2501.13856Jan 23, 202527 pages, 4 figures. Version 3: We resolve Question 1 from the previous version by proving that, for an arbitrary convex body, the Fadell-Rabinowitz index of the space of systoles equals that of the space of centralized systoles, and extend the study of the dynamics of generalized Zoll convex bodies. Minor expository improvements

Systolic $S^1$ -index and characterization of non-smooth Zoll convex bodies

Stefan Matijević

Symplectic Geometry Differential Geometry Dynamical Systems

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Abstract

We define the systolic $S^1$ -index of a convex body as the Fadell-Rabinowitz index of the space of generalized systoles associated with its boundary. We show that this index is a symplectic invariant. Using the systolic $S^1$ -index, we introduce the notion of generalized Zoll convex bodies and prove that this definition coincides with the classical one when the convex body satisfies the uniqueness of systoles property, that is, when through every point passes at most one systole. Moreover, we show that generalized Zoll convex bodies can be characterized in terms of their Gutt-Hutchings capacities, and we prove that the space of generalized Zoll convex bodies is closed in the space of all convex bodies. As a corollary, we establish that if the interior of a convex body is symplectomorphic to the interior of a ball, then the convex body is generalized Zoll, and in particular Zoll if it satisfies the uniqueness of systoles property. Finally, we discuss several examples.