Existence of a positive hyperbolic Reeb orbit in three spheres with finite free group actions

Abstract

Let $(Y,\lambda )$ be a non-degenerate contact three manifold. D. Cristfaro–Gardiner, M. Hutchings, and D. Pomerleano showed that if $c_{1}(\xi =\textrm{Ker}\lambda )$ is torsion, then the Reeb vector field of $(Y,\lambda )$ has infinity many Reeb orbits; otherwise, $(Y,\lambda )$ is a lens space or three sphere with exactly two simple elliptic orbits. In the same paper, they also showed that if $b_{1}(Y)>0$, $(Y,\lambda )$ has a simple positive hyperbolic orbit directly from the isomorphism between Seiberg–Witten Floer homology and Embedded contact homology. In addition to this, they asked whether $(Y,\lambda )$ with infinity many simple orbits also has a positive hyperbolic orbit under $b_{1}(Y)=0$. In the present paper, we answer this question under $Y \simeq S^{3}$ with non-trivial finite free group actions. In particular, it gives a positive answer in the case of a lens space $(L(p,q),\lambda )$ with odd p.