The Conley-Zehnder index of a minimal orbit and existence of a positive hyperbolic orbit

Abstract: As a refinement of the Weinstein conjecture, it is a natural question whether a Reeb orbit of particular types exists. D. Cristofaro-Gardiner, M. Hutchings and D. Pomerleano showed that every nondegenerate closed contact three manifold with $b_{1}>0$ has at least one positive hyperbolic orbit by directly using the isomorphism between ECH and Seiberg-Witten Floer (co)homology. In the same paper, they also asked whether the case of $b_{1}=0$ does. Suppose that $(S^{3},\lambda)$ is non-degenerate contact three sphere with infinity many orbits. In the present paper, we prove the existence of a simple positive hyperbolic orbit on $(S^{3},\lambda)$ under the condition that the Conley-Zehnder index of a minimal periodic orbit induced by the trivialization of a bounding disc is larger than or equal to 3. As an immediate corollary, we have the existence of a simple positive hyperbolic orbit on a non-degenerate dynamically convex contact three sphere $(S^{3},\lambda)$ with infinity many simple orbits. In particulr, this implies that a $C^{\infty}$ generic compact strictly convex energy hypersurface in $\mathbb{R}^{4}$ carries a positive hyperbolic simple orbit.