Minsung Kim: New phenomena on the deviation spectrum of Birkhoff integrals for locally Hamiltonian flows

Abstract: The deviation of the Birkhoff integral for area-preserving flow on compact surfaces was first studied by Forni. He proved that the deviation spectrum was determined by the Lyapunov exponents of a renormalization cocycle so-called Kontsevitch-Zorich cocycle. This deviation result was later proved again by Bufetov and Frączek-Ulcigrai for translation flows and locally Hamiltonian flows for non-degenerate types.

In this talk, we study the spectrum for deviations of Birkhoff integrals for locally Hamiltonian flows beyond the case of Forni where the observable vanishes at the singularities. Our new developments include a better understanding of the asymptotics at singularities (degenerate type) and the appearance of a new deviation spectrum. This is a joint work with Krzysztof Frączek.