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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2506.14027 |
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| _version_ | 1866911007753895936 |
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| author | Ciavattini, Filippo Steele, T. H. |
| author_facet | Ciavattini, Filippo Steele, T. H. |
| contents | Let $\mathcal{M}$ be a compact metric space with no isolated points, and $f:\mathcal{M}\longrightarrow\mathcal{M}$ a homeomorphism. Consider a sequence of shrinking open balls $\{B^i_n\}_{n\in\mathbb{N}}^{i\in\mathbb{N}}$ with centers $\{p_i\}_{i=1}^\infty\subseteq\mathcal{M}$ and radii $\{ρ^i_n\}_{n=1}^\infty$. For every point $x\in\mathcal{M}$ and $n\in\mathbb{N}$, consider which ball the trajectory $\{x,f(x),f^2(x),\dots\}$ of the point first visits. We find that whenever the closure of $\{p_i\}_{i=1}^\infty$ is nowhere dense, and with very minor restrictions on $\{ρ_n^i\}_{n\in\mathbb{N}}^{i\in\mathbb{N}}$, the typical trajectory $\{f^k(x)\}_{k=0}^\infty$ will first visit, for each $i$, the ball $B^i_n$, for infinitely many $n$. This is never the case, should $\{p_i\}_{i=1}^\infty$ be somewhere dense. Keywords: Open Dynamical System, Topological Dynamics, Transitive Homeomorphism, Baire category. MSC2020: 37B05, 37B20, 18F60, 54E52. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14027 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nowhere dense competing holes in open dynamical systems Ciavattini, Filippo Steele, T. H. Dynamical Systems Let $\mathcal{M}$ be a compact metric space with no isolated points, and $f:\mathcal{M}\longrightarrow\mathcal{M}$ a homeomorphism. Consider a sequence of shrinking open balls $\{B^i_n\}_{n\in\mathbb{N}}^{i\in\mathbb{N}}$ with centers $\{p_i\}_{i=1}^\infty\subseteq\mathcal{M}$ and radii $\{ρ^i_n\}_{n=1}^\infty$. For every point $x\in\mathcal{M}$ and $n\in\mathbb{N}$, consider which ball the trajectory $\{x,f(x),f^2(x),\dots\}$ of the point first visits. We find that whenever the closure of $\{p_i\}_{i=1}^\infty$ is nowhere dense, and with very minor restrictions on $\{ρ_n^i\}_{n\in\mathbb{N}}^{i\in\mathbb{N}}$, the typical trajectory $\{f^k(x)\}_{k=0}^\infty$ will first visit, for each $i$, the ball $B^i_n$, for infinitely many $n$. This is never the case, should $\{p_i\}_{i=1}^\infty$ be somewhere dense. Keywords: Open Dynamical System, Topological Dynamics, Transitive Homeomorphism, Baire category. MSC2020: 37B05, 37B20, 18F60, 54E52. |
| title | Nowhere dense competing holes in open dynamical systems |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2506.14027 |