  Consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally
polynomial maps from a finite union of copies of $\C$ to itself. In the space
of suitably normalized maps of this type, the hyperbolic maps form an open set
called the hyperbolic locus. The various connected components of this
hyperbolic locus are called hyperbolic components, and those hyperbolic
components with compact closure (or equivalently those contained in the
"connectedness locus") are called bounded hyperbolic components. It is shown
that each bounded hyperbolic component is a topological cell containing a
unique post-critically finite map called its center point. For each degree $d$,
the bounded hyperbolic components can be separated into finitely many distinct
types, each of which is characterized by a suitable reduced mapping scheme
$\bar S_f$. Any two components with the same reduced mapping scheme are
canonically biholomorphic to each other. There are similar statementsfor real
polynomial maps, for polynomial maps with marked critical points, and for
rational maps. Appendix A, by Alfredo Poirier, proves that every reduced
mapping scheme can be represented by some classical hyperbolic component, made
up of polynomial maps of $\C$. This paper is a revised version of [M2], which
was circulated but not published in 1992.
