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

Abstract

We consider smooth area-preserving flows (also known as locally Hamiltonian flows) on surfaces of genus $g\ge 1$ and study ergodic integrals of smooth observables along the flow trajectories. We show that these integrals display a power deviation spectrum and describe the cocycles that lead the pure power behaviour, giving a new proof of results by Forni [Ann. of Math. (2) 155 (2002), 1–103] and Bufetov [Ann. of Math. (2) 179 (2014), 431–499] 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 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 [Ann. of Math. (2) 173 (2011), 1743–1778] 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 $g\ge 2$.