On the Maslov-type index for general paths of symplectic matrices

TL;DR

This paper introduces a new Maslov-type index for general symplectic paths with arbitrary endpoints, extending existing indices and providing a direct, consistent construction applicable to a broader set of matrices.

Contribution

It defines a novel Maslov-type index for symplectic paths with arbitrary endpoints, generalizing previous indices and avoiding reliance on Lagrangian path indices.

Findings

The index is consistent regardless of the starting point of the path.

The new index generalizes the Conley-Zehnder-Long index to larger degenerate sets.

Comparison with other Maslov indices shows its broader applicability.

Abstract

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 partial generalization of the Conley-Zehnder-Long index in the sense that the degenerate set of symplectic matrices is larger. The method of constructing the index is direct without taking advantage of Maslov index of Lagrangian paths and 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 are verified. We review other versions of Maslov indices and compare them with our definition. In particular, this Maslov-type index can be regarded as a realization of Cappell-Lee-Miller index for a pair of Lagrangian paths from the point of view of index for symplectic paths.