Some Topological properties of Julia sets of maps
Of the form (λz − λ z 2 )
Naoum-Adil, Talb-Iftichar, M. And Al.salami-H.Q.
Abstract
In this work, we study the topological properties of Julia sets of the
quadratic polynomial maps of the form
(λz − λ z )
2
where λ is a non-zero
complex constant .
Introduction
Complex dynamics is the study of iteration of maps which map the
complex plane into itself . In general ,
their dynamics are quite complicated
and hard to explain , but for some classes of maps , many interesting results
can be proved . For example , one often studies the Julia sets of polynomial
maps . The Julia set of the quadratic map of the form ( z 2 + c ) was studied
extensively . From a dynamical systems point of view , all of the interesting
behavior of a complex analytic map occurs on its Julia set , it is this set that
contains the interesting topology [4] .The idea behind the Julia sets is to study
whether the absolute value of a point in the complex plane converges towards
infinity or not , when it is iterated under a map . All the points that do not go
toward infinity , when iterated , are in the Julia set .
1 - Preliminary Definition
Let C be the complex set or complex plane,The complex plane together
with the point at infinity , denoted by ∞ , is called the extended complex plane
, it is topologically equivalent to the Riemann sphere. We put C ∞ =
C U
{ ∞ } .The metric space of the complex plane is the usual metric , while the
metric space of the Riemann sphere is the chordal metric .we use the symbol
fn
to denote n -th iteration
for n ∈ N . f : C → C is smooth , if f is a
C r - diffeomorphism if f is a C r - homeomorphism such that f
-1
is also C r
. A point x ∈ X
period n if f
n
is called a fixed point if f ( x ) = x . It is a periodic with
(x ) = x
of period n for
, but f
m
(x ) ≠ x
for m < n .. Let x be a periodic point
f . The point x is hyperbolic if
attracting periodic point if
( f n )′ ( x ) < 1 and x
( f n )′ ( x )
≠ 1 , x is
is repelling periodic point if
( f n )′ ( x ) > 1 .
Remark (1-1)
The fixed points of Q λ ( z ) = λz − λ z 2 are z = 0 or z =
if z = 0 then Q ′λ (0 ) =
λ −1
. If
λ
λ . If λ < 1 , then z = 0 is attracting fixed point . If
λ > 1 , then z = 0
is repelling fixed point .If
z=
λ −1
λ
then
λ −1
λ − 1⎞
is repelling fixed
Q ′λ ⎛⎜
⎟ = 2 − λ . If 3 < λ or λ < 1, then z =
λ
⎝ λ ⎠
point . If 1 <
λ < 3 , then z =
λ −1
is attracting fixed point . The critical point
λ
for Qλ is 0.5 .
A is said to be completely invariant under f if f ( A) = A = f −1 ( A) .In [4]
proved that J ( f
) is completely invariant
There are many definition of Julia sets:
Definition (1-2) [2]
Suppose f :
C → C is an analytic map . The Julia set is the closure
of all repelling periodic points of f . That is
J ( f ) = closure { all repelling periodic points of f } .
Definition (1-3) [1]
The family
{ f n} is said to be normal on U
f n `s has a subsequence which either
1.converges uniformly on compact subsets of U ,or
if every sequence of the
2. converges uniformly to ∞ on U
Now, we will give the definition of the Fatou set and Julia set :
Definition (1-4) [2]
Let f : C
be a map . The Fatou set ( stable set ) F ( f
→ C
set of points z ∈ C
such that the family of iterates
some neighborhood of z .The Julia set J ( f
set , that is J ( f ) = { z ∈ C : the family
That is J ( f ) ≡ C \ F ( f
)
) is the
{ f n} is normal family in
is the complement of the Fatou
{ f n}n≥0 is not normal at
z }.
).
Also the previous definition can satisfy on the space C ∞ .
