Deviation spectrum of ergodic integrals for locally Hamiltonian flows on surfaces

The talk will consists of a historical introduction to the topic of deviation of ergodic averages for locally Hamiltonian flows on compact surafces as well as some current results obtained in collaboration mainly with Corinna Ulcigrai and Minsung Kim. New developments include a better understanding of the asymptotic of so-called error term (in non-degenerate regime) and the appearance of new exponents in the deviation spectrum (in degenerate regime).