Exercise 27, § 3

Let $X$ be a Hausdorff uniform space and let $G$ be an equicontinuous group of homeomorphisms of $X$.

a) A point $x_0 \in X$ is said to be almost periodic with respect to $G$ if the orbit of $x_0$ with respect to $G$ is relatively compact in $X$. Let $Y$ be the (compact) closure of this orbit in $X$; then $u(Y) \subset Y$ for all $u \in G$. Let $\tilde{u}$ be the restriction of $u$ to $Y$ considered as an element of $\mathscr{C}(Y; Y)$; show that $\tilde{u}$ is a homeomorphism of $Y$ onto itself and that the closure $\Gamma$ in $\mathscr{C}_u(Y; Y)$ of the image of $G$ under the mapping $u \to \tilde{u}$ is a compact group of homeomorphisms (no. 6, Theorem 4, Corollary) which is transitive on $Y$.

b) Suppose that X is complete. Then $x_0$ is almost periodic with respect to G if and only if, for each neighbourhood V of $x_0$ in X, there is a finite number of elements $u_i \in G$ such that, for all $u \in G$, we have $u_i^{-1}(u(x_0)) \in V$ for at least one index i.

c) Suppose that $x_0$ is almost periodic and that G is endowed with a topology compatible with its group structure. With the notation of a), the mapping $u \to \bar{u}$ of G into $\Gamma$ (endowed with the topology of uniform convergence) is continuous if and only if, for each pair of distinct points $x, y$ of Y, there is a neighbourhood U of e in G and a neighbourhood V of y such that the relation $u \in U$ implies $u(x) \notin V$ [use the fact that the topologies induced on $\Gamma$ by those of $C_u(Y; Y)$ and $C_s(Y; Y)$ are the same].