The asymptotic Maslov index and its applications

Abstract

Let $\mathcal{N}$ be a 2n-dimensional manifold equipped with a symplectic structure $\omega$ and $\Lambda(\mathcal{N})$ be the Lagrangian Grassmann bundle over $\mathcal{N}$. Consider a flow $\phi^t$ on $\mathcal{N}$ that preserves the symplectic structure and a $\phi^t$-invariant connected submanifold $\Sigma$. Given a continuous section $\Sigma\to\Lambda(\mathcal{N})$, we can associate to any finite $\phi^t$-invariant measure with support in $\Sigma$, a quantity, The asymptotic Maslov index, which describes the way Lagrangian planes are asymptotically wrapped in average around the Lagrangian Grassmann bundle. We pay particular attention to the case when the flow is derived from an optical Hamiltonian and when the invariant measure is the Liouville measure on compact energy levels. The situation when the energy levels are not compact is discussed.