In this paper we consider families of rational maps of degree $2n$
on the Riemann sphere $\F :
\overline{\bbC
} \rightarrow \overline{\bbC
}$ given by
$$
\F(z) = z^n + \frac{\la}{z^n}
$$
where $\la \in \bbC -\{0\}$ and $n \geq 2$. One of our goals in
this paper is to describe a type of structure that we call a {\it
Cantor necklace\/} that occurs in both the dynamical and the
parameter plance for $\F$. Roughly speaking, such a set is
homeomorphic to a set constructed as follows. Start with the
Cantor middle thirds set embedded on the $x$-axis in the plane.
Then replace each removed open interval with an open circular disk
whose diameter is the same as the length of the removed interval.
A Cantor necklace is a set that is homeomorphic to the resulting
union of the Cantor set and the adjoined open disks.
The second goal of this paper is to use the Cantor necklaces in
the parameter plane to prove the existence of several new types of
Sierpinski curve Julia sets that arise in these families of
rational maps. Unlike most examples of this type of Julia set, the
maps on these Julia sets are structurally unstable. That is, small
changes in the parameter $\la$ give rise to Julia sets on which
the dynamical behavior is quite different. In addition, we also
describe a new type of related Julia set which we call a hybrid
Sierpinski curve.