Invariant Sets and Hyperbolic Closed Reeb Orbits

TL;DR

This paper explores how hyperbolic closed Reeb orbits influence the dynamics on high-dimensional contact spheres, establishing conditions for infinitely many orbits and analyzing orbit isolation properties.

Contribution

It extends known Hamiltonian results to Reeb flows, proving the existence of infinitely many orbits under mild conditions and providing a Reeb analogue of a classical theorem.

Findings

Hyperbolic closed Reeb orbits imply infinitely many simple closed Reeb orbits.

Non-degenerate dynamically convex Reeb pseudo-rotations have no isolated closed orbits.

A Reeb variant of the crossing energy theorem is introduced.

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.