# Polynomial maps

# Quadratic maps

For iterations of a quadratic map *z*_{k+1} =
az_{k}^{2} + bz_{k} + c
by a linear transformation *t = Az + B* (which includes two translations by
*Re(B), Im(B)*, scaling by *|A|* and rotation by *Arg(A)* )
one can exclude two parameters from *a, b, c* and reduce the map
to iterations of *t*_{k+1} = t_{k}^{2} + d
(see the proof in Appendix below).

Quadratic maps *f(z) = P*_{2}(z) have the only
critical point (that is a point where
*P'*_{2}(z) = dP_{2}(z)/dz = 0) and
it is in *z = 0* for the map *f(z) = z*^{2} + c.
The critical points are important due to Fatou theorem: every attracting
cycle for a polynomial or rational function attracts at least one critical
point. Thus, quadratic maps may have the *only* attracting cycle and
testing the critical point shows if there is any attracting cycle.
# Polynomial maps

For a cubic map *f(z) = P*_{3}(z) from 4 parameters one can exclude 2
so it remains two complex parameters. Usually maps *f(z) = z*^{3} -
3a^{2}z + b are used with two critical points at *z = +-a*.
Unfortunately there are two complex parameters (4 real numbers), therefore we
can plot only two dimentional cross-sections of the 4D space. E.g. we can
watch iterations of real cubic maps.
The simplest polynomial maps are *f(z) = z*^{N} + c . They have
degenerate critical point at *z = 0*.

Here you see the *z*^{3}, *z*^{4} Mandelbrot
sets and corresponding Douady rabbits.
# Bifurcations map of the x^{4} Mandelbrot set

I've "measured by scale" *α = 1.69* and *δ = 7.34*.
*Appendix* Let us prove equivalence of the maps *z = az*^{2} +
bz + c and *z = z*^{2} + c directly

*az = a(az*^{2} +bz + c) = (az)^{2} +b(az) + ac =

[(az)^{2} +2(az) b/2 + (b/2)^{2}] -
(b/2)^{2} + ac = (az + b/2)^{2} -(b/2)^{2} + ac

then

*(az + b/2) =
(az + b/2)*^{2} -(b/2)^{2} + ac +b/2 or

*t = t*^{2}+ d, t = az + b/2,
d = - (b/2)^{2} + ac +b/2.

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*updated* 23 Sept 2002