Résume  Consider a smooth areapreserving or locally Hamiltonian flow on a surface S of genus g≥1 with Morse saddles. Birkhoff integrals of smooth observables along the trajectories are well known to display polynomial deviations, a phenomenon conjectured by Kontsevich and Zorich and proved by Forni and AvilaViana for a large class of regular observables. We will present a new proof which allows to consider also the case of observables which are nonzero at the saddle points (based on 'correction' operators a' la MarmiMoussaYoccoz for functions with logarithmic singularities over IETs). The result has an application to the infinite ergodic theory of Rextensions of locally Hamiltonian flows (studied in genus one by Krygin and FayadLemanczyk): we show the existence of ergodic infinite extensions for a full measure set of locally Hamiltonian flows with nondegenerate saddles in any genus g≥2.
The talk in based on joint work with Krzysztof Fraczek.
