On Maslov-type index for general paths of symplectic matrices

In this article, we define an index of Maslov type for general symplectic paths which have two arbitrary end points. This Maslov-type index is a generalization of the Conley-Zehnder-Long index and the method of constructing the index is consistent no matter whether the starting point of the path is identity or not, which is different from the ones for Long's Maslov-type index and Liu's $L_0$-index. Some natural properties for the index still hold. We review other versions of Maslov indices and compare them with our definition. In particular, this Maslov-type index can be looked as a realization of Cappell-Lee-Miller's index for a pair of Lagrangian paths from the point of view of index for symplectic paths.