ARKIV FOR MATEMATIK Band 6 nr 6
1.65 04 I ( ' o m m u n i e a t e d 24 F e b r u a r y 1965 t)y TRYGVE NAGELL a n d LENNART C A R L I ~ : S ( ) N ~
l n v a r i a n t s e t s u n d e r i t e r a t i o n o f r a t i o n a l f u n c t i o n s By HANS BROLIN
C O N T E N T S
I n t r o d u c t i o n . . . 103
Cha,pter I. Main r e s u l t s c o n c e r n i n g t h e i t e r a t i o n of r a t i o n a l f u n c t i o n s 104 1. D e f i n i t i o n s . . . 104
2. T h e s e t F . . . 105
3. On c r i t i c a l p o i n t s . . . . . . . 108
4. T h e s e t F is h o m o g e n o u s . . . 109
5. On l i m i t f u n c t i o n s of t h e i t e r a t e s {Rn(z)} in t h e c o m p l e m e n t of F 111 6. On t h e i n v e r s e f u n c t i o n s {R_n(z)} of t h e i t e r a t e s {Rn(z)} . . . . 112
7. On t h e s t r u c t u r e of t h e c o m p l e m e n t of F . . . l l 4 8. T h e s t r u c t u r e of F , w h e n t h e n u m b e r of a t t r a c t i v e f i x p o i n t s is ~ 2 l l 6 9. On t h e e x i s t e n c e of t a n g e n t s to t h e c u r v e s t h a t lie in F . . . . 118
10. T h e s t r u c t u r e of F w h e n t h e n u m b e r of a t t r a c t i v e f i x p o i n t s e q u a l s 1 . . . 120
C h a p t e r I I . On t h e i t e r a t i o n of p o l y n o m i a l s . . . 124
11. G e n e r a l r e s u l t s . . . 124
12. On t h e i t e r a t i o n of p o l y n o m i a l s of t h e s e c o n d d e g r e e w i t h real c o e f f i c i e n t s . . . 125
13. On t h e i t e r a t i o n of p o l y n o m i a l s of t h e t h i r d d e g r e e w i t h r e a l c o e f f i c i e n t s . . . 131
C h a p t e r I I I . A s y m p t o t i c d i s t r i b u t i o n of p r e d e c e s s o r s . . . 138
14. D e f i n i t i o n s . . . 138
15. T h e c a p a c i t y of t h e set F . . . 138
16. Mass d i s t r i b u t i o n s p r o d u c e d b y i t e r a t i o n of p o l y n o m i a l s . . . . 140
17. E r g o d i e a n d m i x i n g p r o p e r t i e s of p o l y n o m i a l s . . . 142
R e f e r e n c e s . . . 143
B i b l i o g r a p h y . . . 143
I n t r o d u c t i o n
T h e t h e o r y of t h e i t e r a t i o n o f a r a t i o n a l f u n c t i o n R(z) d e v e l o p e d b y F a t o u [ 5 - 6 ] a n d J u l i a [9] t r e a t s t h e s e q u e n c e o f i t e r a t e s { R n ( z ) } d e f i n e d b y
R o ( z ) - z , R , ( z ) = R(z), Rn+I(z)= RI(Rn(z)) , n - 0 , 1, 2, . . . .
A f u n d a m e n t a l r o l e i s p l a y e d h e r e b y t h e s e t F o f t h o s e p o i n t s o f t h e c o m p l e x p l a n e
s . o 1 0 3
H. BROLIN, Inrariant sets under iteration of rational functions
where {RAz)} is not normal. I n the general theory a numl)er of properties of F are deduced. Fatou and Julia have established the possible structures F can have. These structures depend in a very complicated way on the coefficients of R(z). But very little is known about what the possible structures are even for the simplest classes, e.g. the second and the third degree polynonfials with real coefficients. The aim of this thesis is to continue the general investigations concerning F, and to examine the structure of F for the polynomials mentioned above.
Since there exists no modern survey on this subject, we shall devote most of Chapter I to a treatment of the known properties of F and of the theory needed in what fol- lows. A list of references for these known theorelns is added at the end of the paper.
Furthermore, in Chapter I we solve a problem, treated by Fatou under special condi- tions, concerning the Lebesgue measure of F on a line and in the plane under as- sumptions which imply that F is totally disconnected.
In Chapter lI, we consider polynomials. We examine the structure of F for poly- nomials of the second and the third degree. Certain results concerning the second degree polynomials and the polynomial z 3 '-p, p real, have already been established hy Myrberg [10-19].
In Chapter I I I , we define a mass d i s t r i b u t i o n / ~ by placing the mass k -'~ at the k" roots of the equation P ~ ( z ) - % = 0 , where P(z) is a polynomial of degree k and z 0 any point in the plane with at most two exceptions. We prove t h a t ju~-§ under weak convergence, where #* is the equilibrium distribution of F with respect to the logarithmic potential. In proving this, we also establish t h a t the logarithmic capacity of F is positive. Finally, by regarding P(z) as a transformation T on F we prove that T preserves #* and t h a t T is strongly mixing.
The subject of this paper was suggested by Professor Lennart Carleson to whom the author is deeply grateful for his generously given advice and never failing interest.
Chapter I. Main results concerning the iteration of rational functions 1. Definitions
In the investigations of this paper we will use the extended complex plane with the usual topology and the following notation, for a set E.
C E is the complement of E, /~ is the closure of E,
E is the boundary of E,
d(E1, E~) is the distance between the sets E 1 and E 2.
Henceforth R(z) will always denote a non-linear rational function and P(z) a non- linear polynomial. The sequence of iterates {Rn(z)} to be studied is defined by
R o ( z ) = z , R l ( z ) = R ( z ) , R , + I ( z ) - RI(Rn(z)) , n = 0 , 1, 2, ....
Definition 1.1. I] w = Rn(z) we say that w is a successor o[ z and z is a predecessor o/
w, in both cases o[ order n.
Since the fixpoints of the iterates play an important part in iteration theory, we need the following definition.
104
ARKIV FOR MATEMATIK. B d 6 n r 6 Definition 1.2. I/ R,,(:() :~ and R~(u)&~ when p < n , we .~ay that :( is (t /ixpoi~t o/
order ~. The deric,ttice RI~(:~) is called the ntultiplier o/:~.
TILe successor of a fixpoint of order n is a new fixpoint of order n. Furthernlore, tile set {~, R(~.), R2(:r ) ... R,, 1(~)} is called a cycle of order n and all fixpoints of an n-cycle have the same multiplier RI,(=) since Rn(:()=:l-In:. 0 R'(RA(~)).
Definition 1.3. A /ixpoint ~ (or a cycle) o/ order n is called attractive, indi//erent, or repulsive according as [R'~(:~)I < 1 , 1, or > 1 , respecticely. I / R~(~) e 2~~'q, where p and q are integers, we say tkat ~ (or the cycle) is rationally indi//erent.
I n this paper a MS1)ius t r a n s f o r m a t i o n of w -= R(z) is a t r a n s f o r m a t i o n of the follow- ing form
(z, w) -~(Lz, Lw),
where L is linear. I t is easy to see t h a t t h e fixpoints and their nmltipliers are left invariant b y this t r a n s f o r m a t i o n . Finally, we need the following definition.
Definition 1.4. A set E is said to be invariant under R(z) i / R ( E ) = E, and con~pletely in cariant i/ R I(E ) - E = R(E), where R_l(z ) denotes the inverse/unction o~ R(z).
