Remetrizing dynamical systems to control distances of points in time

TL;DR

This paper proves that any continuous dynamical system on a metrizable space can be remetrized to control the growth of distances between points under iteration, allowing for tailored Lipschitz properties.

Contribution

It introduces a method to construct compatible metrics that make iterates of functions Lipschitz with arbitrarily increasing constants, enhancing control over dynamical distances.

Findings

Existence of compatible metrics with prescribed Lipschitz growth

Ability to remetrize any dynamical system for distance control

Enhanced analysis of dynamical behavior through remetrization

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.