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Ratner's property for special flows over irrational rotations under functions of bounded variation

We consider special flows over the rotation by an irrational $α$ under the roof functions of bounded variation without continuous, singular part in the Lebesgue decomposition and the sum of jumps $\neq 0$. We show that all such flows are weakly mixing. Under the additional assumption that $α$ has bounded partial quotients, we study weak Ratner's property. We establish this property whenever an additional condition (stable under sufficiently small perturbations) on the set of jumps is satisfied. While it is classical 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.