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

Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms

In this article we prove that for a smooth fiberwise convex Hamiltonian, the asymptotic Hofer distance from the identity gives a strict upper bound to the value at 0 of Mather's $\beta$ function, thus providing a negative answer to a question asked by K. Siburg in \cite{Siburg1998}. However, we show that equality holds if one considers the asymptotic distance defined in \cite{Viterbo1992}.Comment: 21pp, accepted for publication in Geometry & Topolog