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

Let (Y,λ ) be a non-degenerate contact three manifold. D. Cristfaro–Gardiner, M. Hutchings, and D. Pomerleano showed that if c_1(ξ =Kerλ ) is torsion, then the Reeb vector field of (Y,λ ) has infinity many Reeb orbits; otherwise, (Y,λ ) 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,λ ) 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,λ ) 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 ≃ 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),λ ) with odd p .