Remetrizing dynamical systems to control distances of points in time
Krzysztof Gołębiowski
TL;DR
The paper addresses controlling the evolution of distances under dynamical systems on metrizable spaces by remetrizing the space. It introduces moduli of continuity and finite equicontinuity to formalize when a family of maps can be remetrized so that all iterates (and compositions) have prescribed Lipschitz or continuity growth, bounded by a sequence {a_k} with 1 < a_k and a_k → ∞. The main theoretical contribution is a general remetrization theorem (Theorem 'glowne_twierdzenie') showing how to construct a compatible metric ensuring uniform control over all F^n via associated moduli ω_n, including a submultiplicative sequence {b_n} and a metric hat{d} that realize these bounds; the result recovers the log-growth bound Lip(f^n) ≤ log(n+2) for a single map and extends to groups generated by finitely many transformations. These findings provide a new toolset for analyzing dynamical systems on noncompact spaces by tuning distances to manage orbit divergence, mixing, and equicontinuity properties.
Abstract
The main aim of this article is to prove that for any continuous function $f \colon X \to X$, 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 $(a_k)_{k=1}^{\infty}$ is an arbitrarily fixed sequence of real numbers such that $1 < a_k$ and $\lim\limits_{k\to+\infty}a_k = +\infty$. In particular, any dynamical system can be remetrized in order to significantly control the distance between points by their initial distance.