## Institute for Mathematical Sciences

## Preprint ims92-3

** J. Milnor (appendix by A. Poirier)**
* Hyperbolic Components in Spaces of Polynomial Maps*

Abstract: We 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 which have degree two or more on each copy. In
any space $\p^{S}$ of suitably normalized maps of this type, the
post-critically bounded maps form a compact subset $\cl^{S}$
called the connectedness locus, and the hyperbolic maps in
$\cl^{S}$ form an open set $\hl^{S}$ called the hyperbolic
connectedness locus. The various connected components
$H_\alpha\subset \hl^{S}$ are called hyperbolic components. It
is shown that each hyperbolic component is a topological cell,
containing a unique post-critically finite map which is called
its center point. These hyperbolic components can be separated
into finitely many distinct ``types'', each of which is
characterized by a suitable reduced mapping schema $\bar S(f)$.
This is a rather crude invariant, which depends only on the
topology of $f$ restricted to the complement of the Julia set.
Any two components with the same reduced mapping schema are
canonically biholomorphic to each other. There are similar
statements for real polynomial maps, or for maps with marked
critical points.

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