A special Julia set over double numbers (Q1345439)

The ring \(D\) of double numbers is the set of all \(x+ jy\) where \(x\), \(y\) are real and \(j^ 2= 1\). The author studies in detail the iteration of the function \(f(Z)= Z^ 2\) for \(Z\in D\). For example he determines the sets of \(Z\)-values where \(| f^ n(Z)|\to 0\) and \(| f^ n(Z)|\to \infty\). Here \(f^ n\) denotes the \(n\)th iterate of \(f\) and \(| x+ jy|= \sqrt{| x^ 2- y^ 2|}\).