on the two-torus associated by Giulietti and Liverani to an Anosov diffeomorphism either grow linearly or are bounded, in other words there are no deviations. For this, we use topological invariance of the Artin-Mazur zeta function to exclude resonances outside of the open unit disc. Transfer operators acting on suitable spaces of anisotropic distributions and their Ruelle determinants are the key tools in the proof. As a bonus, we show that for any smooth Anosov diffeomorphism F on the two-torus, the correlations for the measure of maximal entropy and smooth observables decay with a rate strictly smaller than exp(-h_top(F)). We compare our results with related work of Forni.