On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions

TL;DR

This paper analyzes the asymptotic behavior of ergodic integrals for locally Hamiltonian flows on surfaces, extending previous results and proving the existence of ergodic infinite extensions for a broad class of these flows.

Contribution

It provides a new proof of power deviation spectrum for ergodic integrals, generalizes previous results to observables non-zero at fixed points, and establishes ergodic infinite extensions for flows with non-degenerate saddles.

Findings

Power deviation spectrum described for ergodic integrals.

Generalization of results to observables non-zero at fixed points.

Existence of ergodic infinite extensions for flows with non-degenerate saddles.

Abstract

We consider smooth area-preserving flows (also known as locally Hamiltonian flows) on surfaces of genus $g\geq 1$ and study ergodic integrals of smooth observables along the flow trajectories. We show that these integrals display a \emph{power deviation spectrum} and describe the cocycles that lead the pure power behaviour, giving a new proof of results by Forni (Annals 2002) and Bufetov (Annals 2014) and generalizing them to observables which are non-zero at fixed points. This in particular completes the proof of the original formulation of the Kontsevitch-Zorich conjecture. Our proof is based on building suitable \emph{correction operators} for cocycles with logarithmic singularities over a full measure set of interval exchange transformations (IETs), in the spirit of Marmi-Moussa-Yoccoz work on piecewise smooth cocycles over IETs. In the case of symmetric singularities, exploiting former work of the second author (Annals 2011), we prove a tightness result for a finite codimension class of observables. We then apply the latter result to prove the existence of ergodic infinite extensions for a full measure set of locally Hamiltonian flows with non-degenerate saddles in any genus $g\geq 2$.