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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2210.11412 |
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| _version_ | 1866913022676566016 |
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| author | Idrissi, Nizar El Kabbaj, Samir |
| author_facet | Idrissi, Nizar El Kabbaj, Samir |
| contents | A relaxation of the notion of invariant set, known as $k$-quasi-invariant set, has appeared several times in the literature in relation to group dynamics. The results obtained in this context depend on the fact that the dynamic is generated by a group. In our work, we consider the notions of invariant and 1-internally-quasi-invariant sets as applied to an action of a function $f$ on a set $I$. We answer several questions of the following type, where $k \in \{0,1\}$: what are the functions $f$ for which every finite subset of $I$ is internally-$k$-quasi-invariant? More restrictively, if $I = \mathbb{N}$, what are the functions $f$ for which every finite interval of $I$ is internally-$k$-quasi-invariant? Last, what are the functions $f$ for which every finite subset of $I$ admits a finite internally-$k$-quasi-invariant superset? This parallels a similar investigation undertaken by C. E. Praeger in the context of group actions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_11412 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Functions with a maximal number of finite invariant or internally-1-quasi-invariant sets or supersets Idrissi, Nizar El Kabbaj, Samir Dynamical Systems Combinatorics 37D10 A relaxation of the notion of invariant set, known as $k$-quasi-invariant set, has appeared several times in the literature in relation to group dynamics. The results obtained in this context depend on the fact that the dynamic is generated by a group. In our work, we consider the notions of invariant and 1-internally-quasi-invariant sets as applied to an action of a function $f$ on a set $I$. We answer several questions of the following type, where $k \in \{0,1\}$: what are the functions $f$ for which every finite subset of $I$ is internally-$k$-quasi-invariant? More restrictively, if $I = \mathbb{N}$, what are the functions $f$ for which every finite interval of $I$ is internally-$k$-quasi-invariant? Last, what are the functions $f$ for which every finite subset of $I$ admits a finite internally-$k$-quasi-invariant superset? This parallels a similar investigation undertaken by C. E. Praeger in the context of group actions. |
| title | Functions with a maximal number of finite invariant or internally-1-quasi-invariant sets or supersets |
| topic | Dynamical Systems Combinatorics 37D10 |
| url | https://arxiv.org/abs/2210.11412 |