A generalization of imaginary parts of eigenvalues for matrices: Chain rotation numbers

Rotation numbers and chain rotation numbers may be interpreted as a generalization of the imaginary parts for matrices. In dimension two they measure how the solutions of a linear autonomous differential equation rotate in the phase space, and they reduce to the imaginary parts of the eigenvalues of the system’s matrix. In higher dimensions they measure how a two-frame of vectors rotate under the induced flow in the plane which is spanned by the frame. For their calculation, only special sets in the oriented Grassmann manifold of planes are relevant, and to each of these sets corresponds a compact interval of chain rotation numbers. In this paper we will determine these relevant sets and calculate the corresponding sets of chain rotation numbers.