Genus zero transverse foliations for weakly convex Reeb flows on the tight $3$-sphere

TL;DR

This paper investigates weakly convex Reeb flows on the tight 3-sphere, establishing conditions for genus zero transverse foliations with specific index orbits and analyzing entropy in the real-analytic case.

Contribution

It introduces new criteria for genus zero transverse foliations with prescribed index-2 Reeb orbits and examines entropy related to stable/unstable manifolds in real-analytic settings.

Findings

Conditions for genus zero transverse foliations with index-2 binding orbits.

Existence of transverse foliations with additional index-3 binding orbits.

Positive topological entropy in the real-analytic case when stable/unstable manifolds are non-coincident.

Abstract

A contact form on the tight $3$-sphere $(S^3,\xi_0)$ is called weakly convex if the Conley-Zehnder index of every Reeb orbit is at least $2$. In this article, we study Reeb flows of weakly convex contact forms on $(S^3,\xi_0)$ admitting a prescribed finite set of index-$2$ Reeb orbits, which are all hyperbolic and mutually unlinked. We present conditions so that these index-$2$ orbits are binding orbits of a genus zero transverse foliation whose additional binding orbits have index $3$. In addition, we show in the real-analytic case that the topological entropy of the Reeb flow is positive if the branches of the stable/unstable manifolds of the index-$2$ orbits are mutually non-coincident.