Definition (1-5) [2]
Let f : C ∞ → C ∞ be a polynomial of degree n ≥ 2 . Let K ( f
)
denote
the set of points in C whose orbits do not converge to the point at infinity .
{
That is K ( f ) = { z ∈ C : f ( z )
n
}
∞
n =0
is bounded }
. This set is called filled
Julia set .
Definition (1-6) [2]
Let f : C ∞ → C ∞ be a map . The escape set A( ∞ ) of f is all those
points that escape to infinity , that is
A( ∞ ) = { z : f n ( z ) → ∞ as n → ∞ } .
We can say that A( ∞ ) is the basin of attraction of ∞ . Now we can state
another definition for Julia set .
Definition (1-7) [2]
The Julia set is the boundary of the filled Julia set , that
is J ( f ) = ∂K ( f
Julia set of
) . The complement of the basin of attraction of ∞
f . That is C ∞ \ A( ∞ ) = K ( f ) .
is the filled
Theorem (1-8)
Let f : C → C be a polynomial of degree d ≥ 2 . Then the following
statements are equivalent .
1. J ( f
2.
) is the closure of repelling periodic points .
J ( f ) is the complement of the Fatou set
3. J ( f
) is the boundary of the filled Julia set .
Proof :
1 ⇔ 2 see [1]
2 ⇔ 3 From [2] , J ( f ) = ∂A( ∞ ) .Since J ( f ) = ∂K ( f ) . 2.Julia Sets Properties
In this section , we will study the topological properties of Julia sets
Theorem (2-1) [3] (Bottcher theorem )
Let f : C ∞ → C ∞ be a rational map . If z 0 is super attracting fixed
point , f ( z ) = z 0 + a ( z − z 0 ) + . . . , n ≥ 2 and a ≠ 0 , then f is conjugate to
n
ϕ o f o ϕ −1 : w → wn in some neighborhood of z0 .
The proof can be found in [3] .
Definition (2-2) [3]
Let f : C ∞ → C ∞ be a rational map has a super attracting fixed
point at 0 . Let Ω be the basin of attraction of this fixed point . Define
G : Ω − {0} → R by G( z ) = log ϕ ( z ) . If G( z ) < 0 . Then the map is
called the Green’s maps of f .
Let f : C ∞ → C ∞ be a rational map . Define a critical value to be the
image of a critical point. And, let a branch of the inverse map f
−1
( w)
be the
bijection between a neighborhood of w and a neighborhood of z where
f ( z) = w , w
not a critical value of f .
Theorem (2-3)
The Julia set J (Qλ ) , where Q λ ( z ) = λz − λ z 2 , is connected if and
only if there is no finite critical point of Qλ in the basin of attraction Aλ ( ∞ ) .
Proof :
Let
r(z)=1/z,then F λ ( z ) = r o
Qλ
o
r( z ) = r
o
1
λ λ
z2
Qλ ⎛⎜ ⎞⎟ = r ⎛⎜ − ⎞⎟ =
. Thus F λ (0 ) = 0 . Therefore ∞ is super
⎝ z ⎠ ⎝ z z 2 ⎠ λz − λ
attracting fixed point of Q λ and Q λ ( ∞ ) = ∞ , also Q ′λ ( ∞ ) = 0 . Now in a
neighborhood of ∞ and by theorem (2-1) there exists a conformal map
ϕ
ϕ (Q λ ( z ) ) = ϕ ( z ) 2 . . . (*), where ϕ ( z ) = z + O (1) that is the
such that
following diagram commutes :
Qλ ( z )
U∞
U∞
ϕ
z2
U∞
Such that
ϕ Qλ ϕ −1 = g
U∞
and g ( z ) = z 2 . Next since log ϕ ( z )
logarithmic pole at ∞ , log ϕ ( z ) = log z + O(1) ≅ log z = log
has a
2
2
x +y ,
∂ 2 log ϕ ( z ) ∂ 2 log ϕ ( z )
+
= 0 .Thus log ϕ ( z ) is positive and harmonic
∂ y2
∂ x2
.Since
J (Qλ ) = ∂ Aλ ( ∞ )
and
if
z = 1 , then
log ϕ ( z ) → 0
as
z → ∂ Aλ ( ∞ ) .By definition (2-2) , thus log ϕ ( z ) = G( z ) for Aλ ( ∞ ) .Thus
taking the logarithm of the modulus of (*) for
G( z ) , log ϕ (Qλ ( z )) = 2 log ϕ ( z ) , we have
G (Qλ ( z ) ) = 2G( z )
. . . (**) .
