Sierpinski and non-Sierpinski curve Julia sets in families of rational maps

We discuss the dynamics as well as the structure of the parameter plane of certain families of rational maps with few critical orbits. Our paradigm is the family R-t(z) = t(1 + (4/27) z(3)/(1 - z)), with dynamics governed by the behaviour of the postcritical orbit (R-t(n) (t))(n is an element of N). In particular, it is shown that if t escapes (that is, R-t(n) (t) tends to infinity), then the Julia set of Rt is a Cantor set, or a Sierpinski curve, or a curve with one or else infinitely many cut-points; each of these cases actually occurs.