## Institute for Mathematical Sciences

## Preprint ims12-02

** John Milnor, with an appendix by A. Poirier**
*Hyperbolic Components*

Abstract: Consider polynomial maps f : C -> C of degree d >= 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 bouned 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 S_f. Any
two components with the same reduced mapping scheme are
canonically biholomorphic to each other. There are similar
statements for 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.

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