A pointwise limit of maps uniformly close to phase shifts of a periodic map is a phase shift

Let $F$ be a normed group, $T>0$, and let $x:\mathbb{R}\to F$ be continuous and $T$-periodic. Fix $\kappa\in\mathbb{R}$, and let $f_j:\mathbb{R}\to F$ converge pointwise to $L:\mathbb{R}\to F$. Assume that for every $\varepsilon>0$ and all sufficiently large $j$ there is a phase $c_j\in\mathbb{R}$ with

Then there is $c\in\mathbb{R}$ with $L(t)=x(\kappa t+c)$ for all $t\in\mathbb{R}$.

Typical use: $f_j$ are loops on a contact manifold that converge to a limit loop and approach the reparametrizations of a fixed periodic Reeb orbit with a phase depending on $j$; the limit loop is then an exact reparametrization of that orbit.

Formalization Note Periodicity is Function.Periodic x T; pointwise convergence is Tendsto (fun j => f j t) atTop (𝓝 (L t)) for each t; the supremum is written pointwise for all t.