Caleb Martin
Properties of Julia Sets

Let f :
be a polynomial f ( z )  an z n  an 1 z n 1 
 a0 of degree at least 2, and let f k ( z ) be the k-fold
composition of f , so f k ( w)  f ( f ( ( f ( w)) )) denotes the k th iterate of w under f . Then we define the filled-in
Julia set for f
K ( f ) : {z 
: f k ( z ) ½ }.
The Julia set for f is then defined to be the boundary of this set, J ( f ) : K ( f ) . For example, if f ( z )  z 2 ,
then K ( f )  {z :  z   1} and J ( f )  {z :  z   1} , where  z  represents absolute value, since clearly f k ( z )  0 iff
 z   1 and
f k ( z )   iff  z   1 .
Lemma 1: Given f ( z )  an z n  an 1 z n 1 
 a0 with an  0 , there exists some constant r such that if  z   r then
 f ( z )  2  z  .
Proof: Choose r large enough that if  z   r , then ½  an  z   2  z  and
n
 an 1  z     a1  z    a0   ½  an  z  . Then if  z   r ,
n
n 1
 f ( z )   an  z   ( an 1  z     a1  z    a0 )
.
n
 ½  an  z   2  z  .
n 1
n
Further, if  f p ( z )   r for some p, then we can apply Lemma 1 inductively to show that
 f p  k ( z )   2 p  f k ( z )   r , so we must have f k ( z )   .
Some fundamental properties of Julia sets follow quickly from this:
1.
J ( f ) and K ( f ) are nonempty and compact, e.g. closed and bounded
2.
J ( f n )  J ( f ) for all n 
3.
J ( f ) is forward and backward invariant, e.g. f ( J )  J  f 1 ( J )
Proof:
1.
With r given by Lemma 1, clearly J ( f ) and K ( f ) are contained in the neighborhood of the origin
N r (0) and thus are bounded. Suppose z  K ( f ) . Then f k ( z )   , so  f p ( z )   r for some p, and since
polynomials are everywhere continuous,  f p ( w)   r for all w in some small neighborhood N ( z ) . So by
the lemma, f k ( w)   for all such w and therefore w  K ( f ) . So
\ K ( f ) is open and K ( f ) is closed.
J ( f )  K ( f ) is also closed, and thus both sets are compact.
2.
By the lemma, f k ( z )   iff ( f n ) k ( z )  f kn ( z )   . So f and f k have the same filled-in Julia sets
and thus the same Julia sets.
3.
Let z  J ( f ) , so f k ( z ) ½  , and we can find a sequence wn  z such that f k ( wn )   for each n.
Thus the points f k ( f ( wn )) 
\ K ( f ) , but f k ( f ( z ))  K ( f ) , and by continuity, f ( wn ) can be chosen
arbitrarily close to f ( z ) , and thus f ( z )  K ( f )  J ( f ) . It follows that f ( J )  J and
J  f 1 ( f ( J ))  f 1 ( J ) . For the reverse inclusion, take z and wn as before and find a sequence
vn  f 1 ( z ) and vn  f 1 ( wn ) for all n. By similar arguments we get f 1 ( z )  J ( f ) , so it follows that
f 1 ( J )  J and the equalities we want hold.
1
Caleb Martin
We say w is an attracting fixed point of f if f ( w)  w and 0  f (w)  1 . We say w is repelling if f ( w)  1
or f ( w)  1 , and similarly for periodic points. (w is periodic with period k if f k ( w)  w for some k) Note we allow
 to be considered as a normal point. Define the basin of an attracting fixed point w to be the set
B( w) : {z 
: f k ( z )  w} .
The following three statements are given without proof, as they rely on some fairly technical results from
complex variable theory, specifically Montel’s Theorem about normal families of complex analytic functions, which
implies that z  J ( f ) iff the family { f k } is not normal at z. However, they give useful characterizations of general
Julia sets, so are worth mentioning.
In particularly, they rely on the result that for any z  J ( f ) , if U is a small neighborhood of z, then

