Exercise 29, § 3

Let $X$ be a compact space and let $G$ be a topological group operating continuously on $X$, such that every orbit of $G$ is dense in $X$. Show that if, for every continuous real-valued function $f$ on $X$, there exists $x_0 \in X$ such that $s \mapsto f(s.x_0)$ is almost periodic on $G$, then $G$ is equicontinuous (use the fact that X is homeomorphic to a closed subspace of a cube). Consider the converse, when G is abelian.