Dynamical extensions of Zoll to nonsmooth convex bodies

TL;DR

This paper extends the symplectic Zoll property to non-smooth convex bodies, analyzing their dynamical behavior and capacity, and establishing new topological and compactness results in symplectic geometry.

Contribution

It introduces a dynamical extension of the Zoll property to non-smooth convex bodies and relates it to the Ekeland-Hofer-Zehnder capacity, with new classification and compactness results.

Findings

Dynamical behavior of non-smooth action-minimizing closed characteristics is less irregular than expected.

Established equivalence between the dynamical extension and a topological extension of the Zoll property.

Proved an $H^1$-compactness result for the space of non-smooth convex bodies.

Abstract

The symplectic Zoll property of smooth convex domains in the classical phase space has been extensively studied in recent years and, in particular, has been shown to detect local maximizers of the systolic ratio. We propose a dynamical extension of this property to the non-smooth setting, related to the behavior of the Ekeland-Hofer-Zehnder capacity with respect to hyperplane cuts. Under certain hypotheses, we establish its equivalence to a known topological extension of the Zoll property. In this context, we study the space of non-smooth action-minimizing closed characteristics and show that their dynamical behavior is not as irregular as one might first expect. We classify several types of dynamical behaviors and derive a certain $H^1$-compactness result, which is of independent interest.