Misiurewicz points and the M-set
Douglas C. Ravenel
Here are illustrations of M near some of Misiurewicz points.
Preperiodic points are in the center of the pictures.
The images are zoomed 4, 3 and 1.3283 = 2.34 times
respectevely. Some self-similar periodic points with its period are shown too.
In the last figures rotational angle is very close to 120o,
which accounts for the 3-fold rotational symmetry in the picture. In the
center of the picture one has 3 lines meeting, and there are numerous nearby
points where 6 lines meet. At each of the letter points there is a tiny
replica of M.
You see, that preperiodic points explain too spokes symmetry in the largest
antenna attached to a primary bulb.
These pictures have next features in common :
Theorem Let co be a preperiodic point with
period 1. Let cn denote the nearest periodic point
with period n. Then as n approaches infinity
- The preperiodic points are not in black regions of M.
- They exhibit self-similarity, i.e., they look roughly the same at
shrinking the picture centered at preperiodic point by a factor of
|λ| and rotating through the angle
Arg(λ). This becomes more precise as
the magnification increases. The rotational angles of the
sequences are -23.1256o and 119.553o respectively.
This accounts for the slight changes in orientation under successive
magnifications in figures.
- There is a sequence of miniature Ms of decreasing size converging
to the point. Each of them has a periodic point in its main cardioid (see
the theorem below). When we shrink the picture by a factor of
λ the miniature Ms shrink by a factor
of λ2 therefore
nearby miniature Ms shrink faster than the view window, so they
- There is a fourth feature not visible in these pictures: For preperiodic
point co , the Julia set J(co ) near the
point z = co looks very much like the Mandelbrot set
M near co . This is a theorem of Lei, which
we will discuss on the next page.
(cn - c0 )/(cn+1 -
c0 ) → λ = 2h
where h is the fixed point of the critical orbit of
We will use Newton's approximation to find a root of an equation
fCnon(0) = 0
for periodic point cn with period n near
If (cn - co) value is small enough, then
fCo+(Cn-Co)on(0) = fCoon(0) +
(cn - co) d/dc
fCon(0) |C=Co = 0
(we do not prove that we can use this approximation).
For simplicity we will denote
dn = d/dc
fCon(0) |C=Co .
As co is preperiodic with period 1, than
fCoon(0) = h for large enough n, therefore
cn - co = - h/dn .
fCo(n+1)(0) = [fCon(0)]2
+ c, it follows that for large n
dn+1 = 2 h dn + 1
(cn - co )/(cn+1 -
co ) = (h/dn)/(h/dn+1) =
dn+1 /dn = 2h + 1/dn.
The limit of this as n approaches infinity is 2h as claimed,
because dn gets arbitrarily large for large n.
 Douglas C. Ravenel
Fractals and computer graphics
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updated 14 Sep 2012