Now a component of the Fatou set map onto another component of the
Fatou set since otherwise a boundary point ( in element of the julia set ) map
to a point in the interior of a component of the Fatou set . This is a
contradiction because by[4], J (Qλ ) is completely invariant . Next if a bounded
component of Aλ ( ∞ ) exists , some iterates of Q λ maps onto the component
of Aλ ( ∞ ) which contains ∞ . This means that for some z in the bounded
component and integer n , Q λ ( z ) = ∞ .This is contradiction because the
n
iterates of a polynomial are polynomials do not have poles . Thus Aλ ( ∞ ) is
connected . Define a level curve of G( z ) as Λ a = {z : G( z ) = a}, where
a ∈ R ,since for z ∈ Λ a , G (Qλ ( z ) ) = 2G( z ) = 2a , then Qλ ( z ) takes the
level curve Λ a to the level curve Λ 2 a . So Q λ ( z ) ∈ Λ 2 a . Define the exterior
of level curve Λ a to be the set E a = { z :G ( z ) > a } = { z :
ϕ (z )
>
ea } .
Then Q λ ( z ) maps E a two –to-one to E 2 a which is a subset of E a . To
ϕ ( z ) , first consider a neighborhood U = E r of ∞ on which the
extend
theorem (2-1) holds .
Then on E r we can define
2
ϕ ( z ) = ϕ (Qλ ( z ) )
(since z ∈ E r , Q λ ( z ) ∈ E r ) so the right hand side of the equation is
2
defined . Continue in this way defining
ϕ ( z ) on E rn = { z : ϕ ( z ) >
2
r
exp( 2n )} as long as there are no critical point in the extended region . At a
critical point a single –valued analytic map can not be defined . So as n → ∞
∞
,
r
ϕ is defined on U E rn = { z : ϕ ( z ) > exp( 2n )} , that is
2
n =1
∞
U E
n =1
(⇐)
r
2n
= { z : ϕ ( z ) >1 } ={ z : G( z ) >0}= Aλ ( ∞ ) .
Recall that Aλ ( ∞ ) is connected . Now
ϕ is homeomorphism which
maps Aλ ( ∞ ) conformally to the exterior of the unit disk . Since simple
connectivity is preserved by homeomorphism and exterior of the unit disk on
the Riemann sphere is simply connected , Aλ ( ∞ ) must be simply connected
. Thus it follows that ∂ Aλ ( ∞ ) = J (Qλ ) is connected .
