Let $f, g: \mathbb{ R}^2 \to \mathbb{ R}$ be two submersion functions and $\mathscr{ F}(f)$ and $\mathscr{ F}(g)$ be the regular foliations of $\mathbb{ R}^2$ 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 $\mathscr{ F}(f)$ and $\mathscr{ F}(g)$, 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.
Categories: math.DS