# The fixed points and periodic orbits

The first derivative of quadratic map *f(z) = z*^{2} + c at
a fixed point *z*_{∗} = f(z_{∗ })

*λ = f '(z*_{∗ }) = 2z_{∗}

is a complex number called *multiplier* (or the *eigenvalue*)
of the point. In particular, *f ' = 0* for critical points of
*f*. For small enough *ε*

*f(z*_{∗} + ε) =
f(z_{∗ }) + λ ε +
O(ε^{2}),

so a fixed point is either *attracting* or *repelling*
or *indifferent (neutral)* according as its multiplier satisfies
*|λ| < 1* or *|λ| > 1* or *|λ| = 1*.

It is evident that for a period-*n* orbit (or cycle)

*f: z*_{1} → z_{2} →
... → z_{n} = z_{1}

*n* points *z*_{i} (i = 1,2,...,n) are the fixed
points of the *n*-fold iterate *f*^{ on}(z) = f(f(...f(z)))
of map *f*

*z*_{i} = f^{ on}(z_{i }).

There is product formula for multiplier of a period-*n* orbit

*λ = (f *^{on})'
(z_{i }) = f '(z_{1 }) f '(z_{2 }) ... f '(z_{n })
= 2^{ n} z_{1} z_{2} ... z_{n} .
## Basin of attraction of periodic orbit

If *O* is an attracting period-*n* orbit *z*_{i}
(i = 1,2,...,n), we called the *basin of attraction of z*_{i}
the open set of all points *z* for which the successive
iterates *f*^{ on}(z), f^{ o2n}(z),... converge towards
*z*_{i }.
As we know, for any *c* in the interior of a (^{m} / n)
primary bulb of the Mandelbrot set the corresponding Julia set is connected.
It contains infinite sequence of *n* joint in one point pieces
(*n = 2* and *3* for the *(1/2)* and *(1/3)* bulbs in
these pictures).
Interior of any of such piece is attracted under iterations of
*f*^{ on}(z) to one of points of period-*n* orbits. Here
all pieces that are attracted to the point *z*_{1} in the center
of the J-set are painted in "red" color. The other pieces (which are attracted
to the second and third points of orbit) are painted in "blue" and "green".
You see that in any point of the J-set only all *n* colors can
meet together.
And we have one order of colors ("red → blue → green" under
revolutions in the counter-clockwise direction).
One more "daisy" below.

Contents
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Next: Local dynamics at a fixed point

*updated* 12 Sep 2013