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

TL;DR

This paper addresses extending topological entropy to non-smooth dynamical objects (polytopes, $C^0$-contact forms, and $C^0$-Riemannian metrics) by introducing entropy notions via star-shaped smoothings. Using the $C^0$-robust positive entropy results from ADMP on $(S^3,\xi_{\rm tight})$, the authors embed smoothed polytopes into a Weinstein neighborhood to transfer lower bounds to the Reeb flows. They prove the main theorem: there exist starshaped polytopes $P\subset \mathbb{R}^4$ such that every star-shaped smoothing of $\partial P$ yields $h_{\rm top} > 0$, and similarly construct non-differentiable $C^0$-forms with arbitrarily large entropy. These results answer Ostrover–Ginzburg and reveal how combinatorial polygonal data relate to chaotic Reeb/geodesic dynamics, while proposing open questions about realization sets and connections to $C^0$-symplectic topology.

Abstract

We study Reeb dynamics on starshaped hypersurfaces in $\mathbb{R}^4$ arising as smoothings of convex 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 $h_{\rm top}>C$.

Theorems & Definitions (11)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Remark 1.4
• Definition 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 1.9
• proof