Invariant sets and hyperbolic closed Reeb orbits
Year: 2025
Type: article
Abstract: We investigate the effect of a hyperbolic (or, more generally, isolated as an invariant set) closed Reeb orbit on the dynamics of a Reeb flow on the $(2n-1)$-dimensional standard contact sphere, extending two results previously known for Hamiltonian diffeomorphisms to the Reeb setting. In particular, we show that under very mild dynamical convexity type assumptions, the presence of one hyperbolic closed orbit implies the existence of infinitely many simple closed Reeb orbits. The second main result of the paper is a higher-dimensional Reeb analogue of the Le Calvez-Yoccoz theorem, asserting that no closed orbit of a non-degenerate dynamically convex Reeb pseudo-rotation is locally maximal, i.e., isolated as an invariant set. The key new ingredient of the proofs is a Reeb variant of the crossing energy theorem.
Source: Advances in Mathematics
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,790
Awards: CAREER: Periodic Orbits of Hamiltonian Diffeomorphisms and Reeb Flows, G2432568968, New challenges in contact and symplectic topology, Collaborative Research: Floer Theory and Topological Entropy, Collaborative Research: Floer Theory and Topological Entropy, G5821831666, G613124455, G6671297155, G686694141, G7990632658, G8637609037
OpenAlex ID: W4386644362