Dynamics of quadratic functions in cycle planes (Q1198216)

It is well known that all quadratic functions can be reduced by substitution to the form \(f(z) = z^ 2 + w\), whose behavior under iteration is thus representative for all quadratic functions. Now, from a geometric point of view, the plane coordinatized by \(\mathbb{C}\) is the Möbius plane, which is only one of the three classical cycle planes. We will here consider two kinds of generalized complex numbers. They are quadratic extensions of \(\mathbb{R}\) with the following properties: In the case of the ``double numbers'', we adjoin an element \(j \notin \mathbb{R}\) such that \(j^ 2 = 1\). If \(a\), \(b \in \mathbb{R}\), then these numbers \(z = a + bj\) constitute a ring, and they coordinatize the classical Minkowski plane. The second case is that of the ``dual numbers'', involving adjunction of an element \(e \notin \mathbb{R}\) with \(e^ 2 = 0\), producing a ring whose elements \(z = a + be\) coordinatize the classical Laguerre plane. In these rings we will deal with the iteration of the function \(z_ 1 = z^ 2 + w, \dots, z_ n = z_{n-1}^ 2 + w\), and study the question for which values of \(z\) and \(w\) the iteration remains bounded, and for which values the iteration tends toward infinity, where ``infinity'' should mean that at least one of the real and the imaginary parts becomes unbounded. Now, if \(a_ n\) and \(b_ n\) denote, respectively, the real and the imaginary parts of \(z_ n\), then \(\{z:| a_ n | < \infty\), \(| b_ n | < \infty\), \(\forall n\}\) will be called the ``Julia set for \(w\)'', whereas the ``Mandelbrot set'' will be defined as \(\{w : | a_ n | < \infty\), \(| b_ n | < \infty\), \(\forall n\), \(z = w\}\). These definitions stem from the theory of dynamic systems in the case of the usual complex numbers. They are, however, not uniform in the literature. It is well known that the Mandelbrot set is connected, and that Julia sets for points \(w\) inside the Mandelbrot set are connected. In this paper we deal only with Julia sets of these points. However, it is quite possible that the study of the ``Fatou dust'' in the disconnected Julia sets of other points may yield interesting results.