On basin boundaries
The author considers the map \(T= (T_1,T_2): \mathbb{R}^2\to \mathbb{R}^2\) which is defined by \[ (x,y)\mapsto (T_1, T_2) (x,y):= (x^2- y^2+ \lambda+ \varepsilon f_1(x,y), \;2xy+ \varepsilon f_2(x,y)), \tag{1} \] where \(\lambda\in ]-\frac 34, \frac 14[\), \(\varepsilon\in \mathbb{R}\), \(f= (f_1,f_2)\) is a differentiable map, such that when \(\varepsilon\neq 0\), \(T\) is a non-analytic map. Note that the map \(T\) can be considered as a perturbation of the polynomial complex \(z\mapsto z^2+ \lambda\). The purpose of this note is to study a class of basin boundaries which is generated by (1). The author shows that an efficient characterization of a given element of this class can be constructed with segments of differentiable curves.
scientific article; zbMATH DE number 1311103