The fixed points and periodic orbits
The first derivative of quadratic map f(z) = z2 + 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 δ z
f(z* + δ z) =
f(z*) + λ δ z +
O(δ z2) ~ z* + λ δ z
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: z1 → z2 →
... → zn = z1
n points zi (i = 1,2,...,n) are the fixed
points of the n-fold iterate f on(z) = f(f(...f(z)))
of map f
zi = f on(zi).
There is product formula for multiplier of a period-n orbit
λ = (f on)'
(zi) = f '(z1) f '(z2) ... f '(zn)
= 2n z1 z2 ... zn .
Basin of attraction of periodic orbit
If O is an attracting period-n orbit zi
(i = 1,2,...,n), we called the basin of attraction of zi
the open set of all points z for which the successive
iterates f on(z), f o2n(z),... converge towards
As we know, for any c in the interior of a (m / n)
primary bulb of the Mandelbrot set 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
the pictures above).
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 z1 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 [Peitgen&Richter, Schroeder].
And we have one order of colors ("red → blue → green" under
revolutions in the counterclockwise direction).
One more "daisy"
Previous: The Fundamental Dichotomy for J-sets
Next: Local dynamics at a fixed point
updated 22 February 2000