On the minimal number of periodic orbits on some hypersurfaces in R2n\mathbb{R}^{2n}

We study periodic orbits on a nondegenerate dynamically convex starshaped hypersurface in $\mathbb{R}^{2n}$ along the lines of Long and Zhu, but using properties of the $S^1$-equivariant symplectic homology. We prove that there exist at least $n$ distinct simple periodic orbits on any nondegenerate starshaped hypersurface in $\mathbb{R}^{2n}$ satisfying the condition that the minimal Conley-Zehnder index is at least $n-1$. The condition is weaker than dynamical convexity.Comment: To appear in Annales de l'Institut Fourie

Real holomorphic curves and invariant global surfaces of section

In this paper we prove that a dynamically convex starshaped hypersurface in $\mathbb{C}^2$ which is invariant under complex conjugation admits a global surface of section which is invariant under conjugation as well. We obtain this invariant global surface by embedding $\mathbb{C}^2$ into $\mathbb{CP}^2$ and applying a stretching argument to real holomorphic curves in $\mathbb{CP}^2$. The motivation for this result arises from recent progress in applying holomorphic curve techniques to gain a deeper understanding on the dynamics of the restricted three body problem.Comment: 36 page