We introduce two properties: strong R-property and $C(q)$ -property, describing a special way of divergence of nearby trajectories for an abstract measure-preserving system. We show that systems satisfying the strong R-property are disjoint (in the sense of Furstenberg) with systems satisfying the $C(q)$ -property. Moreover, we show that if $u_t$ is a unipotent flow on $G/\Gamma $ with $\Gamma $ irreducible, then $u_t$ satisfies the $C(q)$ -property provided that $u_t$ is not of the form $h_t\times \operatorname {id}$ , where $h_t$ is the classical horocycle flow. Finally, we show that the strong R-property holds for all (smooth) time changes of horocycle flows and non-trivial time changes of bounded-type Heisenberg nilflows.

We prove that neither a prime nor an l-almost prime number theorem holds in the class of regular Toeplitz subshifts. But when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler’s totient function is assumed, then the two theorems hold.

We prove that special flows over an ergodic rotation of the circle under a $C^{1}$ roof function with one discontinuity do not have local rank one. In particular, any such flow has infinite rank.

Let ${\mathcal{T}}=(T_{t}^{f})_{t\in \mathbb{R}}$ be a special flow built over an IET $T:\mathbb{T}\rightarrow \mathbb{T}$ of bounded type, under a roof function $f$ with symmetric logarithmic singularities at a subset of discontinuities of $T$ . We show that ${\mathcal{T}}$ satisfies the so-called switchable Ratner’s property which was introduced in Fayad and Kanigowski [On multiple mixing for a class of conservative surface flows. Invent. Math. to appear]. A consequence of this fact is that such flows are mildly mixing (before, they were only known to be weakly mixing [Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations.J. Mod. Dynam.3 (2009), 35–49] and not mixing [Ulcigrai. Absence of mixing in area-preserving flows on surfaces. Ann. of Math. (2) 173 (2011), 1743–1778]). Thus, on each compact, connected, orientable surface of genus greater than one there exist flows that are mildly mixing and not mixing.

We construct an increasing sequence of natural numbers $(m_{n})_{n=1}^{+\infty }$ with the property that $(m_{n}{\it\theta}[1])_{n\geq 1}$ is dense in $\mathbb{T}$ for any ${\it\theta}\in \mathbb{R}\setminus \mathbb{Q}$, and a continuous measure on the circle ${\it\mu}$ such that $\lim _{n\rightarrow +\infty }\int _{\mathbb{T}}\Vert m_{n}{\it\theta}\Vert \,d{\it\mu}({\it\theta})=0$. Moreover, for every fixed $k\in \mathbb{N}$, the set $\{n\in \mathbb{N}:k\nmid m_{n}\}$ 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.

We consider special flows over the rotation by an irrational $\alpha $ under the roof functions of bounded variation without continuous, singular part in the Lebesgue decomposition and sum of jumps not equal to zero. We show that all such flows are weakly mixing. Under the additional assumption that $\alpha $ has bounded partial quotients, we study the weak Ratner property. We establish this property whenever an additional condition (stable under sufficiently small perturbations) on the set of jumps is satisfied. While it is a classical result that the flows under consideration are not mixing, one more condition on the set of jumps turns out to be sufficient to obtain the absence of partial rigidity, hence mild mixing of such flows.