Remetrizing dynamical systems to control distances of points in time

The main aim of this article is to prove that for any continuous function $f\text{\hspace{0.17em}⁣}:X\to X$, where $X$ is metrizable (or, more generally, for any family $\mathcal{F}$ of such functions, satisfying an additional condition), there exists a compatible metric $d$ on $X$ such that the $n$th iteration of $f$ (more generally, the composition of any $n$ functions from $\mathcal{F}$) is Lipschitz with constant $a_k$ where $(a_k{)}_{k=1}^{\mathrm{\infty }}$ is an arbitrarily fixed sequence of real numbers such that $1<a_k$ and $\underset{k\to +\mathrm{\infty }}{\lim }{a}_k=+\mathrm{\infty }$. In particular, any dynamical system can be remetrized in order to significantly control the distance between points by their initial distance.