**Polynomial-like maps**

*Nuria Fagella*
A *polynomial-like map* of degree *d* is a
holomorphic
map (MathWorld) *f: U' -> U* such that every point in *U*
has exactly *d* preimages
in *U'*, where *U, U'* are open sets isomorphic to disc and
*U* contains *U'* in its interier.

Let *f : U' -> U* be a polynomial-like map. The *filled Julia
set K*_{f} of *f* is defined as the set of points in *U'*
that never leave *U'* under iteration.

**The Straightening Theorem** Let *f : U' -> U* be
a polynomial-like map of degree *d*. Then *f* is hybrid
equivalent (quasi-conformally conjugate) to a polynomial *P* of degree
*d*. Moreover, if *K*_{f} is connected, then *P* is
unique up to (global) conjugation by an affine map.

In particular, a hybrid equivalence implies that corresponding Julia sets are
homeomorphic (MW).
This theorem explains why one finds copies of Julia sets of polynomials
in the dynamical planes of all kinds of functions. If *f* is polynomial-
like of degree two and its Julia set is connected then *f* is hybrid
equivalent to a polynomial *z*^{2} + c for a unique value of
*c*. This may also be true for other families of polynomial-like
maps of degree large than two, as long as the resulting class of polynomials
has a unique representative in each affine class. See [1] for details.

**Example A** The obvious example is an actual polynomial *P* of degree
*d*, restricted to a large enough open set. You see quadratic
Julia set *J(-1)* here. Open sets *U, U'* are enclosed by the
*G, G'*
equipotential curves and *G :=
P(G')*. The triple
*(P|*_{U'} , U', U) is a polynomial like map of degree two.

The theory of polynomial-like mappings of A.Douady and J.Hubbard
explains why the very particular family of polynomials is important for
the understanding of iteration of a much wider class of functions namely
thouse that localy behave as polynomials do.

**Example B.1** Consider the cubic polynomial *P*_{a}(z) =
z^{3} - 3a^{2}z - 2a^{3} - a.
For all value of *a*, the
critical point *w*_{2} = -a is a
fixed (superattracting) point. For *a = - 0.6* the critical point
*w*_{1} = a escapes to infinity.
The open set *U* is enclosed by the equipotential curve
*G*.
Then the preimage of *G* under *P* is
a figure eight curve *G'*. This curve bounds
two connected components *U'* and *V*.
*U'* is the component that contains the critical point
*w*_{2} with a bounded orbit,
*U'* maps to *U* with degree two, i.e. every
point in *U* has exactly two preimages in *U'*.

By the Straightening theorem, *P*_{-0.6}(z) restricted to the
open set *U'* is hybrid equivalent to a quadratic polynomial and hence,
to a polynomial of the form *Q*_{c} = z^{2} + c.
In this case *c = 0*, since *Q*_{o}(z) iz the only
quadratic polynomial of this form with the critical point being fixed.
Note that only the largest component in *U'* corresponds to the filled
Julia set of the polynomial-like map of degree *2*.
Let *f* is the restriction of polynomial *P* to a set *U'*.
As *V* maps to *U* with degree one, hence, there are points in
*U'* that map to *V* and come back to *U'* afterwards never
leaving the set *U*. Such points do not belong to *K*_{f}
since they are not in *U'* at all times but they belong to
*K*_{P} since they do not escape to infinity under iteration.
Therefore *K*_{P} might have more connected components than
*K*_{f} but not larger ones.

**Example B.2** For the polynomial *R*_{a}(z) =
z^{3} - 3a^{2}z + ((9a^{2}-4)^{1/2} + a -
4a^{3})/2 the critical point
*w*_{2} = -a is a point of period
*2* cicle for all *a* values. For *a = -0.75* (the right
image) the critical
point *w*_{1} = a escapes to infinity.
*R*_{-0.75}(z) restricted to *U'* as above, is hybrid
equivalent to a quadratic polynomial *Q*_{-1}(z) with the
critical poin being of period two (see Example A).

**Example C** Let *f(z) = p cos(z)*
is an entire transcendental function, *U'* is the open simply connected
domain

*U' = { |Im(z)| < 1.7,
|Re(z)+p| < 2 }*,

and *U = f(U')*. Since *U'* contains only one critical point
*w = -p*, it
follows that *f* maps *U'* to *U* with degree two.
Hence the triple *{f|*_{U'}, U', U} is a polynomial-like of
degree two. Since the critical point *-p* is
fixed under *f*, the largest component (around
*-p* below) is homeomorphic
to the filled Julia set of *Q*_{o}(z) = z^{2}.

As usual, the fenomena in dynamical plane are reflected in parameter space.
To the right you see the parameter *l*-plane
for the mapping

*f*_{l}(z) =
l cos(z).

There is a copy of the Mandelbrot set with
*l = p* as the
center of its main cardioid.
[1] *Nuria Fagella*
The theory of
polynomial-like mappings - The importance of quadratic polynomials

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*updated* 12 May 2002