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 $S^3$.

arXiv comment11 pages. The typo is fixed