Remark. I f n o t h i n g else is said, R_l(Z ) always means all the inverse branches a n d R~)l(z) one branch.
2. The set F
We shall now introduce t h a t set which is the principal object of our investigations.
Delinition 2.1. The set.F consists o/ those points at which the sequence {R,(z)} is not normal, in the sense o/Montel.
This implies t h a t C F is an open set. Hence the L e m m a 2.1. The set F is closed.
Before characterizing F we m u s t prove t h e Theorem 2.1. F # r
Proo/. Suppose, on the c o n t r a r y , t h a t F = 6 . T h e n {Rn(z)} is nornlal in the whole e x t e n d e d plane a n d there exists a subsequenee {R,~(z)} with a rational limit functioll g(z). B y considering t h e equations R ( z ) - z =0 a n d R 2 ( z ) - z =0 it is easy to see t h a t there exist at least t w o different cycles. T h u s g(z) is n o t constant. Suppose t h a t g(z) is of degree s. Choose p > s. F r o m {R,v_v(z)} we t h e n e x t r a c t a subsequence which tends to a rational function h(z) of degree t > 0 . T h u s Rp(h(z))=g(z), where the degree of Rp(h(z)) is > s , which contradicts t h e hypothesis.
We shall n o w start our investigation of the properties of F .
Theorem 2.2. The set consisting o/the repulsive and rationally indi//erent /ixpoints is a subset o/ F.
105
H. BROLIN, lnvariant s~ts under iteration of rational functions Proo/. It is sufficient to consider f i x p o i n t s of order one.
(a) Let ~ be a r e p u l s i v e f i x p o i n t of order one, a n d for s i m p l i c i t y t a k e ~.: 0. Hence, in a neighl)ourhood of the origin,
R(Z) a l Z 4, (g2Z2-1 .... where
I.,I-l.
Since R , , ( z ) (*~ z # ...
a n d lira ]%1" :- oo.
it is easily seen t h a t {R,,(z)} c a n n o t be nornlal at ~ 0.
(b) Suppose now t h a t :r : 0 is a r a t i o n a l l y indifferent fixpoint of o r d e r one. I t is sufficient to consider t h e case where R ' ( a ) : l, for when R'(~) ,e e~i~' q we consider Rq(z) a n d its iterates. Hence, in a neighl)ourhood of t h e origin,
R , (z) - z -~ *~" a S ' ~ ...
and e v i d e n t l y {R,(z)} c a n n o t be n o r m a l a t ~ O.
Theorem 2.3. The set F is completely invaria~t under R(z).
Pro@ I t is e v i d e n t t h a t if {R,~(z)} converges u n i f o r m l y in a n e i g h h o u r h o o d of
~, t h e n {R,,, i(z)} a n d {R,,,,+~(z)} converge u n i f o r m l y in neighi)ourhoods of R(s a n d R l(~), r e s p e c t i v e l y . Thus, if {R.(z)} is u o r m a l at a p o i n t ~ it is also n o r m a l a t t h e p o i n t s R(~) a n d R 1(~). This implies t h a t C F is c o m p l e t e l y i n v a r i a n t u n d e r
R(z) a n d t h e n F has t h e same p r o p e r t y . This t h e o r e m has t h e following corollary.
@orollary 2.1. The set F does not change, i~ we replace R(z) by a~y iterate R~(z).
L e m m a 2.2. Let ~ be an arbitrary point in F. Then in every neighbourhood o/
the /unctions {Rn (z)} omit at most two values. Moreover, the exceptional points, i/
any, are independent o/ ~ and do not belong to F.
P r o @ I f t h e l e m m a is n o t t r u e , t h e n t h e r e exist a r b i t r a r i l y small n e i g h b o u r h o o d s of ~ in which each Rn(z ) in t h e sequence {R~(z)) onlits a t l e a s t t h r e e values. H e n c e {Rn(z)} is n o r m a l a t ~ (for e x a m p l e see Hille [8] p. 248), which c o n t r a d i c t s t h e as- s u m p t i o n ~ E F .
Consider t h e p o s s i b i l i t y of e x c e p t i o n a l points. Suppose t h a t t h e r e exists one excep- t i o n a l p o i n t a. T h e n a can h a v e no predecessors o t h e r t h a n itself. B y a M6bius t r a n s - f o r m a t i o n we can m o v e a to oo, a n d t h e n t h e t r a n s f o r m e d f u n c t i o n m u s t be a p o l y - n o m i a l .
S u p p o s e n o w t h a t t h e r e exist t w o e x c e p t i o n a l p o i n t s a a n d b. T h e n t h e following two cases are possible.
1 ~ a has no predecessors o t h e r t h a n itself a n d b has no o t h e r predecessors t h a n itself.
2 ~ a a n d b m a k e up a cycle of o r d e r two, a n d all t h e i r predecessors coincide w i t h a or b. B y a M6bius t r a n s f o r m a t i o n we can m o v e a a n d b to 0 a n d oo. Clearly, in t h e
10B
ARKIV FOR MATEMATIK. Bd 6 nr 6 first case the t r a n s f o r n l e d function m u s t be of the fornl M z ~ where M is a c o n s t a n t , a n d in t h e second case of t h e folun M z ~
F r o m this we conclude t h a t t h e e x c e p t i o n a l p o i n t s d e p e n d only on R(z) for 1)y considering t h e t r a n s f o r m e d functions above, we see t h a t t h e e x c e p t i o n a l p o i n t s are a t t r a c t i v e fixpoints of order one or two. Thus, t h e following quite t r i v i a l l e m m a completes t h e proof.
L e m m a 2.3. The seq~ence {R,,(:)} is ~tort~al (~t a~t attmctice [ixpoit~t :r o / ( ~ y order, i.e. ~ C F .
As we know, t h e set F is closed. I t is, however, now possible to prove a m u c h s t r o n g e r result, n a m e l y
Theorem 2.4. The set F is per/ect.
Proo/. Since F is closed it is sufficient to prove t h a t F is dense in itself.
W e first observe t h a t e v e r y ~ C F has a t least one predecessor ~* such t h a t r R,(~). This is e v i d e n t l y t r u e when ~ is not a fixpoint. If ~ is a fixpoint of o r d e r ~, t h e n ~ is in t h e ~-cycle {~, ~1, ~'., ... ~n 1} a n d :~ has a t least one predecessor,
~"_~, such t h a t ~:4:~"-~. Otherwise t h e e q u a t i o n R,(z) - ~ = 0 has a m u l t i p l e root z = s ~, i.e. R ~ ( ~ ) = 0 a n d ~ is an a t t r a c t i v e fixpoint. F u r t h e l a n o r e , ~__, ~ , v - 1, 2 ... ~ , - 1, for otherwise r =s,., some v. Thus we can t a k e ~_~ " ~ to be ~*.
L e t ~ E F a n d choose ~* as above. Since F is c o m p l e t e l y i n v a r i a n t , ~* C F . B y L e m m a 2.2 we conclude t h a t ~ is a n a c c u m u l a t i o n p o i n t of t h e predecessor of ~*, i.e. an ac- c u m u l a t i o n p o i n t of the set F . Thus, F is dense in itself a n d t h e o r e m is proved.
H a v i n g e s t a b l i s h e d T h e o r e m 2.4 we use it to prove t h e converse of L e m m a 2.2.
