Generic uniformly continuous mappings on unbounded hyperbolic spaces
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913416690532352 |
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| author | Ravasini, Davide |
| author_facet | Ravasini, Davide |
| contents | We consider a complete, unbounded, hyperbolic metric space $X$ and a concave, nonzero and nondecreasing function $ω:[0,+\infty)\to[0,+\infty)$ with $ω(0)=0$ and study the space $\mathcal{C}_ω(X)$ of uniformly continous self-mappings on $X$ whose modulus of continuity is bounded above by $ω$. We endow $\mathcal{C}_ω(X)$ with the topology of uniform convergence on bounded sets and prove that the modulus of continuity of the generic mapping in $\mathcal{C}_ω(X)$, in the sense of Baire categories, is precisely $ω$. Some related results in spaces of bounded mappings and in the topology of pointwise convergence are also discussed. This note can be seen as a completion of various results due to F. Strobin, S. Reich, A. Zaslavski, C. Bargetz and D. Thimm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_15277 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generic uniformly continuous mappings on unbounded hyperbolic spaces Ravasini, Davide Functional Analysis General Topology Metric Geometry 54E50, 54E52 We consider a complete, unbounded, hyperbolic metric space $X$ and a concave, nonzero and nondecreasing function $ω:[0,+\infty)\to[0,+\infty)$ with $ω(0)=0$ and study the space $\mathcal{C}_ω(X)$ of uniformly continous self-mappings on $X$ whose modulus of continuity is bounded above by $ω$. We endow $\mathcal{C}_ω(X)$ with the topology of uniform convergence on bounded sets and prove that the modulus of continuity of the generic mapping in $\mathcal{C}_ω(X)$, in the sense of Baire categories, is precisely $ω$. Some related results in spaces of bounded mappings and in the topology of pointwise convergence are also discussed. This note can be seen as a completion of various results due to F. Strobin, S. Reich, A. Zaslavski, C. Bargetz and D. Thimm. |
| title | Generic uniformly continuous mappings on unbounded hyperbolic spaces |
| topic | Functional Analysis General Topology Metric Geometry 54E50, 54E52 |
| url | https://arxiv.org/abs/2308.15277 |