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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.17543 |
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| _version_ | 1866916610801926144 |
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| author | Cima, Anna Gasull, Armengol Mañosa, Víctor Mañosas, Francesc |
| author_facet | Cima, Anna Gasull, Armengol Mañosa, Víctor Mañosas, Francesc |
| contents | We study the dynamics of the piecewise planar rotations $F_λ(z)=λ(z-H(z)), $ with $z\in\C$, $H(z)=1$ if $\mathrm{Im}(z)\ge0,$ $H(z)=-1$ if $\mathrm{Im}(z)<0,$ and $λ=\mathrm{e}^{i α} \in\C$, being $α$ a rational multiple of $π$. Our main results establish the dynamics in the so called regular set, which is the complementary of the closure of the set formed by the preimages of the discontinuity line. We prove that any connected component of this set is open, bounded and periodic under the action of $F_λ$, with a period $\ell,$ that depends on the connected component. Furthermore, $F_λ^\ell $ restricted to each component acts as a rotation with a period which also depends on the connected component. As a consequence, any point in the regular set is periodic. Among other results, we also prove that for any connected component of the regular set, its boundary is a convex polygon with certain maximum number of sides. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_17543 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On some rational piecewise linear rotations Cima, Anna Gasull, Armengol Mañosa, Víctor Mañosas, Francesc Dynamical Systems 37C25, 39A23, 37B10 We study the dynamics of the piecewise planar rotations $F_λ(z)=λ(z-H(z)), $ with $z\in\C$, $H(z)=1$ if $\mathrm{Im}(z)\ge0,$ $H(z)=-1$ if $\mathrm{Im}(z)<0,$ and $λ=\mathrm{e}^{i α} \in\C$, being $α$ a rational multiple of $π$. Our main results establish the dynamics in the so called regular set, which is the complementary of the closure of the set formed by the preimages of the discontinuity line. We prove that any connected component of this set is open, bounded and periodic under the action of $F_λ$, with a period $\ell,$ that depends on the connected component. Furthermore, $F_λ^\ell $ restricted to each component acts as a rotation with a period which also depends on the connected component. As a consequence, any point in the regular set is periodic. Among other results, we also prove that for any connected component of the regular set, its boundary is a convex polygon with certain maximum number of sides. |
| title | On some rational piecewise linear rotations |
| topic | Dynamical Systems 37C25, 39A23, 37B10 |
| url | https://arxiv.org/abs/2306.17543 |