k j
f k (U ) is the entirely of
except for no more than one exceptional point for each j 
.
Theorem 1: Let w be any attracting fixed point of f . Then J ( f )  B(w) .
Theorem 2: J ( f ) is the closure of the repelling periodic points of f .
Theorem 3: If z  J ( f ) , then J ( f ) is exactly the closure of

k 1
f 1 ( z ) .
As a simple example of the first two statements, consider our previous example, f ( z )  z 2 . Attracting fixed
points clearly exist at 0 and  , and J ( f ) is the unit circle, which is exactly B(0) and equivalently, B() .
Moreover, for w  e
i
on the unit circle, iterating f corresponds to rotating by
points at all rational values of
 . Since
is dense in
, the closure of
 , so we have repelling periodic
i
{e :   } is exactly the unit circle.
The Mandelbrot Set
We now focus our attention on the specific class on functions f c ( z )  z 2  c , where c is some complex constant.
For simplicity, we write J c to mean J ( f c ) . By Theorem 3, if we can reliably locate at least one point of J c we get
a convenient way to plot the entire Julia set. Since f c ( z )  z 2  c , its inverse is given by fc1 ( z)   z  c . We can
use a deterministic game to plot iterations of f  f1 , f 2 , where the f i 's correspond to the positive and negative
branches of f c1 each chosen with probability ½. However, it takes many iterations to fill the entire Julia set, as
points tend to cluster around certain regions.
Note that our choice of functions to examine is not as restrictive as it seems. Consider the
transformation g ( z )   z   . Then g 1 ( f c ( g ( z ))  ( 2 z 2  2 z   2    c) /  , so by carefully choosing  and
 we can produce a conjugate map g 1 f c g  f for any degree 2 polynomial f we like. Since g ( z )   z   is
a linear map, this means that the Julia sets J f and J c are geometrically similar to each other.
Now we are able to define the Mandelbrot set,
M : {c  : J c is connected}
Note M is a set in the parameter plane, not the normal complex plane where the Jc’s reside. We are ready to
prove our main result of this section, using the following lemma.
2
Caleb Martin
Lemma 2: Let  be a loop (simple closed curve) in the complex plane.
1.
If c is interior to  , then f c1 () is also a loop, with the inverse image of the interior of  equal to the
interior of f c1 () .
2.
If c lies on  , then f c1 () is a figure-8 with self-intersection point at the origin, such that the inverse
image of the interior of  is equal to the interior of each leaf.
Proof: Since fc1 ( z)   z  c and ( f c1 )( z )  ½(z - c) ½ , these are finite and non-zero when z  c , and thus each
branch of f c1 () is locally smooth provided c   .
1.
Suppose c  Int  . Take any initial value w on  and choose one of the two values of f c1 ( w) . If we let
f c1 ( z ) vary as z traverses  , then f c1 ( z ) traces out a smooth curve. When z returns to the initial value w,
f c1 ( w) takes on its other value, and as z traverses  again, f c1 ( z ) continues along a smooth path and
returns to the first value of f c1 ( w) when z returns to w a second time. Since c   , 0  f c1 () , so
f c ( z )  0 along f c1 () . This implies that z0  f c1 ( ) cannot be a point of self-intersection, since then
f c ( z0 ) would be a self-intersection point of  . So f c1 () is also a loop. Note as z traverses  once, the
rays from 0 to f c1 ( z ) fill half the interior of f c1 () , and as z traverses  again, the rays from 0 to the
other value f c1 ( z ) fill the other half, so the correspondence from f c1 (Int ) to Int f c1 () is one-to-one.
2.
Proof is similar to (1). Let c   , so if  is a piece of curve through c, f c1 () consists of two smooth
pieces of curve through 0 which are perpendicular. As in (1), if we take an initial value w on  , choose a
value of f c1 ( w) , and let z traverse  , then f c1 ( z ) traces out a closed loop, hitting 0 at z  c , then taking
on the other value of f c1 ( w) and tracing out another, disjoint, closed loop before closing when z  w
again. For z on the first loop, rays from zero fill the interior, and similarly for z on the second loop, so we
have the desired correspondence between f c1 (Int ) and each leaf of Int f c1 () .
Theorem 4: Let M be the Mandelbrot set. Then the following are equivalent definitions:
M  {c 
:{ f ck (0)}k 1 is bounded}
 {c 
: f ck (0) ½  as k  }
Proof: If we let r  2 , then we can apply Lemma 1 to show that these two definitions are equivalent to each other.
We show that if { f ck (0)}k 1 is bounded, then J c is connected, and for the reverse inclusion, we show that
if f ck (0)   , then J c is disconnected. We write 1 for f c1 () , and   k for ( f c1 ) k ()  f c k () .
( ). Assume { f ck (0)}k 1 is bounded, and let 0 be a circle large enough to completely enclose { f ck (0)}k 1 , with the
additional properties that all points outside 0 are in B() and that f c1 ( 0 ) lies in the interior of 0 . Since
c  fc (0) , we must have c inside 0 . By Lemma 2.1, f c maps Int  1 to Int  0 , so since f c (c)  f c 2 (0) , we
must also have c inside 1 . We proceed inductively, applying the lemma at each stage. For k 
1
c
( k 1)  f (k ) , and at each k, c  Int   k since f
k 1
c
(0)  Int  0 , so f (0)  Int  1 , so f
k
c
k 1
c
,
(0)  Int  2 , and
so on. Thus we construct an infinite series of nested loops, each contained in the interior of the last.
Let K n   n  Int  n , and let K 

