Theorem. Let $G$ compact group and $T(x)=ax$. Then $T$ is ergodic iff $\{a^n\}_{n=-\infty}^\infty$ is dense in $G$. In particular, if $T$ is ergodic, then $G$ is abelian.
This is one direction of Theorem 1.9 in Peter Walters' An Introduction to Ergodic Theory and as far as I can tell it contains a mistake. The direction that I don't understand goes:
Suppose $T$ is ergodic. Let $H$ be the closure of the orbit of $a$. If $H \neq G$ then there exists a character $\hat{\gamma} \in \hat{G}$ with $\gamma=1$ on $H$ but $\gamma$ is not constant $1$. Then $\gamma(Tx) = \gamma(ax) = \gamma(a)\gamma(x) = \gamma(x)$ so $\gamma$ is not a.s. constant, contradicting ergodicity of $T$.
My problem with this is that Walters only defines $\hat{G}$ for abelian $G$. So what has he left out that is needed for this proof to go through? He clearly isn't assuming $G$ is abelian because that is one of the conclusions.