Theorem 2.5. Let z be any poi~t in the plane with at most two exceptions. Then a point ~ belongs to F i/a~uf only i / ~ is an accumulation point o] the predecessors o / z .
Proo]. The necessity follows from L e m m a 2.2. Suppose t h a t ~ satisfies the condition s u p p o s e d t o be sufficient. Choose a p o i n t ~ E F . T h e n ~ is a n a c c u m u l a t i o n p o i n t of t h e predecessors of ~, i.e. of p o i n t s in F . Since F is perfect it follows t h a t ~ E F , which was to be p r o v e d .
W e shall m a k e use of the following corollary in l a t e r sections.
Corollary 2.2. I / q E F and Pq is the set o/predecessors o / q , then F = P q . Proo/. This follows i m m e d i a t e l y from T h e o r e m 2.4 a n d 2.5.
W e end this first c h a r a c t e r i z a t i o n of F with t h e
Theorem 2.6. I / t h e set{ F contai~s i~terior points, then F is ide~tical with the extended pla~e.
Proo/. Suppose t h a t ~E F is an i n t e r i o r point. T h e n t h e r e exists a n e i g h b o u r h o o d O of ~ such t h a t O c F ~ T h e n a n y p o i n t z different from t h e e x c e p t i o n a l p o i n t s of T h e o r e m 2.5 has p r e d e C s s o r s in ~O, which implies t h a t z E F . F u r t h e r m o r e , it is e a s y to see t h a t e x c e p t i o n a l p o i n t s c a n n o t exist in this case.
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H . BROLIN, Invariant sets under iteration of rational functions
_Remark. I n 1918 Lattd constructed a rational function for which the corresponding set F consists of the whole plane. Thus, this case can really occur. (For example see Cremer [3], p. 199.)
3. On critical points
As we have already mentioned, in this paper we shall be concerned chiefly with finding the properties of the set F. In these investigations, however, the critical points of the inverse functions {R_=(z)} will be of great importance. Therefore we have to discuss their relationship to iteration theory.
Definition 3.1. I / the equation R , ( z ) - c = O has a multiple root, then c is called a critical point o/ the inverse/unction R_n(z ). Hence/orth C will denote the set o/critical points o/all/unctions {R_,(z)}.
We begin by establishing two simple but important results.
Lemma 3.1. The critical points o/R_,(z) consist o/ the critical points o/R_l(z ) and their successors o/order 1, 2, 3 ... n - 1 .
Proo/. Divide the equation R,(z)=c as follows:
R,_l(X ) = c (3.1)
R(z) =x. (3.2)
Equation (3.2) has a multiple root if and only if x is one of the critical points of R 1 ( z ). By (3.1) the successors of order n - 1 of these points are critical points of R_n(z ). Now treat the equation Rn_l(X ) =c in the same way as Rn (z)= c. Repeating this procedure n - 1 times will complete the proof.
I t is quite trivial that the critical points of R_l(Z ) are the first, order successors of the zeros of R'(z). This fact yields the following lemma.
Lemma 3.2. I / R(z) is o/degree d, then N, the number o/ critical points o/ R_l(Z), satisfies the inequality N < 2 ( d - 1).
In this paper a domain always means an oPen connected set. We need the follow-
ing definition. 9
Delinition 3.2. 1. The immediate attractive set A*(~t) o / a first order attractive fix- point r162 is the maximal domain o/normality o/{R~(z)} which contains ~. The attractive set A(~) o/r is defined by
A(ot) = {z 12am= R,(z)= a}.
2. Let (r162 be an attractive cycle o/order n. Then the immediate attractive set A*((ak}) o/the cycle is defined'by
A*({=~}) = U A * ( ~ ) ,
k
108
ARKIV FOR MATEMATIK. Bd 6 nr 6 where A*(ctk) is the maximal domain o/normality containing ak, and the attractive set A({ak} ) o/the cycle is defined by
A({ak})={zl{ak} is the cluster set ol {Rn(z)}}.
Remark. F r o m the definition of A*(~) it is obvious t h a t if zEA*(~r t h e n lira, ~ :r Rn(z)= ~. Furthermore, b y Corollary 2.2 it is easy to see t h a t ~ A ( a ) = F.
The following theorem establishes the influence of the critical points on the n u m b e r of attractive fixpoints.
Theorem 3.1. I f {ak} is an attractive cycle, then there exists at least one critical point c o I R_l(z), such that ceA*({~ck}).
Proo/. Suppose first t h a t ~ is an attractive fixpoint of order one. Then choose a neighbourhood U of a such t h a t U c A*(~) and an inverse branch R*l(Z) which satisfies R * l ( ~ ) = a . Further, introduce t h e functions {R*n(z)} defined in U b y R*n(Z)=
R*I(R*-(~-I)(Z)). Thus, if no critical point of R l(Z ) belongs to A*(a), t h e n the functions {R*_n(z)} are meromorphic in U. Since each R*_=(z) omits at least three values in U, for example the set F , {R*_~(z)} is normal in U. That, however, contradicts the fact t h a t ~ is a repulsive fixpoint of the function R*l(z).
I f ~ belongs to an attractive cycle {~, ~1, ~2, ..-, %}, then we define the functions {R*_~(z)} b y
R*-I(~) = ~ , R*~(~) = ~p-1, .-., R*_~(~) = ~.
N o w we use t h e same a r g u m e n t as above and t h e theorem is proved.
I t is evident t h a t t h e indifferent fixpoints m u s t be considered as exceptional points in the iteration theory. I n characterizing the set F, however, we cannot omit the rationally indifferent fixpoints, which have the same influence on the structure of F as do t h e attractive fixpoints. B u t since a complete t r e a t m e n t of these exceptional points takes more space t h a n the general case a n d does n o t involve a n y special dif- ficulties, we will without proof summarize some of their properties, n a m e l y those of importance for the following.
Theorem 3.2. 1% I f ar is a rationally indi//erent /ixpoint, then there exists an im- mediate attractive set A *(~) which is a union o/maximal domains where Rn(z ) is normal, each o/which has ~r as a boundary point.
2 ~ A*(zt) contains at least one critical point o/R_l(z).
3 ~ The number o] indif/erent/ixpoints is/inite.
4. The set F is homogenous
W e shall now prove an equivalent definition of F, which was the start point of Julia's investigations.
Theorem 4.1. The set F is identical with the closure o/ the set o/repulsive/ixpoints.
We shall need the following lemma in the proof.
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H. BROLXN, Inrariant sets under iteration of rational functions Lemma 1.1. Every ~ E F is a~ accumulation point o/[ixpoints.
Proo] o/Lemma 4.1, Let ~ E F . # oo, be different front the poles and the critical points of Re(z). Then there exists a neighbourhood U of ~ where the d e inverse branches of R 2 ( z ) are bounded, holomorphic and have different ranges. Choose three of these branches R%_(z), R('2?,(z) and R~a.~(z).
Now suppose, on the contrary, t h a t R , ( z ) - z + O , every zE U, and every ~. This implies t h a t
R,,(z) + R~(z), R,,(:) 4- R(2~(z), R,,(z) # RC~(z).
if zEU, every n. Otherwise R,.~(z)=z for some zE U. We introduce the functions
/~,,(z)
R.2(z) R('~(z)-R("_~(z) (~) Rn(z) R(I~(z) cf'(Z)=R,,(z) n~"-~(z) RC~(z) R(l~(z) Rn(z) ~ ( 2 ) 1~~ ' Q ( Z ) 9Each function q,,(z) omits in U the values O, 1, ~ . Thus the sequence {q,~(z)} is normal in U. B u t
(2) ( l )
9:,(z) Q(z)
and we conclude t h a t {R,(z)} is norlnal in U, contradicting $ E F. Thus, the lemma is proved.