j 0
K j . By our choice of 0 , any point outside K n for some n must
iterate to infinity, so it follows that B() 
\ K . But then the filled-in Julia set K c is exactly K.
So B()  ( \ K )  K  Kc  J c . From a basic result in topology, a nested sequence of compact
connected sets has connected intersection and boundary, so we see that J c is connected, as desired.
3
Caleb Martin
(). Assume { f ck (0)}k 1 is unbounded, i.e. f ck (0)   . Then let 0 be a circle large enough that all points outside
0 are in B() , f c1 ( 0 ) lies in the interior of 0 , and that for some value of p, we have f c p 1 (c)  f c p (0)   0
with k  p  f ck (0)  Int  0 , and k  p  f ck (0)  Ext  0 . We begin just as in (1), constructing a series of
nested loops { k } , but when we get to k  p , where c lies on the curve   ( p 1) instead of interior to it and
Lemma 2.1 no longer applies. From now on we apply Lemma 2.2, so we get that   p is a figure-8, with J c
contained in its interior. Since we have the one-to-one correspondence between inverse images given by the
lemma, each leaf of the figure-8 contains points of J c , and thus J c is disconnected.
Note we can actually do better than this and make our original definition of M more robust. If J c is
disconnected, we can inductively apply Lemma 2.2 to show that after some   n , there is an infinite series of figure8’s contained in each leaf of the previous curve 1 n , all containing nonempty subsets of J c . Thus J c consists of
infinitely many disconnected sets, and is actually a Cantor dust (each of these sets consists only of a single point),
but that requires some extra analysis to prove explicitly.
See Appendix A for an illustration of the general idea behind this proof.
Periodic Orbits and the Structure of M
The large cardioid that comprises the bulk of the Mandelbrot set corresponds to those values of c where f c ( z )
has an attracting fixed point. Julia sets in this region are deformed circles, with the point c  0 giving a the unit
circle as we saw before, and J c becoming increasingly complex as c approaches the boundary of the cardioid. We
use this characterization to solve for its boundary explicitly.
If f c ( z ) has an attracting fixed point at z  rei , f c ( z )  z 2  c  z and 0  f c ( z )  2r  1 . Clearly z on the
boundary of such points satisfies 2r  1 , or z  ½ei , 0    2 . Substituting back into f c , we get 4c  2ei  e2i ,
which indeed traces out the cardioid as  traverses 0, 2  . Reparametrizing, we can model the equation as
{Re(c)  ½cos( ) -¼cos(2 ), Im(c)  ½sin( ) -¼sin(2 )} .
Similarly, we can solve explicitly for the circular bulb to the left of the main cardioid by noting that it
corresponds to values of c where f c ( z ) has a period-2 orbit.
If f c ( z ) has a period-2 orbit, then f c 2 ( z ) has an attracting fixed point. Then for z in the circular region,
f c2 ( z )  ( z 2  c) 2  c  z . Factoring, we get
[( z 2  c) 2  c]  z  ( z 2  z  1  c)( z 2  z  c)
Roots of the second term on the right are just the fixed points of f c ( z ) , which makes sense because fixed points of
f c ( z ) are also fixed under f c 2 ( z ) . Let z1 , z 2 be the roots of the first term on the right. The product of the two roots
is the constant term, so we get z1 z2  1  c . Since they are period-2, z12  c  z2 and z22  c  z1 . Differentiating
gives us ( f c2 )( z1 )  4 z1 ( z12  c)  4 z2 ( z12  c)  ( f c2 )( z2 )  4 z1 z2 . Since they are attracting, we know that
0  ( f c2 )( zi )  1 , and equating these gives the condition 0  4(1  c)  1 , the boundary of which is indeed a circle of
radius ¼ centered at c  1 .
4
Caleb Martin
Note the correspondence shown below between points in the parameter plane where the bulbs of the Mandelbrot
set are attached to the largest adjacent bulb and the bifurcation points of the logistic map. In the large cardioid, we
correspond to stable orbits of G ( x) , in the period-2 bulb, we correspond to period-2 orbits, and similarly all the way
down the period-double cascade in the logistic map until it dissolves into chaos and the Mandelbrot set is a segment
of the Re(c) axis. The cardioid of the smaller copy of M centered around c  1.75 corresponds to the stable
