Real analytic nonexpansive maps on polyhedral normed spaces
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914917949374464 |
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| author | Lins, Brian |
| author_facet | Lins, Brian |
| contents | If a real analytic nonexpansive map on a polyhedral normed space has a nonempty fixed point set, then we show that there is an isometry from an affine subspace onto the fixed point set. As a corollary, we prove that for any real analytic 1-norm or $\infty$-norm nonexpansive map on $\mathbb{R}^n$, there is a positive integer $q$ such that the period of any periodic orbit divides $q$ and $q$ is the order, or twice the order, of a permutation on $n$ letters. This confirms Nussbaum's $2^n$ Conjecture for $\infty$-norm nonexpansive maps in the special case where the maps are also real analytic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16671 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Real analytic nonexpansive maps on polyhedral normed spaces Lins, Brian Dynamical Systems Functional Analysis Primary 47H09, 47H10, Secondary 39A23, 46T20 If a real analytic nonexpansive map on a polyhedral normed space has a nonempty fixed point set, then we show that there is an isometry from an affine subspace onto the fixed point set. As a corollary, we prove that for any real analytic 1-norm or $\infty$-norm nonexpansive map on $\mathbb{R}^n$, there is a positive integer $q$ such that the period of any periodic orbit divides $q$ and $q$ is the order, or twice the order, of a permutation on $n$ letters. This confirms Nussbaum's $2^n$ Conjecture for $\infty$-norm nonexpansive maps in the special case where the maps are also real analytic. |
| title | Real analytic nonexpansive maps on polyhedral normed spaces |
| topic | Dynamical Systems Functional Analysis Primary 47H09, 47H10, Secondary 39A23, 46T20 |
| url | https://arxiv.org/abs/2407.16671 |