Absence of mixing in area-preserving flows on surfaces

Abstract

We prove that minimal area-preserving flows locally given by a smooth Hamiltonian on a closed surface of genus $g\geq2$ are typically (in the measure-theoretical sense) not mixing. The result is obtained by considering special flows over interval exchange transformations under roof functions with symmetric logarithmic singularities and proving absence of mixing for a full measure set of interval exchange transformations. As a corollary, minimal flows given by multi-valued Hamiltonians on higher genus surfaces which are minimal and have only simple non-degenerate saddles are typically not mixing.