period-3 region that arises in the bifurcation plot, and similarly for the period-doubling cascade following it, and so
on. It is theoretically possible to solve for the bounds of any bulb on the Mandelbrot set using similarly descriptive
conditions on periodic orbits of f c , but the complexity of the relevant equations scale up inordinately quickly for
efficient computation. As we move along the negative Re(c) axis, the next largest copy of M will a period-4 region
in its cardioid, and so on. Note the periods of the bulbs on all these smaller copies are exactly the same as that of the
corresponding bulb on the whole of M, except multiplied by the period of main cardioid in the copy.
M, plotted in black on the parameter plane Im(c)  Re(c) :
Bifurcation diagram of the map g ( x)  x 2  c :
5
Caleb Martin
Extending this, all of the bulbs on the Mandelbrot set can be put in a correspondence with the existence of
periodic orbits for a given period, but this correspondence turns out to be extraordinarily detailed. For example, in
the equation previously given for the boundary of the main cardioid, 4c  2ei  e2i , we observe that k 
, the
point given by   2 / k is exactly where a period-k bulb is connected to the main cardioid. (We just demonstrated
this for the k  2 bulb) Thus we see that as we travel towards the origin along the boundary of the main cardioid
starting at c  0.75 , the next largest bulb (the topmost one) is the period-3 bulb, the next largest the period-4, and
so on. Immediately we know we can find values of c which give stable periodic orbits of any given period, a
hallmark of chaos, though we are in parameter space and thus ranging over a family of related functions instead of
ranging over initial values for a single function. Moreover, given any two bulbs of period p and q, the period of the
largest bulb between them is always given by p  q .
Right: The primary bulbs of M, labeled
according to their period. Note the
behavior as described above.
We can also determine the period of a bulb just by looking at it, either in the parameter plane or by looking at a
Julia set for some c value in the bulb. In a period-k bulb, there will always be a large spindly arm extending away
from the bulb that separates into several smaller arms. The number of arms that meet at this intersection is always
exactly the period, k. Moreover, the filled-in Julia set at c will have a large central region symmetric about the
origin, pinching down on either side to a single point where several smaller buds meet. Since Julia sets are fractals,
self-similarity assures that this happens on all of these secondary buds as well, but the largest ones are sufficient to
determine the period. As expected, there are exactly k buds meeting at each of these points (so removing any one of
them disconnects K c into k disjoint pieces.
6
Caleb Martin
We illustrate this below for several bulbs on the Mandelbrot set:
Period 3
Period 5
Period 4
Period 7
7
Caleb Martin
The same behavior, this time using Julia sets to determine period:
Period 2, c  1.037  0.17i
Period 4, c  0.295  0.55i
Period 5, c  0.52  0.57i
Period 7, c  0.624  0.435i
8
Caleb Martin
This correspondence between location and periodic orbits goes considerably deeper. Instead of assigning each
bulb an integer k that corresponds to the existence of a stable period-k orbit of f c for values of c in the bulb, we
assign each bulb a rotation number m / n 
2 i / k
to the main cardioid at c  ½e
4 i / k
 ¼e
. For the bulbs we first described (i.e. the large primary bulbs attached
for integer values of k), we say m / n  1/ k . For rotation numbers, the
denominator n corresponds to the period, and if we let {z0 , z1 ,
the number such that f c ( zk )  zk  m for k  1,
, zn } be the stable orbit, we let the numerator m be
,n .
We can still determine rotation numbers by looking at the bulb in the Mandelbrot set or the Julia set of and
appropriate value of c. In the Mandelbrot set, consider the spike complex that we used to determine period. The
shortest spike counterclockwise from the main one connected to the bulb is the mth one. Analogously, in the Julia
