Consider the family of rational maps <IMG SRC="/fbpe/img/proy/v21n1/for01.jpg" WIDTH=472 HEIGHT=38>and the hyperboliccomponent A1 = f w : fw has an attracting fixed point g : We prove that <IMG SRC="/fbpe/img/proy/v21n1/for02.jpg" WIDTH=202 HEIGHT=23>iparabolic parameter with corre-sponding multiplier a primitive <IMG SRC="/fbpe/img/proy/v21n1/for03.jpg" WIDTH=298 HEIGHT=24>there exists a hyperbolic component Wq; attached to A1 at the point w0; which contains w ¡ values for which fw has an at-tracting periodic cycle of period q. 1991 Mathematics Subject Classification : Primary 30D05, 58F23