(⇒ )
Assume that there exist z 0 ∈ ∂ Aλ ( ∞ ) , z 0 is a finite critical of Q λ ( z )
. Let G( z 0 ) = r 0 and consider Λ r 0 . Differentiating (**) at z 0 yields
⎛ ∂ G (Q ( z ) )⎞. Q ′ ( z ) = 2 ∂ G( z ) , since z is a critical point of Q ,
0 ⎟
0
0
0
⎜
λ
λ
λ
∂z
⎝ ∂z
⎠
∂
∂
∂
Q′λ ( z 0 ) = 0 , thus ⎛⎜ G (Qλ ( z 0 ) )⎞⎟. Q ′λ ( z 0 ) = 2 G( z 0 ) = 0 = 2 G( z 0 ) ,
∂z
∂z
⎝ ∂z
⎠
thus
∂
G( z 0 ) = 0 .So z 0 is a critical point of G( z ) . Thus the level curve
∂z
Λ r 0 consists of at least two simple closed curves that meet at the critical point
z 0 . Within each of these simple curves there exist points in the Julia set. If
not , G( z ) is harmonic and positive on a non-empty region V within one of
the simple curves and the maximum principle applied to G( z ) and - G( z 0 )
gives G( z ) ≤ r 0 and - G( z 0 ) ≤ − r 0 for all z ∈ V . So G( z ) ≡ r 0 on V . Let
be the analytic map with real part equal to G( z ) . Then by
f
uniqueness theorem , G( z ) ≡ r 0
on
the
Aλ ( ∞ ) . This contradicts that
Aλ ( ∞ ) → ∞ . Thus J (Qλ ) is disconnected . Therefore there is no finite
critical point of Qλ in Aλ ( ∞ ) . Proposition (2-4)
If Qλ has a critical point in Aλ ( ∞ ) , then J (Q λ ) has uncountably many
components.
Proof :
Let z 0 be a critical point for Qλ .Let w be an element of the backward
orbit of z 0 , that is Q λ ( w) = z 0 for some n . Then by theorem (2-3),that G
n
defined on this theorem , G (Qλ ( w) ) = 2n G( w) , or G( w) = 2− n G (Qλ ( w) ) .
n
G ( w ) = 2n G ( z 0 ) .
Thus
n
Differentiate
both
sides
to
get
∂
∂
G( w) = 2n G( z 0 ) = 0 ,so that w is a critical point of G( z ) .Thus any
∂z
∂z
level curve consists of at least two simple closed curves that meet at the
critical point w . Since the choice of w was arbitrary , the level curves split in
each of the w ,so follow the splitting by assigning 0 to the left branch and 1 to
the right branch .Since there are uncountably many sequences of 0`s and 1`s
there are uncountably many components of J (Q λ ) . Definition (2-5) [1]
A set is totally disconnected If it contains no intervals .
Theorem (2-6)
Let Q λ ( z ) = λz − λ z 2 . If Qλ (0.5) → ∞ , then J (Q λ ) is totally
n
disconnected .
Proof :
Since ∞ ∈ F (Qλ ) and F (Q λ ) is open , there exists a neighborhood D∞
of ∞ such that D∞ ⊂ F (Q λ ) . And since ∞ is an attracting fixed point of Qλ
, Qλ ( D∞ ) ⊂ D∞ .
Let D = C ∞ \ D∞ . Then D is an open set and J (Qλ ) ⊂ D .
Now, since Qλ (0.5) → ∞ by assumption , choose k large enough so that
n
Qλk maps 0.5 to D∞ . Thus for n ≥ k , there is no critical value of Qλn in D ,
−n
and all the branches of the inverse map Q λ are defined and map D in D . (
Else there exists z ∈ D such that w = Qλ ( z ) ∈ (C ∞ \ D ) = D∞ . Now
−n
Q λn ( w) = z ∈ D , but Qλn : D∞ → D∞ implies that z ∈ D∞ . This contradicts
the choice of z ∈ D ) .Let z 0 ∈ J (Q λ ) , then Qλ ( z 0 ) ∈ J (Q λ ) since the Julia
n
set is completely invariant under Qλ from [4] . Define f n to be the branch of
the inverse map Q λ which maps Qλ ( z 0 ) to z 0 . That is , f n (Qλ ( z 0 ) ) = z 0 .
−n
n
n
{ f n} are uniformly bounded on D .Note that by
above , { f n} is uniformly bounded on a neighborhood
Since f n maps D into D ,
modifying the integer k
of
{ f n}
D . Thus
is normal on
D .Now for all z ∈ D I Aλ ( ∞ ) ,
f n ( z ) accumulates on J (Qλ ) since f n ( z ) → ∂ Aλ ( ∞ ) (except for z = ∞ ) .
