The almost-periodic homeomorphism conjecture

Let $X$ be a connected manifold, let $\cl G$ be a closed subgroup of $\operatorname{Homeo}(X)$, and let $f\in\cl G$ be almost periodic, meaning that the subgroup $\langle f\rangle=\{f^k\mid k\in\mathbb{Z}\}$ has compact closure. Almost-periodic homeomorphism conjecture. There exists $q\in\mathbb{N}$ such that $f^q$ is contained in the image of a homomorphism $\mathbb{R}\to\cl G$. The paper presents this as a consequence of the validity of the Hilbert–Smith conjecture, rather than as an independently resolved result.