Topologically Versal Deformations of Matrices; Codimension at most Two

1. Introduction. The reduction of a matrix to its Jordan normal form is an unstable operation in that both the normal form itself and the reducing mapping depend discontinuously on the elements of the original matrix. For example, the matrix trivially reduces to itself in Jordan form, but there are arbitrarily small perturbations of this matrix that reduce to the form which is certainly not a small perturbation of the original matrix, and moreover the reducing mapping is not a small perturbation of the identity. In [1], Arnol'd derives the simplest possible normal forms to which families of matrices may be linearly reduced in a ‘stable’ manner. In this paper, we consider a ‘topological' version of the problem, using the classification of matrices up to topological conjugacy given in [8] and the classification of linear dynamical systems up to orbital equivalence given in [9].