Notes on Minimality and Ergodicity of Compact Abelian Group Extensions of Dynamics

W. Parry [5] introduced the notion of a G-extension of a topological dynamics, where G is a compact abeliaii group, and gave necessary and sufficient conditions for a G-exteiisioii of a minimal (respectively uniquely ergodic) topological dynamics to be minimal (uniquely ergodic). In the first part of this paper a proof of the Minimality Theorem of W. Parry without his "simple free" condition is given. In the purely measuretheoretic case W. Parry [6] introduced the notion of G-extension of type fi, where tf" is an automorphism of G, and spectrally analysed it. In the second part of this paper a necessary and sufficient condition for a Gextension of an ergodic measure-preserving dynamics to be ergodic is shown. As particular cases of this result we have well-known necessary and sufficient conditions for a translation, a group-automorphism and an affine transformation on a compact group to be ergodic.