In this article, we develop the Yoccoz puzzle technique to study a family of
rational maps termed McMullen maps. We show that the boundary of the immediate
basin of infinity is always a Jordan curve if it is connected. This gives a
positive answer to a question of Devaney. Higher regularity of this boundary is
obtained in almost all cases. We show that the boundary is a quasi-circle if it
contains neither a parabolic point nor a recurrent critical point. For the
whole Julia set, we show that the McMullen maps have locally connected Julia
sets except in some special cases.Comment: Complex dynamics, 51 pages, 13 figure