A New Simple Class of Rational Functions whose Julia Set is the whole Riemann Sphere

The paper first gives sufficient conditions on the criticalpoints and the Schwarzian derivative of a real rational functionR such that the Julia set of R is . Further, it is shown that under mild conditions on another realrational function with possibly non-empty Fatou set, the Julia set of R is the whole Riemann sphere again. Then familiesof rational functions are given whose Julia set is and whose critical points are not necessarily preperiodic.Concrete examples were previously available only for the preperiodiccase. Finally, it is demonstrated that the methods presentedalso apply to the construction of polynomials whose Julia setsare dendrites and whose critical points in the Julia set arenot necessarily preperiodic.