New Phenomena in Deviation of Birkhoff Integrals for Locally Hamiltonian Flows

The uploaded paper proves a full deviation spectrum for ergodic integrals of locally Hamiltonian flows on a minimal component: a decomposition into global Bufetov–Forni terms ui(T,x) tied to the positive KZ exponents and new local cocycles c_{σ,α}(T,x) tied to jets at saddles, with sharp exponents b(σ,k)=(m_σ−2−k)/m_σ and sub-polynomial remainders, both pointwise and in L1 (see Theorem 1.1 and (1.3)–(1.12) in the PDF ). The paper’s strategy reduces ergodic integrals to Birkhoff sums via a special-flow representation over IETs, introduces new Banach spaces P_a for polynomial singularities, imposes a Filtration Diophantine Condition (FDC) that holds for a.e. IET, and constructs correction operators to neutralize KZ-growth before obtaining upper and lower bounds (Sections 2–7, 9–10 ). The candidate solution outlines the same architecture: special-flow/IET reduction, a regular/singular splitting, Forni–Bufetov expansions for the regular part, local jet analysis at saddles yielding exponents b(σ,k), and renormalization-based upper/lower bounds with sub-polynomial error. Minor gaps are that it does not name the paper’s FDC explicitly and sketches a “linear elimination” of global terms instead of the paper’s correction operator R_{σ,k}, but these do not change the core argument. Overall, both reach the same result with substantially the same proof strategy.