On roundness of rotation sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911200198000640 |
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| author | Perrot, Boris Boroński, Jan Clark, Alex |
| author_facet | Perrot, Boris Boroński, Jan Clark, Alex |
| contents | Motivated by the question whether a round disk can be realized as the rotation set of a torus diffeomorphism, we study the roundness of rotation sets of a parametric family of torus diffeomorphisms $F_ρ$, where the parameter $ρ$ ranges over irrational numbers in $(0,1)$. Each $F_ρ$ is a Kwapisz-like diffeomorphism with a 2-dimensional non-polygonal rotation set
$$Λ'_ρ= \operatorname{conv}\left(\left\{(\pm\frac{\lceil mρ\rceil}{m+n+1}, \pm\frac{\lceil nρ\rceil}{m+n+1}): m, n \in \mathbb{N} _0, \lceil mρ\rceil - mρ<ρ,\lceil nρ\rceil - nρ<ρ\right\}\right)$$ whose extreme point set contains exactly four (two-sided) accumulation points. We define the roundness of $Λ'_ρ$ as the ratio $R_ρ=\frac{\operatorname{Area}(Λ'_ρ)}{πρ^2}$, and give its upper and lower bounds in terms of $ρ$.
$R_ρ$ is neither monotone nor continuous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08235 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On roundness of rotation sets Perrot, Boris Boroński, Jan Clark, Alex Dynamical Systems Motivated by the question whether a round disk can be realized as the rotation set of a torus diffeomorphism, we study the roundness of rotation sets of a parametric family of torus diffeomorphisms $F_ρ$, where the parameter $ρ$ ranges over irrational numbers in $(0,1)$. Each $F_ρ$ is a Kwapisz-like diffeomorphism with a 2-dimensional non-polygonal rotation set $$Λ'_ρ= \operatorname{conv}\left(\left\{(\pm\frac{\lceil mρ\rceil}{m+n+1}, \pm\frac{\lceil nρ\rceil}{m+n+1}): m, n \in \mathbb{N} _0, \lceil mρ\rceil - mρ<ρ,\lceil nρ\rceil - nρ<ρ\right\}\right)$$ whose extreme point set contains exactly four (two-sided) accumulation points. We define the roundness of $Λ'_ρ$ as the ratio $R_ρ=\frac{\operatorname{Area}(Λ'_ρ)}{πρ^2}$, and give its upper and lower bounds in terms of $ρ$. $R_ρ$ is neither monotone nor continuous. |
| title | On roundness of rotation sets |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2510.08235 |