Abstract: | We discuss the dynamics as well as the structure of the parameterplane of certain families of rational maps with few criticalorbits. Our paradigm is the family Rt(z) = (1 + (4/27)z3/(1– z)), with dynamics governed by the behaviour of thepostcritical orbit (Rn())n. In particular, it is shown thatif escapes (that is, Rn() tends to infinity), then the Juliaset of R is a Cantor set, or a Sierpiski curve, or a curve withone or else infinitely many cut-points; each of these casesactually occurs. |