sets, the smallest of the n buds counterclockwise from the main one is the mth one.
Thus referring back to our previous examples and reading left-right, top-bottom, the rotation numbers of the
Mandelbrot set bulbs are given by 1/3, 1/4, 2/5, and 3/7. In the Julia set examples, rotation numbers are given by 1/2,
1/4, 1/5, and 1/7. Note for Julia sets, it is difficult to tell if a bulb is primary or not, and in general one must check
where the c value actually lies in the parameter plane to check this.
It is easy to see that points where bulbs attach to the main cardioid of the Mandelbrot set are dense on its
boundary. Amazingly, the set of points where the bulb of rotation number m / n 
attaches to the cardioid are
ordered precisely by the natural order of the rationals. Since M is symmetrical with respect to the Re(c) axis, we can
imagine traversing the around boundary of the main cardioid starting and ending at the origin, and at every rational
point q  m / n we will hit a bulb of rotation number m / n .
Also, note that we can generalize rotation numbers of primary bulbs to bulbs of arbitrary order by analogous
definitions, where a k-ary bulb defined recursively as a bulb attached by a point to a (k-1)-ary bulb and primary
bulbs the ones touching the main cardioid by assigning k-ary bulbs a k-tuple of rotation numbers in
k
.
9
Caleb Martin
To expand on this, we can still calculate rotation numbers from the rotation numbers of nearby bulbs in a
similar fashion as we had for periods. Given bulbs with rotation numbers m1 / n1 and m2 / n2 , the rotation number of
the largest bulb between them is not given by m1 / n1  m2 / n2 , but rather by (m1  m1 ) / (n2  n2 ) , the naïve
definition of addition in the rational numbers. There is an explanation for this occurrence, however. This type of
addition is relevant in number theory, where it is termed Farey addition, and it describes the relationship that given
two (proper) fractions m1 / n1 and m2 / n2 , the resulting sum (m1  m1 ) / (n2  n2 ) is the (proper) fraction strictly
between m1 / n1 and m2 / n2 with lowest denominator. This makes sense with our description of how the rotation
numbers ordered the bulbs along the boundary of the main cardioid in the same manner that
is ordered as a field.
Right: an illustration of Farey addition for
rotation numbers of bulbs in the
Mandelbrot set, (e.g.
(1  1) / (2  3)  2 / 5 )
Finally, because of this structure of how rotation numbers and periods are located around the boundary of the
cardioid, we can provide geometric trajectories around the parameter plane that represent nearly any algebraic
sequence we wish:
Right: For example, we can represent
the familiar Fibonacci sequence
F (0)  F (1)  1 , F (n)  F (n  1)  F (n  2)
by starting with F (2) the period-2 bulb, and
then taking F (3) to be the period of the
next largest bulb clockwise around the
boundary, F (4) the period of the next bulb
counterclockwise, and so on, so we always
consider the largest bulb between the
previous two.
10
Caleb Martin
The Mandelbrot Set and π
The following procedure describes an embedding of  in the structure of the Mandelbrot set, which needless to
say, is very surprising, considering that M is generated by the dynamics of a quadratic system. Suppose we wish to
verify that the period-2 bulb is indeed only attached to the main cardioid at a single point, namely c1  0.75  0i .
Since M is symmetrical with respect to the Re(c) axis, we need only consider approaching c1 from one direction
along the line L1 ( )  c1  i for real values of  , and verify that as   0 and, we always have L( )  M .
Similarly, suppose we wanted to verify that the valley of the main cardioid indeed comes to a sharp point and
reaches zero width only at the point c2  ¼ . Using the same approach, this is equivalent to considering approaching
c2 along the line L2 ( )  c2   , and verifying that as   0 and L2 ( )  c2 , we always have L( )  M .
We know if f ck (0)  2 for any k, then 0 must escape to infinity, so c  M . Using a computer, we iterate f ck ( z )
for c  {c1   i, c2   } and record the number of iterations required before f ck ( z )  2 and let  decrease towards 0.
Results for   10 n are shown:


