From the introduction of the paper: “The main purpose of this paper is to give new results on the basin structure of noninvertible maps of a plane (two-dimensional endomorphisms) and their bifurcations. Especially, we explain bifurcations in which a regular basin boundary is changed into a fractal boundary, the phenomenon of fractalization. These new results have been suggested by the study of a particularly interesting example, a quadratic noninvertible map \(x \to ax + y\), \(y \to x^ 2 + b\). We use this example to illustrate the new results.
Another purpose of this paper is to make known some earlier results in a new formalism, and make explicit some aspects which previously were implicit.
We consider a class of two-dimensional noninvertible maps \(T : \mathbb{R}^ 2 \to \mathbb{R}^ 2\), having a curve LC which separates the plane into two regions, the points of which have different numbers of rank-1 preimages.
The text develops especially a study of the case for which LC separates \(\mathbb{R}^ 2\) into two open regions \(Z_ 0\) and \(Z_ 2\), a point \(X\) belonging to \(Z_ 2\) having two distinct preimages (or antecedents) of rank-1, and a point \(X\) of \(Z_ 0\) being without preimages”.