New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows

Abstract

Abstract We consider smooth locally Hamiltonian flows on compact surfaces of genus g ≥ 1 {g\geq 1} to prove a deviation formula of Birkhoff integrals for smooth observables. Our work generalizes results of [G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 2002, 1, 1–103] and [A. I. Bufetov, Limit theorems for translation flows, Ann. of Math. (2) 179 2014, 2, 431–499] (and a recent result in [K. Frączek and C. Ulcigrai, On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions, preprint 2021]) which prove the existence of the deviation spectrum of Birkhoff integrals for observables whose jets vanish at sufficiently high order around fixed points of the flow. They showed that ergodic integrals display a power deviation spectrum with exactly g positive exponents related to the positive Lyapunov exponents of the cocycle so-called Kontsevich–Zorich, a renormalization cocycle over the Teichmüller flow on a stratum of the moduli space of translation surfaces. Our paper extends the study of the deviation spectrum of ergodic integrals beyond the case of observables whose jets vanish at sufficiently high order around fixed points. We prove the existence of some extra terms in the deviation spectrum related to the non-vanishing of some derivatives of the observable at fixed points. The proof of this new phenomenon is inspired by tools developed in the recent work of the first author and Ulcigrai for locally Hamiltonian flows having only (simple) non-degenerate saddles. In full generality, due to the occurrence of (multiple) degenerate saddles, we introduce new methods of handling functions with polynomial singularities over a full measure set of interval exchange transformations.