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

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 $h_{\rm top}>C$.

💡 Research Summary

The paper investigates the dynamical complexity of two classes of non‑smooth geometric objects: star‑shaped polytopes in ℝ⁴ and continuous (nowhere‑differentiable) Riemannian metrics on closed surfaces. The central invariant is topological entropy h_top, which measures exponential orbit growth and is a standard indicator of chaotic behavior. Since Reeb flows on the boundary of a polytope or geodesic flows of a merely continuous metric are not defined in the classical sense, the authors adopt a “smoothing” approach: a non‑smooth object is approximated by a sequence of C³‑smooth objects, and the entropy of the original object is defined as the lim inf of the entropies of the approximating sequence. This definition is motivated by recent work (Alves–Dahinden–Meiwes–Pirnapasov, J. Eur. Math. Soc.) establishing C⁰‑stability of topological entropy for Reeb flows in dimension three.

Main results.

1. Theorem 1.7 (Polytopes). There exist star‑shaped polytopes P⊂ℝ⁴ such that for every star‑shaped C³‑smoothing {M_j} of ∂P the associated Reeb flow has positive topological entropy. In other words, the entropy of ∂P, defined via Definition 1.1, is strictly positive. This answers a question raised independently by Ginzburg and Ostrover concerning the existence of polytopes whose Reeb dynamics are intrinsically chaotic. The proof proceeds by selecting a contact form α on the tight contact 3‑sphere (S³, ξ_tight) whose Reeb flow has positive entropy and is C⁰‑robust (ε‑close contact forms also have entropy >c). By identifying α with a smooth star‑shaped hypersurface M⊂ℝ⁴ (α=λ₀|_M) and taking a small tubular neighborhood U≈(−δ,δ)×M, any hypersurface in U that is a graph over M inherits a contact form e^{f}α which stays within ε of α. Choosing a polytope P inside U that can be written as such a graph guarantees that every smoothing of P lies in U and thus its Reeb flow inherits the same positive entropy bound.

2. Theorem 1.8 (Continuous metrics and contact forms). For any constant C>0 there exist (i) a continuous, nowhere‑differentiable Riemannian metric g_C on a closed surface S with area 1 and h_top(g_C)>C, and (ii) a continuous, nowhere‑differentiable contact form α_C on any closed cooriented contact 3‑manifold (M,ξ) with h_top(α_C)>C. The construction uses a smooth metric g with h_top(g)>3C (e.g., a metric of large curvature) and the C⁰‑stability theorem from