Topological equivalence of submersion functions and topological equivalence of their foliations on the plane: The linear-like case

Abstract

Let f,g:R2→R be two submersion functions and F(f) and F(g) be the regular foliations of R2 whose leaves are the connected components of the levels sets of f and g, respectively. The topological equivalence of f and g implies the topological equivalence of F(f) and F(g), but the converse is not true, in general. In this paper, we introduce the class of linear-like submersion functions, which is wide enough in order to contain non-trivial behaviors, and provide conditions for the validity of the converse implication for functions inside this class. Our results lead us to a complete topological invariant for topological equivalence in a certain subclass of linear-like submersion functions.