On the representation of measurable and continuous dynamical systems by Lipschitz functions
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911559010222080 |
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| author | Gutman, Yonatan Huo, Qiang |
| author_facet | Gutman, Yonatan Huo, Qiang |
| contents | Two representations theorems are presented:
1. Any Borel action of a second countable locally compact group $G$ on a standard Borel space $X$ admits an injective $G$-equivariant Borel map into the shift space of $1$-Lipschitz functions from $G$ to the unit interval $Lip_1(G)$.
2. Any continuous action of $\mathbb{R}^k$ ($k\in \mathbb{N}$) on a metrizable compact space $X$ admits an injective $G$-equivariant continuous map into $Lip_1(\mathbb{R}^k)$ if the fixed point set $Fix(X,\mathbb{R}^k)$ embeds into $[0,1]$ and $(X,\mathbb{R}^k)$ is \textit{weakly locally free}, that is $\mathbb{R}^k$ acts freely outside the
fixed point set.
The first theorem generalizes a theorem from 1973 by Eberlein for $\mathbb{R}$-flows. The second theorem generalizes a Lipschitz refinement of the Bebutov-Kakutani theorem proven by Gutman, Jin and Tsukamoto in 2019. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_14653 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the representation of measurable and continuous dynamical systems by Lipschitz functions Gutman, Yonatan Huo, Qiang Dynamical Systems 37A05, 37B05, 54H20 Two representations theorems are presented: 1. Any Borel action of a second countable locally compact group $G$ on a standard Borel space $X$ admits an injective $G$-equivariant Borel map into the shift space of $1$-Lipschitz functions from $G$ to the unit interval $Lip_1(G)$. 2. Any continuous action of $\mathbb{R}^k$ ($k\in \mathbb{N}$) on a metrizable compact space $X$ admits an injective $G$-equivariant continuous map into $Lip_1(\mathbb{R}^k)$ if the fixed point set $Fix(X,\mathbb{R}^k)$ embeds into $[0,1]$ and $(X,\mathbb{R}^k)$ is \textit{weakly locally free}, that is $\mathbb{R}^k$ acts freely outside the fixed point set. The first theorem generalizes a theorem from 1973 by Eberlein for $\mathbb{R}$-flows. The second theorem generalizes a Lipschitz refinement of the Bebutov-Kakutani theorem proven by Gutman, Jin and Tsukamoto in 2019. |
| title | On the representation of measurable and continuous dynamical systems by Lipschitz functions |
| topic | Dynamical Systems 37A05, 37B05, 54H20 |
| url | https://arxiv.org/abs/2505.14653 |