Preservation of immersed or injective properties by composing generic   generalized distance-squared mappings
  Any generalized distance-squared mapping of equidimensional case has singularities, and their singularity types are wrapped into mystery in higher dimensional cases. Any generalized distance-squared mapping of equidimensional case is not injective. Nevertheless, in this paper, it is shown that the non-singular property or the injective property of a mapping is preserved by composing a generic generalized distance-squared mapping of equidimensional case. 
Shunsuke Ichiki and Takashi Nishimura
math.GT