Proo/ o/ Theorem 4.1. B y Theorem 3.1 and 3.2 the n u m b e r of attractive and indifferent fixpoints is finite. Since the repulsive fixpoints belong to F all application of L e m m a 4.1 proves the theorem.
By using the previous theorem, we get a simple proof of the following f u n d a m e n t a l result concerning F.
Theorem 4.2. Let E be a closed set containing none o/the exceptional points o/Theorem 2.5. I / ~ E F, then there exists/or every neighbourhood U o/~ an N such that E c RN( U).
Proo/. According to Theorem 4.1 it is sufficient to consider the repulsive fixpoints.
Let ~ be a repulsive fixpoint of order n and choose a ncighbourhood U of ~ such t h a t U c R , ( U ) ~ R 2 , ( U ) = ... ~ R , , ( U ) ~ .
We see from Theorem 2.5 t h a t every z E E belongs to some R~n(U), and since E is closed we can extract a finite subcovcring from {R~,(U)}. If we then choose N equal to the largest index used in this covering, we get E ~ RN(U), and the theorem is proved.
Since F is invariant under R(z) this theorem yields the
Theorem 4.3. I / D is any domain such that D (1 F = F* # r then there exists an integer N such that F = RN(F*).
Remark. Instead of the formulation above we might say t h a t F is "rationally ho~7~ogeveous " .
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ARKIV F6R MATEMATIK. B d 6 n r 6
5. On limit functions of the iterates {R,(z)} in the complement of F
H e n c e f o r t h G a n d G,. will a l w a y s d e n o t e m a x i m a l domains, where {R,,(z)} is normal.
L e m n m 5.1. I / the number o/ limit /unctions o/ ~r is finite, then erery limit /unction is a constant.
Proo/. If lira . . . . R~,,(z)=/(z), uniformly, in some U,., t h e n lira . . . . R~,, j,(z):=:
Rh(l(z)).
According to the a s s u u l p t i o n t h e r e exist integers h a n d N such t h a t Rh(/(z))- Rh+N(/(Z)). W e conclude t h a t / ( z ) is a c o n s t a n t .Theorem 5.1. I / l i m , _ ~ R=,(z) - a , uniformly, in a domain D and i/ el t F, theft ,~
is an attractive/ixpoint.
Proo/. I f D~n--R~,~(D), t h e n t h e sequence {D~,, } converges u n i f o r n d y to z :: (t.
Thus t h e r e exist two d o m a i n s D%. ~ a n d D% such t h a t D% qc D%. B y t a k i n g
we get. for z ED~v q, Observing t h a t
lira Rl~,,(z )- lim Rh%%(z)=a.
Rh4 Z,,(z) -- Rh(Rp,,(z)) (5.1)
we ol)tain, b y t a k i n g limits in (5.1),
a = R h ( a ) .
Thus, a is a f i x p o i n t of o r d e r h. Since a(~F it camlot, however, be a r e p u | s i v e or r a t i o n a l l y indifferent fixpoint. Moreover, if a is a n indifferent fixpoint, not r a t i o n a l , t h e n
R.h(a) =a, IRSh(a)l =1
a n d no c o n s t a n t l i m i t f u n c t i o n can exist in a n e i g h h o u r h o o d of z - a . We conclude t h a t a m u s t be an a t t r a c t i v e f i x p o i n t a n d t h e t h e o r e m is proved.
A m o r e difficult p r o b l e m has been t o decide w h e t h e r n o n - c o n s t a n t limit functions can exist. I t was f i n a l l y p r o v e d in 1942 b y Carl L u d w i g Siegel [20] t h a t t h i s case a c t u a l l y can occur. I n t h e n e x t section we shall e s t a b l i s h a condition due to F a t o u , which excludes t h e p o s s i b i l i t y of n o n - c o n s t a n t l i m i t functions. O t h e r results n e e d e d are s t a t e d in t h e following theorem.
Theorem 5.2. I / l i m ~ _ . ~ Rn,,(z)=/(z) in a domain G and i / / ( z ) is not constant and / ( G ) - G*, then there exists a subsequence which converges to the limit/unction F ( z ) - z in G*. Furthermore, there exists an iterate Rk(z), which maps G* one to one onto itsel/.
Proof. Since {R~(z)} is n o r m a l in G*, we can e x t r a c t from the sul)sequence {Rn~ .~l-n,(Z)} a n o t h e r subsequence {Rn' ~ n'~(Z)} = {Rm~(z)} which t e n d s u n i f o r m l y to a function F(z) on e v e r y c o m p a c t s u b s e t of G*. Observing t h a t
R,,;~ l_n'(Rn;,(z)) = Rn;+i(z ) (5.2)
111
H. BROLIN, Invariant sets under iteration of rational functions we obtain, by taking the limits in (5.2),
P(l(z)) =l(z)
and we conclude that F(z) = z.
Now if zoEG*, there exists an iterate Rk(z) such that Rk(zo)EG*. I f R~(G*)=Gk, then Gk N G*#r Since Gk and ,G* are maximal domains of normality of {Rn(z)}, we have Gk=--G*.
I t remains to prove that the mapping is one to one. Since lim~_~r162 Rm~(Z)=z, the assumption Rk(Zl)= Rk(z~) implies
z l = lim
Rmv(zl)=
lim Rm_k(Rk(zx) )= limRay(Z2)=Z
2 and the theorem is proved.Remark. A domain such as G* is called a singular domain.
6. On the
inverse functions{R_~(z)} of the iterates {Rn(z)}
Clearly, a more detailed investigation of F has to make use of Theorem 2.5. Then a good knowledge of the behavior of theinverse functions {R_n(z)} is needed. There- fore this section is devoted to these functions.
We begin, however, by treating the following closely related question.
Let a be any point in the plane other than the exceptional points of Theorem 2.5, and form the set Pa of predecessors of a. Let P~ be the derived set of Pa and in- clude a in P~ when a has an infinite number of predecessors which coincide with a.
If a E F then by Theorem 2.4 and 2.5 F=P'~ and if a ~ F then at least FcP'~. The question now is whether or not F = Pa can occur when a ~ F. The following result holds.
Theorem 6.1. F = P'~ i/ and only i/ a does not belong to the set o/attractive/ixpoints or to a singular domain.
Proo/. Consider two points a and b such that b eP'a and b ~ F. Let {a_~} be a sequence of predecessors of a such that lim~_~o a_~=b. Since b ~ F , the sequence {R~,(z)-a} is normal in a ncighbourhood U of b.
We can extract a subsequence {Rnr which converges uniformly in U.
Since R~,(a_~')-a = 0 we conclude that
lira R,;(b) - a = O.
Thus a is an accumulation point of the successors of b. Moreover, since b ~ F, it follows that a ~ F.
Hence, if b belongs to a domain where the iterates {R,(z)} have only constant limit functions, then by Theorem 5.1 a is an attractive fixpoint of some order. If, however, b belongs to a domain where there exist non-constant limit functions, then by Theorem 5.2 a belongs to a singular domain. The necessity is obvious from Theorem 5.2 and the properties of attractive fixpoints.
