On real singularities with a Milnor fibration

In this article we study the singularities defined by real analytic maps (R-m, 0) --> (R-2, 0) with an isolated critical point at the origin, having a Milnor fibration. It is known [14] that if such a map has rank 2 on a punctured neighbourhood of the origin, then one has a fibre bundle phi : Sm-1 - K --> S-1, where K is the link. In this case we say that f satisfies the Milnor condition at 0 is an element of R-m. However, the map phi may not be the obvious map f/parallel tofparallel to as in the complex case [14, 9]. If f satisfies the Milnor condition at 0 is an element of R-m and for every sufficiently small sphere around the origin the map f/parallel tofparallel to defines a fibre bundle, then we say that f satisfies the strong Milnor condition at 0 is an element of R-m. In this article we first use well known results of various authors to translate "the Milnor condition" into a problem of finite determinacy of map germs, and we study the stability of these singularities under perturbations by higher order terms. We then complete the classification, started in [20, 21] of certain families of singularities that satisfy the (strong) Milnor condition. The simplest of these are the singularities in R-2n similar or equal to C-n of the form {Sigma(i=1)(n) z(i)(ai) (z) over bar (bi)(i) = 0, a(i) > b(i) greater than or equal to 1}. We prove that these are topologically equivalent (but not analytically equivalent!) to Brieskorn-Pham singularities.