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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 lensspace or three sphere with exaxtly two simple elliptic orbits. In the samepaper, they also showed that if $b_{1}(Y)>0$, $(Y,\lambda)$ has a simplepositive hyperbolic orbit directly from the isomorhphism between Seiberg-WittenFloer homology and Embedded contact homology. In addition to this, they askedwhether $(Y,\lambda)$ with infinity many simple orbits also has a positivehyperbolic orbit under $b_{1}(Y)=0$. In the present paper, we answer thisquestion 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$S^{3}$.

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