112
ARKIV FOR MATEMATIK. B d 6 nr 6 We shall now consider the inverse functions (R_,(z)) which are algebraic functions.
Before proving some fundamental lemmas we recall that C denotes the set of critical points of the functions (R~(z)}.
Lemma 6.1. A n y in/inite set o/branches (R~)~p(z)}, meromorphic in a domain D, is normal in D.
Proo/. By considering the equation R ( z ) - z = 0 it is easy to see that there exists a t least one fixpoint ar different from the exceptional points of Theorem 2.5. This point :r has two predecessors ~r and a-2 of order one and two such that cr162162 2~=:r If ~ D , then each function R('_)~p(Z), 2~>2, evidently omits the values cr cr 1 and :r in D, and thus (R~)~p(z)} is normal in D. If teED, then by considering the equations R ( z ) - z = O and R~(z)-z=O, it is easy to see that there exists at least one more fixpoint fl of order one or a cycle (Yl, ~'e) of order two, in both cases different from t h e exceptional points (cf. Baker [8]). We can repeat the discussion concerning ~, and consequently there remains only the case where all the fixpoints mentioned above belong to D. But then we can divide D into a finite number of overlapping subsets, such that (R~)~p(z)~ is normal in each of these sets.
Since this implies the normality of (R(~)~p(z)} in all D, the lemma is proved.
Lemma 6.2. I] the domain D is simply connected and i/ D A C = r then the set o]
]unctions (R_~(z)} is a normal/amily in D.
Proo/. This follows immediately from Lemma 6.1.
Lemma 6.3. Let E be a closed set which contains no accumulation point o/the suc- cessors o/a point outside F. I / E ~ = R n(E), then the sequence ( En} converges uni/ormly t o F .
Proo/. Suppose that the lemma is false. Then there exists a sequence of increasing integers (~n} and a sequence of points (z (n)} outside an e-neighbourhood U of F such that Ra,(z (~)) =~(n), where ~(n) e E. Evidently (z (~)} has an accumulation point z (~ also outside U. Thus (R~(z)} is normal in a neighbourhood of z (~ I t is then easy to see that there exists a subsequenee of (R~Jz)} which, according to uniform convergence and the fact t h a t E is closed, in z (~ tends to a point ~(0)E E. This con- tradicts Our assumption and so the lemma is proved.
We now state the main result of this section.
T h e o r e m 6.2. Let {R?~.(z)} be any in]inite set o/ inverse branches, which are mero.
morphic in a domain D. We suppose that D is not a subset o / a singular domain and that F is not identically equal to the whole plane. Then (R~)~ (z)} is normal in D and every convergent subsequence o / ( R ~ ( z ) } tends to a constant.
Proot. B y Lemma 6.1 {R~n(z)} is normal in D. Furthermore, it is evident that there exists a domain D 1 c D such t h a t D 1 satisfies the conditions of Lemma 6.3.
Thus, in D 1 the values of the functions {R~n(z)} converge uniformly to F, i.e. to 1 1 3
ti. BROLIN, Invoriant sets under iteration of rational functions
a set containing 11o interior points. Since the c o n v e r g e n t subsequences t e n d t o m e r o m o r p h i c functions, t h e y m u s t be c o n s t a n t s a n d the t h e o r e m is p r o v e d .
I t is now possible to p r o v e t h e t h e o r e m m e n t i o n e d e a r l i e r concerning the 11021- existence of singular domains.
Theorem 6.3. I/the set (~ does not divide the pla~e, then there exist ~o singular do,~oi~s.
Proo]. Suppose, on the c o n t r a r y , t h a t t h e r e exists a singular d o m a i n G*, a l t h o u g h C does n o t divide t h e plane. F u r t h e r m o r e , let G* 1 be a n y of the n o n - s i n g u l a r d o m a i n s R I(G* ). If we choose z'CG* a n d z"EG~I we can, according to the a s s u m p t i o n , find a s i m p l y connected d o m a i n D containing z' a n d z" a n d such t h a t D f3 C - 4 .
I n D t h e functions {R ,,(z)} m a k e a n o r n l a l family. Since G* is a s i n g u l a r d o m a i n t h e r e exists a subsequence {R~!~,,,.(z)}, which in a n e i g h b o u r h o o d of z' t e n d s to a non- c o n s t a n t l i m i t function, h i a n e i g h b o u r h o o d of z", however, {R("~,.(z)} t e n d s to ~ c o n s t a n t . This is impossible a n d t h e t h e o r e m is proved.
However, we can s t a t e a m o r e useful condition, which also can be used to p r o v e a t h e o r e m concerning t h e v a l u e s of R'~(z) on F .
Theorem 6.t. I / F (1 C -4, thel~ there exist ~o singular domains.
Proo/. Suppose t h e r e exists a singular d o m a i n G* a n d t h a t C d i v i d e s t h e p l a n e . Since there a h v a y s exist non-singular domains, b y T h e o r e m 4.2 a n y n e i g h b o u r h o o d s of a p o i n t : e ?G* c o n t a i n non-singular c o m p o n e n t s . Thus we can choose z', z" a n d D as in the proof of T h e o r e m 6.3 a n d t h e n use t h e same a r g u n m n t .
Theorem 6.5. I / F N C - 4 , then /or each k> 1 there exists an integer h such that [R~(z)[ > k > l i / z e F and n ~ h .
Proo/. I f d(F, C ) - 5 > 0 , we cover F b y a finite n m n b e r of circles D, w i t h r a d i i of l e n g t h r < & Set D = UD~ a n d suppose t h a t F is bounded. The f u n c t i o n s {R_=(z)}
are m e r o m o r p h i c in D,. a n d t h u s c o n s t i t u t e a n o r m a l f a m i l y . A c c o r d i n g to T h e o r e m 6.4 no s i n g u l a r d o m a i n s can exist, so t h e n all l i m i t functions are constants. C o n s e q u e n t l y we conclude t h a t t h e functions {R'_,(z)} converge u n i f o r m l y to zero in D,,. This implies t h a t for each k > l , t h e r e exists an i n t e g e r h such t h a t
[R2 (z)]<k 2, if z f i D , n>~h, i.e. [R'~(z)[>k, if z e F , n>~h, which was to be p r o v e d .
Remark. I f R,~(z)--->a in a d o m a i n t h e n it follows t h a t a f i C , ([6], pp. 60-61.) T h u s if F f ) 0 = r t h e n b y T h e o r e m 5.1 a n d 6.4 t h e l i m i t functions of {R,(z)}
are a t t r a c t i v e fixpoints.
7. On the structure of the complement of F
I n t h i s section we shall discuss how the set F d i v i d e s t h e plane. W e need t h e follow- ing t h e o r e m .
Theorem 7.1. The number o/ simply connected domains, which are completely in- variant under R(z) is at most 2.
F o r t h e proof we n e e d t h e following l e m m a . 1 ! 4
ARKIu FOR MATEMATIK. Bd 6 nr 6 l~emma 7.1. I[ the domai~ D is simply cont~ected at~d completely iJ~variant under ct r.tion.1/u~ctio~ R(z) o/ degree d, the~ D contains . t le.sl d 1 critical poiJds o/ R l(z).
Remark. I t is a l w a y s u n d e r s t o o d t h a t t h e critical p o i n t s have to be r e p e a t e d as m a n y t i m e s as t h e i r o r d e r indicates.
