Let $(X,d)$ be a (bounded, complete, even polish) metric space, and let $G$ be the group of isometries of $(X,d)$ with the uniform metric $d_u$ defined by $d_u(f,g):=\sup_{x\in X}d(f(x),g(x))$ for all $f$ and $g$ in $G$.
For a fixed positive integer $n$, consider the map $G\rightarrow G$ given by $f\mapsto f^n$ ($f$ composed with itself $n$ times). Is this map uniformly continuous? Is it Lipschitz continuous? In that case, what is a constant?
Since $G$ is a topological group, it is easy to see it is continuous. But i haven't managed to find uniform continuity, and I'm starting to doubt whether it is actually uniformly continuous.
I would appreciate any insight.