On disjointness properties of some smooth flows
Special flows over some locally rigid automorphisms and under $L^2$ ceiling functions satisfying a local $L^2$ Denjoy–Koksma type inequality are considered. Such flows are proved to be disjoint (in the sense of Furstenberg) from mixing flows and (under some stronger assumption) from weakly mixing flows for which the weak closure of the set of all instances consists of indecomposable Markov operators. As applications we prove that
$\bullet$ special flows built over ergodic interval exchange transformations and under functions of bounded variation are disjoint from mixing flows;
$\bullet$ ergodic components of flows coming from billiards on rational polygons are disjoint from mixing flows;$\bullet$ smooth ergodic flows of compact orientable smooth surfaces having only non-degenerate saddles as isolated critical points (and having a “good” transversal) are disjoint from mixing and from Gaussian flows.