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

TL;DR

This paper demonstrates that certain starshaped polytopes and non-differentiable Riemannian metrics can be smoothed to produce Reeb and geodesic flows with arbitrarily high positive topological entropy, answering longstanding questions.

Contribution

It establishes the existence of polytopes and metrics whose smoothed versions retain high topological entropy, extending stability results to new geometric settings.

Findings

Existence of starshaped polytopes with smoothing leading to positive topological entropy

Construction of non-differentiable metrics with arbitrarily high topological entropy upon smoothing

Positive topological entropy is stable under smoothing in these geometric contexts

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 $h_{\rm top}>C$.