Title: Stable components in the parameter plane of meromorphic functions of finite type,
Abstract: We study the parameter planes of certain one-dimensional, dynamically-defined
slices of holomorphic families of meromorphic transcendental maps of finite
type for which infinity is not an asymptotic value. Our planes are defined by
constraining the orbits of all but one of the asymptotic values. We study the
structure of the regions in the parameter plane where the dynamics have a free
asymptotic value that is attracted to an attracting periodic orbit. The tangent
family is an example that has been studied in detail, \cite{KK}, and our goal
is to understand in the general context of meromorphic functions to what extent
the structure one sees in parameter plane of the tangent family is generic.
For the analogous dynamically defined slices of parameter spaces for rational
maps, there are Mandelbrot-like components that have a unique {\em center} that
can be used to give a combinatorial descripton of the components. No such
center exists for our slices, no matter how carefully they are chosen. We can
show, however, that there is an analogous concept, a "virtual" center, that
plays essentially the same role as a center would have. It lies on the boundary
of the component and corresponds to a map for which there is a "virtual cycle",
namely, an iterate of the free asymptotic value that lands at a pole. These
virtual centers are dense in the bifurcation locus of our one-dimensional
families.