  In this paper, we consider the family of rational maps $$\F(z) = z^n +
\frac{\la}{z^d},$$ where $n \geq 2$, $d\geq 1$, and$\la \in \bbC$. We consider
the case where $\la$ lies in the main cardioid of one of the $n-1$ principal
Mandelbrot sets in these families. We show that the Julia sets of these maps
are always homeomorphic. However, two such maps $\F$ and $F_\mu$ are conjugate
on these Julia sets only if the parameters at the centers of the given
cardioids satisfy $\mu = \nu^{j(d+1)}\la$ or $\mu = \nu^{j(d+1)}\bar{\la}$
where $j \in \bbZ$ and $\nu$ is an $n-1^{\rm st}$ root of unity. We define a
dynamical invariant, which we call the minimal rotation number. It determines
which of these maps are are conjugate on their Julia sets, and we obtain an
exact count of the number of distinct conjugacy classes of maps drawn from
these main cardioids.
