Interval Exchange Transformations with Vanishing Sah–Arnoux–Fathi Invariant

The paper is focused on the key ergodic properties (minimality, unique ergodicity, and weak mixing) of interval exchange transformations (IETs) with vanishing Sah–Arnoux–Fathi (SAF) invariant. The lengths of intervals of this kind of IETs satisfy some nontrivial rational relations; therefore, one cannot apply the classical results by M. Keane, H. Masur, W. Veech, and others to examine their ergodic properties, so dealing with such maps requires the use of new tools. In the present paper we apply some advanced methods of symbolic dynamics to construct the first examples of weakly mixing pseudo-Anosov foliations on orientable surfaces such that the SAF invariant of the corresponding IETs vanishes. The paper also contains a short survey of the known results and presents the statements of several open problems and challenging conjectures.