A reciprocal transformation is the composition of a measure-preserving map with a scaling involution that swaps two unequal-measure sets, yielding a non-singular map whose measure distortion is a single ratio. Its discrete Maharam extension is an infinite measure-preserving map on X x Z that cancels the rescaling by shifting levels. Students learn how conservativity and ergodicity of the reciprocal map are studied through its first-return map, and that the extension is ergodic exactly when the map is ergodic of Krieger type III with parameter one over the scaling ratio.
Two abstract constructions in ergodic theory. A 'reciprocal transformation' is the composition F = ΦT of a measure-preserving transformation T with a 'scaling involution' Φ that swaps two sets of une…