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arXiv · 2602.08521

Polytopes and $C^0$-Riemannian metrics with positive $h_{\rm top}$

Marcelo R. R. Alves·Matthias Meiwes
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Abstract

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 $h_{\rm top}>C$.

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Marcelo R. R. Alves, Matthias Meiwes. 2026-02-09. Polytopes and $C^0$-Riemannian metrics with positive $h_{\rm top}$. https://arxiv.org/abs/2602.08521

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