Arnold and Kochergin mixing conservative flows on surfaces stand as the main and almost only natural class of mixing transformations for which higher order mixing has not been established, nor disproved. Under suitable arithmetic conditions on their unique rotation vector, of full Lebesgue measure in the first case and of full Hausdorff dimension in the second, we show that these flows are mixing of any order. For this, we show that they display a generalization of the so called Ratner property on slow divergence of nearby orbits, that implies strong restrictions on their joinings, which in turn yields higher order mixing. This is the first case in which the Ratner property is used to prove multiple mixing outside its original context of horocycle flows and we expect our approach will have further applications.

Let T = (T f t ) t∈R be a special flow built over an IET T : T → T of bounded type, under a roof function f with symmetric logarithmic singularities at a subset of discontinuities of T . We show that T satisfies so-called switchable Ratner's property which was introduced in [4]. A consequence of this fact is that such flows are mildly mixing (before, they were only known to be weakly mixing [31] and not mixing [32]). Thus, on each compact, connected, orientable surface of genus greater than one there exist flows which are mildly mixing and not mixing.

Let (X, T ) be a topological dynamical system. We show that if all invariant measures of (X, T ) give rise to measure theoretic dynamical system that are rigid then (X, T ) satisfies Sarnak's conjecture on Möbius disjointness. We show that the same conclusion also holds if there are countably many invariant ergodic measures, and they all give rise to rigid measure theoretic dynamical systems. This recovers several earlier results and immediately implies Sarnak's conjecture in the following new cases: for almost every interval exchange map of d intervals with d ≥ 2 and almost all translation flows, for all 3-interval exchange maps, and for absolutely continuous skew products over rotations. The latter two are improvement of earlier results of respectively Chaika-Eskin, Wang and Huang-Wang-Ye. We also discuss some purely arithmetic consequences for the Liouville function.

We construct an increasing sequence of natural numbers (m n ) +∞ n=1 with the property that (m n θ[1]) n 1 is dense in T for any θ ∈ R \ Q, and a continuous measure on the circle µ such that lim n→+∞ T m n θ dµ(θ) = 0. Moreover, for every fixed k ∈ N, the set {n ∈ N : k ∤ m n } is infinite. This is a sufficient condition for the existence of a rigid, weakly mixing dynamical system whose rigidity time is not a rigidity time for any system with a discrete part in its spectrum.