This paper solves an open question by

*A. Katok* and

*J.-P. Thouvenot*; see, for example, Section 6.3.2 of [“Spectral properties and combinatorial constructions in ergodic theory”, in: Handbook of dynamical systems. Volume 1B. Amsterdam: Elsevier. 649–743 (2006;

Zbl 1130.37304)]. The statement of the main result is the following: Let

$S$ be a closed surface of genus

$g\ge 2$ and let

${\Phi}:\mathbb{R}\times S\to S$ be a flow given by a multi-valued Hamiltonian associated to a smooth closed differential 1-form

$\eta $. If

${\Phi}$ has only simple saddles and no saddle loops homologous to zero then

${\Phi}$ is not mixing for a typical such form

$\eta $.