Misiurewicz points and the M-set
Douglas C. Ravenel
Here are illustrations of M near some of Misiurewicz points.
The images are zoomed 4, 3 and 1.3283 = 2.34 times
respectevely. Preperiodic points are in the center of the pictures.
Some self-similar periodic points with its period are shown too.
click one of these 3 images to Zoom it in 4 times
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 c0 , the Julia set J(c0) near the
point z = c0 looks very much like the Mandelbrot set
M near c0 . 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.
You can explore preperiodic point vicinity and find "periods" of miniature
"Animated Julia explorer with Orbits 350+350"
 Douglas C. Ravenel
Fractals and computer graphics
Previous: Periodic and preperiodic points in M
Next: M and J-sets similarity. Lei's theorem
updated 22 June 2002