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.

arXiv commentThe majority of the main results are included in arXiv 2307.02122