We describe the infinite interval exchange transformations obtained as a composition of a finite interval exchange transformation and the von Neumann-Kakutani map, called the rotated odometers. We show that with respect to Lebesgue measure on the unit interval, every such transformation is measurably isomorphic to the first return map of a rational parallel flow on a translation surface of finite area with infinite genus and a finite number of ends. We describe the dynamics of rotated odometers by means of Bratteli-Vershik systems, and derive several of their topological and ergodic properties. In particular, we show that every rotated odometer has a unique minimal subsystem, and that there exist rotated odometers whose minimal subsystem does not factor onto the dyadic odometer.
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