This music score was submitted for the Kaleidoscope 2020 Call for Scores, an open access collaboration with the UCLA Music Library.

This music score was submitted for the Kaleidoscope 2020 Call for Scores, an open access collaboration with the UCLA Music Library.

This music score was submitted for the Kaleidoscope 2020 Call for Scores, an open access collaboration with the UCLA Music Library.

In this paper we give explicit characterizations, based on the cutting and spacer parameters, of (a) which rank-one transformations factor onto a given finite cyclic permutation, (b) which rank-one transformations factor onto a given odometer, and (c) which rank-one transformations are isomorphic to a given odometer. These naturally yield characterizations of (d) which rank-one transformations factor onto some (unspecified) finite cyclic permutation, (d′) which rank-one transformations are totally ergodic, (e) which rank-one transformations factor onto some (unspecified) odometer, and (f) which rank-one transformations are isomorphic to some (unspecified) odometer.