P r o o / o / L e m m a 7. l. We can o m i t t h e two quite t r i v i a l eases where D is identieal with the whole p l a n e a n d where D has o n l y one b o u n d a r y point. Suppose f u r t h e r t h a t z ~ D.
If a E D, t h e n D contains t h e d roots of t h e e q u a t i o n R(z) , 0. E v i d e n t l y t h e r e exists a J o r d a n curve 7 sufficiently close t o g D for Y-1 : kJ R(!;~(7) to enclose all these roots. F r o m t h e a r g u m e n t p r i n c i p l e it follows t h a t the curve Y-1 is g e n e r a t e d by d inverse branches, which are p e r m u t e d c y c l i c a l l y when z runs t h r o u g h 7 d times. T h u s 7 n m s t enclose a t least d I critical p o i n t s of R l(Z ) ami t h e l e m m a is p r o v e d .
Proo/ o/ Theorem 7,1. W e know t h a t if R(z) is a r a t i o n a l function of d e g r e e el, t h e n R l(z) has a t n m s t 2(d l) critical points. The conclusion of the t h e o r e m now follows from L e m m a 7.1.
We now consider t h e possible n u m b e r of c o m p o n e n t s (;,. of CF, i.e. tile n u m h e r of m a x i m a l d o m a i n s of n o r m a l i t y of {R~(z)J.
Theorem 7.2. I / t h e ~umber o/disjoint compo~e~d,s o/ C F i,~ /h~ite, the, it i.~ either 1 o r 2 .
Pro@ If t h e n u m b e r N of d i s j o i n t c o m p o n e n t s of C F is finite, it follows t h a t e v e r y c o m p o n e n t G, n m s t be c o m p l e t e l y i n v a r i a n t u n d e r some i t e r a t e R~(z). F u r t h e r m o r e , if N ~ 2, t h e n e v e r y c o m p o n e n t is s i m p l y connected, or else t h e r e exists at least one m u l t i p l y c o n n e c t e d c o m p o n e n t G,, which c o n t a i n s a closed curve s e p a r a t i n g b o u n d a r y p o i n t s of a n o t h e r c o n n e c t e d d o m a i n G~,#G,,. The t h e o r e m t h e n follows from Theo- rem 7.1.
To g e t f u r t h e r i n f o r m a t i o n a b o u t t h e c o m p o n e n t s of C F we now r e t u r n to t h e i m m e d i a t e a t t r a c t i v e set A*(~) a n d t h e a t t r a c t i v e set. A(~), a u d consider t h e i r c o n n e c t i v i t y . W e recall t h a t A*(~) d e n o t e s t h e largest c o n n e c t e d set contain- ing t h e first o r d e r att, r a e t i v e f i x p o i n t :r where l i m ~ _ , ~ c R , ( z ) = a a n d t h a t A ( ~ ) : { z l l i m ~ + ~ Rn(z) ~}.
Theorem 7.3. The immediate attractive set A*(~) i,s, either ~imply com~ected r o/
in]inite connectivity.
Proo/. Since ~ is a first o r d e r a t t r a c t i v e f i x p o i n t t h e r e exists a circular disk r with ~ as centre a n d such t h a t for z C o>
IR(z) 0 < k < , ,
If o ) , ~ - R_,(r t h e n
( O C O ) 1 C ( O 2 C ..,C(O_nC.
Let E~ be t h e l a r g e s t c o n n e c t e d s u b s e t of (,J_,~ which c o n t a i n s g. H e n c e E 1C E~ c . . . ~ E~ c ...
a n d it follows t h a t
A * ( ~ ) - lim E , .
l l 5
H . BROLIN, lnt'ariant sets under iteration of rational functions
Suppose now t h a t A*(x) is m u l t i p l y com~ected, i.e. t h a t some Eh is n m l t i p l y con- nected. T h e n the 1)oundary of E~ consists of q disjoint closed curves. F o r simplicity we assume t h a t z= ~ r We m a y also assume t h a t ~:e> N C - : r If the b o u n d a r y curves are denoted by 7'1~,, ,+t2} ... 7!ql,, t h e n Rh(7}"},)=~(,) , v - - l , 2 .... , q. Thus, if 2 C'+,"',,, we have ;1=: R~g;)C ()(,>. Let ;1 describe a curve are outside (o which t e r m i n a t e s at a point *h C F . T h e n it follows t h a t ~ describes a curve are inside y<';~ ternlinating at a point t l E F . Thus we conclude t h a t each 7<~'~ encloses points belonging to F.
I n an analogous w a y it is then possible to prove t h a t if ~Eh+,z_ 1 consists of q'~
closed curves, each enclosing points belonging to F, t h e n the same holds for t'Eh.,~
with q'+ replaced b y qn.~. B y induction the t h e o r e m then follows.
Since F ~A(~) =#A(/~)---gA(7 ) - . . . , where ~, fl, 7 "-" are a t t r a c t i v e fixpoints, this t h e o r e m has the following corollary.
Corollary 7.1. Suppose that ~ is an attractice /ixpoint o / o r d e r one and that A*(:r = A(~) is ~tot s i m p l y connected. Then i/there e x i s t / i r s t order attractive/ixpoints fl, ~, ....
other than :r A*(fl)~-A(fi), A * ( v ) ~ A ( y ) .... and each A(fi), A ( y ) .... consists o / s i m p l y co~ ~ecteel compo~e~ts.
8. The structure o f F , when the number o f attractive f i x p o i n t s is >~ 2 The last sections of the first c h a p t e r will be d e v o t e d to a more detailed investiga- tion of the structure of F. We begin here b y proving t h a t u n d e r quite general condi- tions F consists of J o r d a n curves. The n u m b e r of these curves can be either one or infinity.
Theorem 8.1. I / R(z) has t w o / i r s t order attractive/ixpoints ~ and fi and i/A*(:() = A(~.) and A*([~)- A(fl), then F is a Jordan curve.
Proo/. F r o m T h e o r e m 7.3 a n d Corollary 7.1 it follows t h a t b o t h A(~) a n d A(fi) are simply connected. I f the degree of R(z) is d then, b y L e m m a 7.1, b o t h A(~) a n d A(fi) contain d - 1 critical points a n d consequently there exist, no a t t r a c t i v e or rationally indifferent fixpoint other t h a n :r a n d ft. Since F ~ C r then, according to T h e o r e m 6.5, given k > 1, we can find an integer h such t h a t
IR;(~)l>k>l for ~ F , n~>h.
I t is no restriction to suppose t h a t h = 1, i.e.
I R ' ( z ) ] > k > l for z E F (8.1)
a n d to take :r = 0 and f l - 2 . We can choose two J o r d a n curves 7 a n d ~o in the follow- ing w a y (for example see the proof of Theorem 7.3)
(i) y e A ( O ) a n d ~ o c A ( ~ ) .
(ii) The critical points belonging to A(0) are inside T a n d those belonging to A ( ~ ) are outside ~o.
(iii) 7-1 = R-I(~) encloses 7 a n d o) encloses (o_ 1 = R_l((O ).
B y L e m m a 6.3 the sequences {7-,} and {(o_~} of the predecessors of 7 a n d ~o b o t h t e n d u n i f o r m l y t o F .
