  We study the parameter planes of certain one-dimensional, dynamically-defined
slices of holomorphic families of entire and meromorphic transcendental maps of
finite type. Our planes are defined by constraining the orbits of all but one
of the singular values, and leaving free one asymptotic value. We study the
structure of the regions of parameters, which we call {\em shell components},
for which the free asymptotic value tends to an attracting cycle of
non-constant multiplier. The exponential and the tangent families are examples
that have been studied in detail, and the hyperbolic components in those
parameter planes are shell components. Our results apply to slices of both
entire and meromorphic maps. We prove that shell components are simply
connected, have a locally connected boundary and have no center, i.e., no
parameter value for which the cycle is superattracting. Instead, there is a
unique parameter in the boundary, the {\em virtual center}, which plays the
same role. For entire slices, the virtual center is always at infinity, while
for meromorphic ones it maybe finite or infinite. In the dynamical plane we
prove, among other results, that the basins of attraction which contain only
one asymptotic value and no critical points are simply connected. Our dynamical
plane results apply without the restriction of finite type.
