Asymptotic Homotopical Complexity of an Infinite Sequence of Dispersing $2D$ Billiards
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912452442062848 |
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| author | Simanyi, Nandor |
| author_facet | Simanyi, Nandor |
| contents | We investigate the large scale chaotic, topological structure of the trajectories of an infinite sequence of dispersing, hence ergodic, $2D$ billiards with the configuration space $Q_n=\mathbb{T}^2 \setminus \bigcup_{i=0}^{n-1} D_i$, where the scatterers $D_i$ ($i=0,1,\dots,n-1$) are disks of radius $r<<1$ centered at the points $(i/n, 0)$ mod $\mathbb{Z}^2$. We get effective lower and upper radial bounds for the rotation set $R$. Furthermore, we also prove the compactness of the admissible rotation set $AR$ and the fact that the rotation vectors $v$ corresponding to admissible periodic orbits form a dense subset of $AR$. We also obtain asymptotic lower and upper estimates for the sequence $h_{top}(n)$ of topological entropies and precise asymptotic formulas for the metric entropies $h_μ(n,r)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16284 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic Homotopical Complexity of an Infinite Sequence of Dispersing $2D$ Billiards Simanyi, Nandor Dynamical Systems 37D05 We investigate the large scale chaotic, topological structure of the trajectories of an infinite sequence of dispersing, hence ergodic, $2D$ billiards with the configuration space $Q_n=\mathbb{T}^2 \setminus \bigcup_{i=0}^{n-1} D_i$, where the scatterers $D_i$ ($i=0,1,\dots,n-1$) are disks of radius $r<<1$ centered at the points $(i/n, 0)$ mod $\mathbb{Z}^2$. We get effective lower and upper radial bounds for the rotation set $R$. Furthermore, we also prove the compactness of the admissible rotation set $AR$ and the fact that the rotation vectors $v$ corresponding to admissible periodic orbits form a dense subset of $AR$. We also obtain asymptotic lower and upper estimates for the sequence $h_{top}(n)$ of topological entropies and precise asymptotic formulas for the metric entropies $h_μ(n,r)$. |
| title | Asymptotic Homotopical Complexity of an Infinite Sequence of Dispersing $2D$ Billiards |
| topic | Dynamical Systems 37D05 |
| url | https://arxiv.org/abs/2501.16284 |