A Classification of 1-parameter Families of Map Germs U\ 0 -> 1r,0 with Applications to Condensation Problems

In his seminal paper [1] Arnold discusses, amongst other things, the problem of the evolution of galaxies. The basic model he considers is the following. Consider a medium of non-interacting particles in IR with an initial velocity distribution v = v(x) and a density distribution p = p(x), where v and p are a smooth vector field and a smooth function, respectively, on IR. The inertial motion of particles defines a time-dependent map gt: U 3 -> U given by gt(x) = x+tv(x). If v(x) is not too badly behaved at infinity the map gt is a diffeomorphism for small values of t. At some time /, however, singularities occur and the critical values of the map correspond to points of condensation of particles, where the density becomes infinite. (The values of the initial density distribution p are not relevant of course, as far as this condensation process is concerned, provided we assume that p is a strictly positive function.) After condensation takes place the physical assumption that there is no interaction is no longer realistic. We shall follow Arnold in assuming that the particles are non-interacting even after the generation of singularities, in the hope of obtaining some of the grosser features of the resulting condensing surfaces. In the theory expounded by Arnold (which he attributes to Zeldovitch) he assumes that the vector field v(x) is a gradient field, so v = VS for some potential S. One then has models for the generation of the condensing surfaces using results of Zakalyukin (see [2,14,7,8,4] for details). We offer the following quotation from Arnold [3, p. 42] on the question of whether or not the assumption of non-interaction is justified. 'According to astrophysicists, when the radius of the universe was a thousand times smaller than at present the large scale distribution of matter in the universe was practically uniform and the velocity field was practically potential. The subsequent movement of particles led to the formation of caustics, i.e. singularities of density and clusters of particles. Up to the formation of pancakes the density remained small so that the particle medium could be assumed to be non-interactive. After this the medium can be assumed to be non-interactive if neutrons account for a significant part of the mass of the universe; if, however, most of the mass is in protons and neutrons then deductions from the geometry of caustics and their metamorphoses must be treated with caution since the medium then ceases to be non-interactive.' Of course 'practically' potential is, nevertheless, not potential. It therefore seems to be of interest to drop this assumption on the velocity distribution and to investigate the generic changes in the resulting condensing surfaces for a general vector field v. To do this we need to classify generic 1-parameter families of map germs U,0 -*• U,0, and most of this paper is devoted to this classification. Our classification uses recent