! 16
AItKIV F~hl MATEMATIK. Bd 6 nr 6 N o w consider the J o r d a n curves {7-,,}. To get a p a r a m e t r i c representation of t h e m we proceed as follows. Map the d o u b l y connected d o m a i n D b o u n d e d b y )'-1 and 7-2 eonformally and one to one onto a circular ring b o u n d e d b y C1 and C._,. Let a radius r of C2 cut. C1 at a a n d C~ at b. The inverse m a p p i n g function m a p s the sul)arc r,b of r onto an are ).AB which consequently cuts Y-i a n d 7~2 at right angles. By letting ,' run t h r o u g h the circular ring we get a corresponding covering of D by orthogonal trajectories 2aB. B y successively m a p p i n g these trajectories by R l(z) we get a cover- ing of the domains between the curves (7-',, V-a), (7-3, 7-~) .... , so t h a t it corresponds to every point on e v e r y 7-,~ one a n d only one o r t h o g o n a l trajectory, i.e. an are t h a t cuts ),,~ at right angles.
L e t 7-1 have t h e p a r a m e t r i c f o r m z =zl(t), 0 < t < 1 a n d z(0) z(1). Then give every 7--~ a p a r a m e t r i c form such t h a t points on the same t r a j e c t o r y have the s a m e / - v a l u e . If the m a x i m a l length of t h e trajectories between 7-,~ a n d 7-(,,+~) is i,,,1, t h e n l>y (S.1) l , + l < k -~. l~, a n d we o b t a i n
lim z,,(t) = z(t), uniformly.
n - - > ~
Since {z,(t)} are continuous functions it follows t h a t z(t) is continuous a n d thus F--= {z(t)10 < t < 1} is a continuous curve.
I t remains t o prove t h a t z=z(t) is simple. Suppose t h a t ~ - . z ( t l ) - z ( t . z) a n d that, t l:@t~. T h e n there exist t w o different, trajectories ),t, a n d 2t~ which ternfinatc at ;~.
Hence )-t, a n d ),t~ t o g e t h e r with y_, b o u n d a s i m p l y connected d o m a i n ~) and 2 ~ ~-2, tt < t < t 2, whence {z(t) I t~ < t <t2} c ~2 U {~}. We now observe t h a t we can t r e a t the era'yes {(o_.} in the same w a y as {7_,}, i.e. ~o= has a p a r a m e t r i c form y-=y,(t) and lim,_~ ~ y~(t) = y(t), uniformly, where F = {y(t) I 0 -<< t ~ 1 }. Since {y(t) } 0 < t .~ 1 } n ~ = we obtain t h a t {z(t) lt ~ <t <t2} = {~}. Thus ~ is n o t a double-point. We conclude that, F is a J o r d a n curve, which is w h a t we wished to show.
Theorem 8.2. Suppose that
(i) the number o/ attractive ]ixpoints is >~ 2, (if) one and only one o/them,/~, has A*(fl)=A(~), (iii) F f3 C = r
Then F contains an infinite number el Jordan curves.
Proo/. F r o m T h e o r e m 7.3 and Corollary 7.1 it follows t h a t t h e assumptions imply the existence of at least o n e , a t t r a c t i v e fixpoint a such t h a t A*(o:)#A(~) a n d such t h a t A*(~) is s i m p l y connected. L e t ~,A*(a)=F~ a n d let R51(z) be the branches of R_l(z ) for which R I(A*(a)) c A*(a). T h e n Rff I(F~) = F~ a n d f u r t h e r m o r e R(F~) = F~.
T h u s b y using R~_l(z) instead of R_l(z ) we can prove, as in t h e proof of T h e o r e m 8.1, t h a t F , is a J o r d a n curve. B y t h e n t a k i n g all t h e predecessors of F~ we get an infinite n u m b e r of J o r d a n curves t h a t belong to F .
Remark 8.1. F a t o u [6], pp. 300-303 p r o v e d t h a t F is a J o r d a n curve, if R(z) has a first order a t t r a c t i v e fixpoint ~ a n d a first order r a t i o n a l l y indifferent fixpoint fi such t h a t A * ( a ) = A ( a ) a n d A*(fl)=A(fl). I n this case fl m u s t satisfy R'(fi)=: + I a n d R " ( ~ ) # O , i.e. A*(fl) consists of only one m a x i m a l d o m a i n of n o r m a l i t y of {Rn(z)}.
See F a t o u [5], pp. 191-221 a n d J u l i a [9], pp. 223-237.
117
tt. BROLIN. lnrariant sets under iteration of rational functions
It is. h o w e v e r , p o s s i b l e to get m o r e d e t a i l e d i n f o r m a t i o n a b o u t t h e J o r d a n c u r v e s in The<)rem S.1 a n d S.2. T h e n e x t s e c t i o n shows t h a t t h e s e c u r v e s d o n o t h a v e t a n - g e n t s at a n y i)oint.
9. On the existence o f tangents to the curves that lie in F
T h e o r e m 9.1. Let :( be a~ attractive/irst order/ixpoint o/ R(z). Suppose that A*(oQ i.~" .~'imply <'o~ected and that F~ N 0 - r where F~ - g~A *(c(). Then, i/ F~ is not a circle or ,t strrtight li~e F~ does ~mt have a tangent at any poi~Tt.
F o r t h e p r o o f we n e e d t h e f o l l o w i n g i m p o r t a n t l e m m a .
L e m m a 9.1. Let ~ be an attractive first order /ixpoint o/ R(z). I/ A*(~) is simply con~ected a~d i/ ~:A*(~) : F: is an. analytic Jordan curve or arc, then F: is either a circle or a~ (~rc o / a circle (straight line or a segment).
Pro@ As u s u a l R(z) is of d e g r e e d. W e b e g i n b y m a p p i n g A*(~) e o n f o r m a l l y a n d one t o one o n t o It] < 1 a n d so t h a t z ~ + , t - O . L e t t h e i n v e r s e m a p p i n g f u n c t i o n be z -h(t). Since F is a n a n a l y t i c , J o r d a n c u r v e or arc, h(t) is m e r o m o r p h i e in ]t I < r 1 w h e r e r ~ > l . M o r e o v e r , if o>-{t I I t] 1} t h e n F~=h((o). F u r t h e r n m r e , R ( z ) m a p s A*(~) o n t o itself q t i m e s . T h u s
h_ 1 ( R(h(t) ) = q)(t) (9. l )
m a p s t h e u n i t d i s k o n t o i t s e l f q t i m e s . T h e n q)(t) m u s t be a B l a s c h k e p r o d u c t , i.e.
q P a , - - t I A ] = I , l<~p~q<<.d"
qJ(t) = A . t ' . I] 1 2 t - S f
v = l
(9.2) W e wish t o p r o v e t h a t h(t) is a r a t i o n a l f u n c t i o n . F r o m (9.1) we g e t
R(h(t) - h(~(t)). (9.3)
F o r I tl > r > 1 a n d r < r 1 W e h a v e I~(t) l >/c. It I, w h e r e k > 1 is a c o n s t a n t i n d e p e n d e n t of r 1. S i n c e {t I It I <krr}c{cf(t)l It[ < r l } , b y (9.3) we c a n c o n t i n u e h(t) a n a l y t i c a l l y to It[ < kr~. w h e r e a n y s i n g u l a r i t i e s of h(t) are poles. F o r if ~c l(t) h a s a c r i t i c a l p o i n t in 1 <; ttl <krl, l e t t m o v e a l o n g a closed p a t h in 1 ~ It[ < k r I s u c h t h a t t h e c r i t i c a l p o i n t is i n s i d e t h e p a t h . A s s u m e t h a t t h e p a t h s t a r t s a n d e n d s a t t o E co. T h e n a b r a n c h
~!~(t)
m o v e s a l o n g a p a t h in 1 4 It I < r~ f r o m ~(')l(t0) = t~ E ~o t o ~(~)l(t0) = t~ E ~o,w h e r e t I =~t 2. B u t R(h(q~_l(t)) = h(t) a n d t h u s for e a c h t o E o) we h a v e R(h(tl) ) = R(h(t2) ) = h(to) a n d we c o n c l u d e t h a t h(t) h a s no a l g e b r a i c s i n g u l a r i t i e s in I t ] < k r 1. T h u s , b y (9.3), we can c o n t i n u e h(t) a n a l y t i c a l l y t o t h e w h o l e p l a n e .
