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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.14055 |
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| _version_ | 1866909228080300032 |
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| author | Hoseana, Jonathan |
| author_facet | Hoseana, Jonathan |
| contents | We study the dynamics of the one-dimensional quasi-affine map $x\mapsto \left\lfloor λx +μ\right\rfloor$, providing a complete description of the map's periodic points, and of the limit points of every $x\in\mathbb{R}$ under the map, for all real parameter values. Specifically, we establish the existence of regions of parameter values for which the map possesses $n$ fixed points for all $n\in\mathbb{N}_0\cup \{\infty\}$, an explicit formula for the number of 2-cycles possessed by the map, and the $ω$-limit set of any $x\in\mathbb{R}$ under the map, which, depending on the parameter values, is either a singleton of a fixed point, a 2-cycle, $\{-\infty,\infty\}$, $\{\infty\}$, or $\{-\infty\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14055 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Dynamics of One-Dimensional Quasi-Affine Maps Hoseana, Jonathan Dynamical Systems 37E99, 37C25 We study the dynamics of the one-dimensional quasi-affine map $x\mapsto \left\lfloor λx +μ\right\rfloor$, providing a complete description of the map's periodic points, and of the limit points of every $x\in\mathbb{R}$ under the map, for all real parameter values. Specifically, we establish the existence of regions of parameter values for which the map possesses $n$ fixed points for all $n\in\mathbb{N}_0\cup \{\infty\}$, an explicit formula for the number of 2-cycles possessed by the map, and the $ω$-limit set of any $x\in\mathbb{R}$ under the map, which, depending on the parameter values, is either a singleton of a fixed point, a 2-cycle, $\{-\infty,\infty\}$, $\{\infty\}$, or $\{-\infty\}$. |
| title | The Dynamics of One-Dimensional Quasi-Affine Maps |
| topic | Dynamical Systems 37E99, 37C25 |
| url | https://arxiv.org/abs/2406.14055 |