Two irrationally elliptic closed orbits of Reeb flows on the boundary of star-shaped domain in $\mathbb{R}^{2n}$
Xiaorui Li, Hui Liu, Wei Wang
TL;DR
The paper addresses the stability and multiplicity of closed Reeb orbits on boundaries of star-shaped domains in $\\mathbb{R}^{2n}$ under dynamical convexity. It develops a framework based on Maslov-type index iteration, common index jumps, and local Floer/equivariant local symplectic homology to control the growth and visibility of orbits, and to relate orbit iterations to their indices. The main result shows that if the Reeb flow is dynamically convex and has exactly $n$ prime closed orbits, then at least two of these orbits are irrationally elliptic, extending the finite-multiplicity regime and clarifying stability properties in higher dimensions. The approach combines precise index-theoretic arguments with generating-function models and local-homology techniques to derive a robust mechanism for identifying elliptic orbits and understanding the impact of convexity hypotheses on the Reeb dynamics.
Abstract
There are two long-standing conjectures in Hamiltonian dynamics concerning Reeb flows on the boundaries of star-shaped domains in $\mathbb{R}^{2n}$ ($n \geq 2$). One conjecture states that such a Reeb flow possesses either $n$ or infinitely many prime closed orbits; the other states that all the closed Reeb orbits are irrationally elliptic when the domain is convex and the flow possesses finitely many prime closed orbits. In this paper, we prove that for dynamically convex Reeb flow on the boundary of a star-shaped domain in $\mathbb{R}^{2n}$ ($n \geq 2$) with exactly $n$ prime closed orbits, at least two of them must be irrationally elliptic.