Polytopes and $C^0$-Riemannian metrics with positive $h_{\mathrm{t}\mathrm{o}\mathrm{p}}$

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 $\mathrm{\partial }P$ 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 $h_{\mathrm{t}\mathrm{o}\mathrm{p}}>C$ in the sense that for any smoothing of $g$ the associated geodesic flows have $h_{\mathrm{t}\mathrm{o}\mathrm{p}}>C$.