Realizations of rotations on an indecomposable compact monothetic group

By the classical Halmos‐von Neumann theorem, each compact monothetic group corresponds to an ergodic dynamical system with discrete spectrum. For such groups we prove two results. We first construct a compact monothetic group which does not split into a direct product of a connected and a totally disconnected compact monothetic group. Then we present a measure preserving dynamical system on the unit square being isomorphic to a rotation on this indecomposable group.