The absence of mixing in special flows over rearrangements of segments (Q1898519)

Let \(X= ([0, 1), \mu)\) be a Lebesgue measurable space, \(F: X\to \mathbb{R}^+\) a measurable function such that \(\int Fd\mu= 1\), and \(S\) an invertible measure-preserving transformation of \(X\). To \(F\) and \(S\) one associates a flow \((T_t)\) acting on the space \(M= \{(x, y)\in \mathbb{R}^2|x\in X, 0\leq y\leq F(x)\}\). It is known [\textit{A. V. Kochergin}, Dokl. Akad. Nauk SSSR 205, 515-518 (1972; Zbl 0262.28015)] that when \(F\) has bounded variation, the flow \((T_t)\) does not have the mixing property. In the paper under review one generalizes the mentioned result to the case when \(X\) is a union of semi-intervals and \(S\) is a rearrangement of the given set of intervals. More precisely, one defines the so-called ND-approximation of the transformation \(S\) and proves that a special flow with recurrence function \(F\) of bounded variation, constructed over a transformation with ND-approximation, does not have mixing property.