ON A GENERALIZATION OF THE MORSE INDEX

Abstract

This chapter describes an index for the isolated invariant sets of a flow on a compact metric space that, in the case of elementary singular points above, contains (exactly) the information of the Morse index. This index is defined in terms of special isolating neighborhoods called isolating blocks. A singular point of an ordinary differential equation is called elementary if the eigenvalues of the linearized equations all have non-zero real parts. In this case, the set of orbits that tends to the singular point has dimension equal to the number of eigenvalues with negative real part; the unstable manifold has the complementary dimension. Either of these dimensions determines the flow near the singular point up to conjugacy; thus, their importance is not overemphasized by giving them a special name. It is customary to call the dimension of the unstable manifold the Morse index of the singular point. Elementary singular points are examples of invariant sets of the flow that are isolated in the sense that they admit a closed neighborhood in which they are maximal. Any such neighborhood will be called an isolating neighborhood for the invariant set. The definition implies that any larger invariant set is quite a bit larger.