Then let f
{ f n } of { f n} .Now f maps
f n ( z ) → w ∈ J (Qλ ) and f n → f ( z ) implies
m
be the limit of some subsequence
D I Aλ ( ∞ ) into J (Qλ ) since
m
f ( z ) = w ∈ J (Qλ ) by the uniqueness of limits . Then by open map see[1]
f is constant since J (Qλ ) = ∂ Aλ ( ∞ )
the boundary of an open set has
empty interior . (If z in the interior of the boundary of an open set U then
there exist a neighborhood of z contained entirely in ∂U .But for any
ε >0 ,
there exists z1 ∈ D( z, ε ) I U , by the definition of the boundary set . This
contradicts that U is open ) . Now , diam
. (Suppose not .
ε >0 and { f nm} such that diam { f nm ( D )} ≥ ε . { f nm} is
Then there exists
normal so there exists
that f n m j → f
{ f n ( D )} → 0
{f } , a subsequence , and
nm j
f a limit map , such
uniformly . By the argument in the previous paragraph ,
f ≡ w0 , a constant . Thus for a fixed branch , f nm j , with j ≥ j 0 ,
f nm j ( z ) − w0 <
ε
3
for all z ∈ D , Then diam
{f
nm j
}< 23ε
(D)
, contradiction
( )
( )
) . Then since f n is continuous , f n D ⊆ f n ( D ) , and f n D has diameter
tending to zero . Next , by invariance of Fatou set , ∂D ⊂ F (Qλ ) implies that
f n (∂D ) ⊂ F (Qλ ) and it is disjoint from J (Qλ ) . Now recall that for
z ∈ J (Qλ )
,
fn
was
f n (Qλn ( z 0 ) ) ⊂ f n ( D )
chosen
since
so
f n (Qλn ( z 0 ) ) = z 0
that
Qλn ( z 0 ) ∈
J (Qλ ) ⊂ D
.
Also
and
,
f n ( D ) ⊂ f n (D ) , so z0∈ f n (D ) for all n . Now diam { f n (D )} → 0 implies
that { z 0} must be a connected component of . To see this recall that D
consists of elements of J (Qλ ) and the boundary which is in F (Qλ ) . For any
ε >0 ,choose k such that diam { f k (D )}< ε . Within this disc f k (D ) , the
boundary is mapped to a curve that winds around elements of the Julia set in
the interior . Thus only points in the Julia set are within 2ε of each other will
be elements of a connected component of J (Qλ ) . But since
ε can be
chosen arbitrarily small , eventually all points of the Julia set will be separated
by the Fatou set , ∂ f k ( D ) for large enough k .By definition (2-6) , J (Qλ ) is
totally disconnected . Corollary (2-7)
Let z be the critical point of Q λ ( z ) = λz − λ z 2 . If Qλ ( z ) → ∞ ,
n
then J (Qλ ) is totally disconnected . Otherwise, {Qλ ( z ) } is bounded , and
n
J (Qλ ) is connected .
Proof :
Since z = 0.5 is the critical point of Q λ ( z ) .If Qλ (0.5) is bounded then
n
0.5 ∈ Aλ ( ∞ ) and J (Qλ ) is connected from theorem (2-3) .Next , if
Qλn (0.5) → ∞ then by theorem (2-6) J (Qλ ) is totally disconnected . References
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edition , Addison-Weseley , 1989 .
[2] Falconer , K.J. , Fractal Geometry , John Wiley & Sons Ltd. ,
England , 1990 .
[3] Milnor , J. , Dynamics in one complex variable :Introductory
Lectures ,
IMS 90-5 ,SUNY stony Brook ,1990 .
[4] Devaney ,R. L., Cantor and Sierpinski , Julia and Fatou
Topology Meets Complex Dynamics,(2003) ,(to appear).
:Complex