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. Hutshings and D. Pomerleano showed that if $c_{1}(\xi=\mathrm{Ker}\lambda)$ 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 exaxtly 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 isomorhphism 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 nontrivial finite free group actions, especially lens spaces $(L(p,q),\lambda)$ with odd $p$ as quotient spaces of $S^{3}$.