N (c1 )
0.1
0.01
0.001
0.0001
0.00001
0.000001
34
317
3144
31417
314162
3141594
0.1
0.01
0.001
0.0001
0.00001
0.000001
N (c2 )
8
20
97
912
991
3140

1E-7
1E-8
1E-9
1E-10
1E-11
1E-12
N (c2 )
9933
31414
99344
314157
993457
3141625
Here N (c j ) represents the number of iterations required until fckj (0)  2 for each c j . Near c1  0.75  0i , we
see that lim N (c1 )     , and near c2  ¼ , lim N (c2 )    . In both cases, the convergence is exponential, but
 0
 0
we show twice as many values of  for N (c2 ) so the convergence in every other case is more clearly observed.
While the following does not comprise and explicit proof that lim N (c2 )    , and says nothing about why 
 0
should arise in the structure of M, it at least gives a numerical argument as to where this limit behavior comes from.
Since letting L2 ( )  c2 gives a trajectory entirely along the Re(c) axis, we show this only for c2 . Similar
methods could be applied around c1 , but the calculations would no longer be involving only real numbers and thus
become much more complicated.
11
Caleb Martin
The basic idea here is we exploit the previously discovered connection between key structural points in the
Mandelbrot set and bifurcation points of real-valued functions. First consider f ck (0) for c  c1 as a series {xn }
in
, given by
xk 1  xk2  ¼ x   , where x0  0 .
We interpret this as a local linear approximation for the equation x(t )  f ( x(t )) , so the iterated system can be
expressed as xk 1  xk   x f ( xk ) for some function f that depends on  , and step size  x . Consider  x  1 .
Then
f ( xk )  xk 1  xk  xk2  xk  ¼   .
Equating these results, we obtain the differential equation x(t )  x 2 (t )  x(t )  ¼   . Letting our initial conditions
be x(0)  0 , we want to solve for when x(t ) reaches 1, which is the number of steps required before xk  1 .
Separating variables, we obtain

