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arxiv: 2410.13093 · v4 · pith:OSY544LQnew · submitted 2024-10-16 · 🧮 math.SG · math.DS

Closed Orbits of Dynamically Convex Reeb Flows: Towards the HZ- and Multiplicity Conjectures

Erman Cineli , Viktor L. Ginzburg , Basak Z. Gurel This is my paper
classification 🧮 math.SG math.DS
keywords closedorbitsreebflowprimecontactconvexdomain
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We study the multiplicity problem for prime closed orbits of dynamically convex Reeb flows on the boundary of a star-shaped domain in $\mathbb{R}^{2n}$. The first of our two main results asserts that such a flow has at least $n$ prime closed Reeb orbits, improving the previously known lower bound by a factor of two and settling a long-standing open question. The second main theorem is that when, in addition, the domain is centrally symmetric and the Reeb flow is non-degenerate, the flow has either exactly $n$ or infinitely many prime closed orbits. This is a higher-dimensional contact variant of Franks' celebrated $2$-or-infinity theorem and, viewed from the symplectic dynamics perspective, settles a particular case of the contact Hofer-Zehnder conjecture. The proofs are based on several auxiliary results of independent interest on the structure of the filtered symplectic homology and the properties of closed orbits.

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