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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2602.08521 |
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| _version_ | 1866911660231360512 |
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| author | Alves, Marcelo R. R. Meiwes, Matthias |
| author_facet | Alves, Marcelo R. R. Meiwes, Matthias |
| contents | We study Reeb dynamics on starshaped hypersurfaces in $\mathbb{R}^4$ arising as smoothings of starshaped polytopes. Using the $C^0$--stability of positive topological entropy for Reeb flows in dimension three from our joint work with Dahinden and Pirnapasov, we show that there exist starshaped polytopes $P$ such that for any starshaped smoothing of $\partial P$ the associated Reeb flows have positive topological entropy. This answers a question of Ostrover and Ginzburg.
Similarly, we show that given a closed surface $M$ and a number $C>0$, there exist continuous and non-differentiable Riemannian metrics $g$ on $S$ with $h_{\rm top}>C$ in the sense that for any smoothing of $g$ the associated geodesic flows have $h_{\rm top}>C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08521 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polytopes and $C^0$-Riemannian metrics with positive $h_{\rm top}$ Alves, Marcelo R. R. Meiwes, Matthias Dynamical Systems Symplectic Geometry 37B40 53D25 52B11 We study Reeb dynamics on starshaped hypersurfaces in $\mathbb{R}^4$ arising as smoothings of starshaped polytopes. Using the $C^0$--stability of positive topological entropy for Reeb flows in dimension three from our joint work with Dahinden and Pirnapasov, we show that there exist starshaped polytopes $P$ such that for any starshaped smoothing of $\partial P$ the associated Reeb flows have positive topological entropy. This answers a question of Ostrover and Ginzburg. Similarly, we show that given a closed surface $M$ and a number $C>0$, there exist continuous and non-differentiable Riemannian metrics $g$ on $S$ with $h_{\rm top}>C$ in the sense that for any smoothing of $g$ the associated geodesic flows have $h_{\rm top}>C$. |
| title | Polytopes and $C^0$-Riemannian metrics with positive $h_{\rm top}$ |
| topic | Dynamical Systems Symplectic Geometry 37B40 53D25 52B11 |
| url | https://arxiv.org/abs/2602.08521 |