| Résume | Consider a smooth area-preserving -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 Avila-Viana for a large class of regular observables. We will present a new proof which allows to consider also the case of observables which are non-zero at the saddle points (based on 'correction' operators a' la Marmi-Moussa-Yoccoz for functions with logarithmic singularities over IETs). The result has an application to the infinite ergodic theory of R-extensions of locally Hamiltonian flows (studied in genus one by Krygin and Fayad-Lemanczyk): we show the existence of ergodic infinite extensions for a full measure set of locally Hamiltonian flows with non-degenerate saddles in any genus g≥2.
The talk in based on joint work with Krzysztof Fraczek.
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