t
x(u )
du   du .
0
x(u )  x(u )  ¼  
t
2
0
Substituting x(u)du by dx , and t by x(t ) on the left integral, we get
dx
t .
x  x ¼ 
1
1
 x
arctan 

An antiderivative of the integrand on the left is given by


2



x (t )
0
gives
1
 x(t )
arctan 


  2 
1
2

 , so evaluating the integral above

 1
 1 
arctan 

  t . We originally wanted to solve for the value of t when


2  
x(t )  1 , so making that substitution, we obtain
 1 
2 arctan 
t.

2  
1
Since lim arctan    , letting   0 gives exactly limt    , as desired.

 0
Right: Cobweb plot of g ( x)  x 2  ¼ for x0  0 ,
illustrating the tangent bifurcation point located
at c  ¼ , which corresponds to f ck (0) .
12
Caleb Martin
Similarity between Julia sets and M
Consider the following picture:
In the top two pictures, we have the Mandelbrot set, overlaid by the result of magnification about the point
z  0.745429  0.113008i in the parameter plane.
In the bottom two pictures, we have the Julia set for c  0.745429  0.113008i , overlaid by the result of
magnification about this same point in the complex plane. Note that the top zoom into the Mandelbrot is by a factor
of ~ 106 , while the bottom zoom into the corresponding Julia set is by a factor of ~ 105 and rotated by ~ 55 . Still,
the similarity cannot be a coincidence, and it is immediately obvious that something very bizarre and very deep is at
work, since the two plots not only are displaying different objects, but are plotted in a completely different plane.
13
Caleb Martin
In general, this phenomenon is not well-understood, but at certain points in the plane, rigorous results have been
proven. For complex values of   rei , we can consider multiplication by  to represent magnification by r  1 and
rotation through  . We say a set A is self-similar at z if r  0 such that N ( z )  A  N ( z )   A for all 0    r .
In the case of M and J c , the local structures are too complex to demonstrate this kind of self-similarity, but the limit
objects of repeated magnification may. So we extend this and say a set A is asymptotically self-similar at z if there is
some scaling factor  , a small radius r and a limit object L which is self-similar at the origin and satisfies
lim N (0)   n ( A  z )  N (0)  L , where A  z is elementwise subtraction. For example, a circle passing through
n 
the origin is asymptotically self-similar all points on it. Under any scaling factor, the intersection of the circle is an
arc which straightens out under repeated scaling and has limit object equal to the line segment from  to  .
For the iterated system zn 1  zn2  c , define an initial point z 0 to be preperiodic if there are constants m, p  1
such that zm  zm  p . Note m cannot be 0, we want to pick out those initial values which are strictly eventually
periodic. We define a parameter c 
to be a Misiurewicz point if z0  0 is preperiodic. We have:
1.
If c is a Misiurewicz point, then the corresponding periodic orbit is repelling
2.
If c is a Misiurewicz point, then Kc  J c , i.e. the Julia set has no interior and thus forms a dendrite
(deformed line segment with 0 or  -ly many branches)
3.
Misiurewicz points are dense in the boundary of M
By a theorem of Tan Lei (1989), if c is a Misiurewicz point, then the Julia set J c and the Mandelbrot set are
asymptotically self-similar at z  c with the same scaling factor    i  m f c ( zi ) , and moreover, the associated
m p
limit objects are the same up to some rescaling and rotation.
Consider the point c  2 . Then the orbit of 0 is given by 0  2  2  2  ... , so c is a Misiurewicz point. J 2
is given by the line segment [-2, 2] on the Re(c) axis, and in the parameter plane, c  2 is the leftmost tip of
the Mandelbrot set. Then clearly, zooming in on z  2 in the complex plane and c  2 in the parameter
plane both give limit objects of a ray pointing right. Any scalar greater than 1 is sufficient for  .
For a more interesting example, consider c  i , orbit given by 0  i  1  i  i  1  i  ... , so i is a
Misiurewicz point, and the scaling factor is given by
   i  m fc ( zi )   i 1 2 zi  4i(1  i)  4  4i  4 2ei / 4 .
m p
2
Thus we should expect that continually zooming into J i about z  i and zooming into M about c  i by a
factor of 4 2 and rotating through 45 should produce similar images. Indeed, we see this in the following:
14
Caleb Martin
Illustration of behavior near the Misiurewicz point c  i :
In the top figures, we start with the full Mandelbrot set and zoom in on the marked box three times, centered at
c  i , each time by a factor of 4 2 and rotating through 45 . In the bottom figures, we start with the Julia set J i
and zoom in on the marked box, centered at z  i , three times by the same magnification and rotations. Notice that
after zooming in by a factor of only (4 2)3  180 , the objects look almost the same. Also, every 8 times we zoom in
again, we will end up with the object, since 8( / 4)  2 .
Finally, there is one point we have glossed over here. By self-similarity of M, we should expect to find
miniature copies of M everywhere in it, seemingly contradicting our result that M looks like J on small scales, since
clearly J can contain no such structures. However, when you zoom in according to powers of R, these copies of M
scale down in size proportionally to R2, so they quickly disappear in the limit object.
15
Caleb Martin
Other Notes of Interest
Since c  M  f c k (0)    k : f c k (0)  2 , we can apply this line of reasoning in reverse to define a
(decreasing) series of curves that each enclose M given by M k  {c 
log 2 f cn (0)
 2k } .
n 
2n
: lim
Graphically, these curves approach and become increasingly
convoluted for large values of k. It is conjectured that M is
actually given by the limit behavior
M  

