Real Basic Quadratic Iteration Sequences

Ifwiththen the number and nature of the fixed points and the intersection of the keep setwith the real axis depends on the value of


  1. No fixed real points if

  2. A single fixed point if

  3. Two real fixed pointsif

Ifthen the fixed points are complex soThe coefficients ofare real so the roots are complex conjugates and since there is no intersection with the real axis, the keep set is not connected. Further, the intersection of the keep set with the imaginary axis is also zero, since

Ifthenhas either 1 or 2 real fixed points sois non – empty. In fact, ifthenis the interval

Ifthenconsists of the closed intervalfrom which a sequence of disjoint, non – empty, open subintervals have been removed. In particular,Furthermore, there is no intersection ofwith the imaginary axis sincesois outside the interval above andis not in the keep set.