Dynamical extensions of Zoll to nonsmooth 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.