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

Norbert Steinmetz

• 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 Rt(z) = (1 + (4/27)z3/(1 � z)), with dynamics governed by the behaviour of the postcritical orbit (Rn())n. In particular, it is shown that if escapes (that is, Rn() tends to infinity), then the Julia set of R is a Cantor set, or a Sierpiski curve, or a curve with one or else infinitely many cut-points; each of these cases actually occurs.