k 0
M k* , where Int M k*  M k ,
but this is not known. Note that this means M can be obtained
by collapsing arcs on a circle, and thus is equivalent to the
conjecture that M is pathwise connected. ( M is only known
to be connected) Shown at the right are the first few lemniscates
in the series M1 through M6.
By a theorem of Shishikura (1994), we have that M has Hausdorff dimension 2. Thus the structure of the
Mandelbrot set in a sense is “maximally” complex, since the (fractal) dimension of the boundary is exactly equal to
the dimension of the set itself.
As a final curiosity, consider generalizing the Mandelbrot set to reflect f c ( z )  z n  c for values of n other than
2. For n  1 , we just get {0}. For n  2 , we get something similar to M, except the main cardioid is no longer a
cardioid but exhibits radial symmetry about the rays  n 1 , the (n  1) st roots of unity. Thus, we can relate the
generalized Mandelbrot set of “order” n to the dihedral group of order 2(n  1) . If we replace z n by its conjugate
z n , the resulting Mandelbrot set still obeys this property, but instead of the large central region having bulges along
 n 1 , we have thin bars along  n 1 . (note the shift in index)
Above: generalized Mandelbrot sets for z 2 , z 3 , and z 4 ; (top) z 2 , z 3 , and z 4 (bottom).
16
Caleb Martin
Appendix A
In the top figure, we see a graphical interpretation of the preceding proof for connected Julia sets, with a series
of loops squeezing down onto the limit curve, J c , and on the bottom figure, we see an illustration of the proof for
disconnected Julia sets, where eventually we get a series of nested figure-8 curves, forcing J c to be disconnected.
References
Crownover, Richard. Introduction to Fractals and Chaos. University of Missouri-Columbia, Jones and Bartlett
Publishers: 1995.
Demidov, Evgeny. The Mandelbrot Set: Anatomy. <http://www.ibiblio.org/e-notes/MSet/Contents.htm>. 27
October 2006.
Falconer, Kenneth. Fractal Geometry. University of St. Andrews, UK, Wiley and Sons, Ltd: 2003.
Lei, Tan. Similarity Between the Mandelbrot set and Julia sets: Communications in Mathematical Physics, vol. 134,
Number 3 (1990), 587-617.
Peitgen, Jurgens, and Saupe. Chaos and Fractals. Springer-Verlag, Inc: 1992.
Equations typeset using MathType 6.
17