Rotation on a compact abelian group

A compact abelian rotation is the probability-preserving system

R_a:K\to K,\qquad R_a(x)=x+a,

where $K$ is a compact metrizable abelian group with normalized Haar measure and $a\in K$ .

Characters and ergodicity

For a character $\chi$ , forward pullback gives $U_{R_a}\chi=\chi(a)\chi$ . Since characters form a Fourier basis, the rotation is ergodic exactly when $\chi(a)\ne1$ for every nontrivial character. Equivalently, the cyclic subgroup generated by $a$ is dense in $K$ .

Group actions by translations

More generally, translations by elements of a subgroup $H\subseteq K$ act ergodically exactly when $\overline H=K$ , or equivalently when no nontrivial character is one on every element of $H$ . This is a joint condition on the translating subgroup.