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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2412.03711 |
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| _version_ | 1866913858966257664 |
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| author | Gołębiowski, Krzysztof |
| author_facet | Gołębiowski, Krzysztof |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03711 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remetrizing dynamical systems to control distances of points in time Gołębiowski, Krzysztof General Topology 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. |
| title | Remetrizing dynamical systems to control distances of points in time |
| topic | General Topology |
| url | https://arxiv.org/abs/2412.03711 |