**Quadratic-like maps and Renormalization**

*Nuria Fagella*
**Example D** Sometimes a polynomial-like map is created as some iterate of
a function restricted to a domain. For example, for *Q*_{c}(z) =
z^{2} + c, c_{o} ~ -1.75488 and

*U' = { |Im(z)| < 0.2, |Re(z)| < 0.2}*

the polynomial *Q*_{Co}^{o3} maps *U'* onto a
larger set *U* with degree *2*. The triple
*( Q*_{Co}^{o3}|_{U'} , U', U ) is a
polynomial-like map of degree two (or quadratic-like map).

A polynomial is *renormalizable* if restriction of some of its iterate
gives a polynomial-like map of the same or lower degree.

You see below the Mandelbrot set and a magification of its homeomorphic
copy near *c*_{o}.

For periodic point *c*_{0} = -1.75488 with period *3*
(see *"airplane"* below) the critical point is fixed under iterations of
*Q*_{c0}^{o3} therefore
the filled Julia set of the quadratic-like map is homeomorphic to circle.
For periodic point *c*_{1} = -1.77289 with period
*6* we have *Q*_{c1}^{o6}(0) = 0. In this case
*Q*_{c1}^{o3} and *Q*_{c1}^{o6} are
renormalizable. The critical point is periodic of period two under iterations
of *Q*_{c1}^{o3} therefore the filled Julia set of the
quadratic-like map is homeomorphic to the Julia set *z*^{2} - 1
(in square). For *Q*_{c1}^{o6} the critical point is
fixed so the renormalized polynomial is *z*^{2} (the greatest bulb
in the center)

**Example E** For *c = -1.401155...* the map *Q*_{c} is
the Feigenbaum polynomial, that is the limit of the cascade of period doublings
in the real axis. For any *n* the polynomial
*Q*_{c}^{o2n} is renormalizable and all these
renormalizations are hybrid equivalent to itself.
Renormalization of *Q*_{c}^{o2} is shown to the left
and below.

**Example F** Let *c = 0.419643 + 0.60629i* is a Misiurewicz point
in the boundary of the Mandelbrot set. For this map *z = 0* becomes
periodic of period two after three iterations (see the picture). Since
*Q*_{c}^{o2} is renormalizable, *z = 0* is fixed
after two iterations of the renormalized map. Hence, the renormilized
filled Julia set is hybrid equivalent to *z*^{2} - 2 , i.e.
a quasiconformal image of the interval *[-2, 2]* (curve *2-0-4*
to the left).
[1] *Nuria Fagella*
The theory of
polynomial-like mappings - The importance of quadratic polynomials

Contents
Previous: Polynomial-like maps
Next: Rabbit's show

*updated* 30 Dec 2013