C o n s i d e r t h e b e h a v i o u r of h(t) a t t - o o . S u p p o s e f i r s t t h a t ~(t) h a s s o m e f i n i t e poles, i.e. 9)(t)=~Atq. If ~(t) h a s a pole a t z - b , t h e n in a n e i g h b o u r h o o d O of z = o~
one b r a n c h q~!')l(t) t a k e s its v a l u e s in a n e i g h b o u r h o o d U of z = b. Thus,h(t) - R(h(~(~)l(t)) h a s t h e r a n g e R(h(U)) in O, i.e. h(t) h a s a t m o s t a pole of f i n i t e o r d e r a t t = o~.
I t r e m a i n s t o s t u d y t h e b e h a v i o u r of h(t) a t t = o% w h e n ( v ( t ) - A t q. W e c a n t a k e A = 1 a n d t h u s we h a v e t h e f u n c t i o n a l e q u a t i o n
R(h(t)) =h(tq). (9.4)
I l S
ARKIV F/3R MATEMATIK. Bd 6 n r 6 S u p p o s e t h a t h(to)= ~, w h e r e t0:#0. T h u s , b y (9.4)
h(t~)") = Rn(h(to) ) = R,(oO = o~.
If 0 satisfies 0 q"= 1~ t h e n
h[ (Oto)q" ] = R,(h(Oto) ) = ~. (9.5) I t follows f r o m (9.5) t h a t :r h a s a n i n f i n i t e n u m b e r of predecessors h(Oto) o n a n arc c o n t a i n i n g ~. T h e s e predecessors h a v e a as a n a c c u m u l a t i o n p o i n t , w h i c h is impos- sible since :r is a n a t t r a c t i v e f i x p o i n t . T h u s , h(t)= ~ h a s n o r o o t s o t h e r t h a n t = 0 . Since, h o w e v e r , t h i s e q u a t i o n h a s r o o t s t @ 0 w h e n ~ h a s predecessors o t h e r t h a n itself, we c o n c l u d e t h a t ~ c a n h a v e n o predecessor o t h e r t h a n itself. B y a M 6 b i u s t r a n s f o r m a t i o n we m o v e ~ t o z : oo. T h e n t h e t r a n s f o r m e d f u n c t i o n will be a p o l y - n o m i a l P(z). F r o m t h e d i s c u s s i o n a b o v e we c o n c l u d e t h a t h(t) h a s o n l y o n e f i n i t e pole, n a m e l y t = 0 . T h u s i n It[ > 1, h(t) h a s n o s i n g u l a r i t i e s .
N o w set m a x
In(0[
= M(r).Itl =r Since i n (9.2) q - d , we g e t f r o m (9.4)
M ( r a ) <. B . (M(r)) a, (9.6)
w h e r e B is a p o s i t i v e c o n s t a n t . Choose A > 1 a n d a n i n t e g e r m s u c h t h a t ~t > B a n d M(2) < 2 m. T h e n b y (9.6)
M(;t a) < B . (;ta) m < ()~)~ +1 a n d M (2 av) < B 1 + a +... + a v- 1. ().av)m <: ().a~) rn + 1.
T h u s we c o n c l u d e t h a t h(t) h a s a t m o s t a pole of o r d e r m + 1 a t t = ~ a n d h(t) is a r a t i o n a l f u n c t i o n .
W e n o w c o n s i d e r t h e possible cases:
(a) h(t) is a l i n e a r f u n c t i o n . T h e n F , is a circle or a s t r a i g h t l i n e a n d t h e l e m m a is p r o v e d i n t h i s case.
(b) h(t) is of degree p >~ 2. L e t t h e c o m p l e m e n t of t h e u n i t disk i n t h e t - p l a n e be o)e a n d set h(coe)= D. B y (9.3)
R,(h(t) ) =h(~vn(t)).
S i n c e cfn(t)-->~ i n 6% we h a v e
l i m Rn(z) = h(co ) = fl, z E D.
n --~ o o
I f n o w h(t) is of degree ~>2, t h e n A*(cr f) D : # r T h u s fl = ~ a n d we o b t a i n A*(~) = D, i.e. F 0 A*(~) is e q u a l t o t h e e x t e n d e d p l a n e .
Moreover, we a s s e r t t h a t h(t) is of degree 2. L e t z 0 E F be n o critical p o i n t of h_l(z).
I f h(t) is of degree p > 2 t h e n z 0 h a s t h e p predecessors t 0, to, to', ... o n (o. Move z a l o n g t h e n o r m a l t o F a t z 0. T h e n t h e c o r r e s p o n d i n g t - v a l u e s m o v e a l o n g t h e n o r m a l s t o co a t to, to, to', ..., r e s p e c t i v e l y . I f p > 2 , t h e n t h e r e exists a zEA*(~) t o w h i c h a t l e a s t t w o t-values i n I t [ < 1 c o r r e s p o n d . T h a t is i m p o s s i b l e a n d t h u s t h e degree of h(t) is 2.
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H. BROLIN, Invariant sets under iteration of rational functions
B y a Mhbius transformation, we move the endpoints of F to z = 0 a n d z = co.
Then m a p A*(r162 onto I m t > 0 instead of I t] < 1 and so t h a t z=O~-*t=O and z =
~ t = ~ . Thus we conclude t h a t h(t) =At 2 and consequently F is a segment. T h a t completes the proof.
Proo/o/ Theorem 9.1. We m a y assume t h a t A*(:r162 If A*(~t)#A(~), then in our proof below, we merely replace F with F~ and R n(z) with R~_n(z), where R~_n(z) is the restriction of R=(z) to A*(~).
Let D be a domain such t h a t D N C = r and 0r r D. B y L e m m a 6.2 all the functions {R ~(z)} make a normal family in D. Moreover, b y Theorem 6.2, every convergent subsequence of {R_n(Z)} tends to a constant, which is a point in F. B y Theorem 6.1, such a subsequence corresponds to every point in F.
L e t ~ fi F a n d let {R~)~,(z)} be a sequence which converges to ~ in D. Furthermore, let ~* be an accumulation point of the successors {~,~} = {R,,(~)} of ~. Since C N F = r we can deform D so t h a t ~*E D. Thus if tm-+~*, m---> ~ , t h e n R-m(~m) - $ = 0 for every m, where {R re(z)} is extracted from {R~)~(z)}. Set
R-re(z) - ~ (9.7)
/.,(z) R ' ~ ( ~ ) "
The functions {Ira(z)} are univalent in D and ]m(~m)=0 ]~($m)= 1. We obtain b y a distortion theorem b y Koebe (see for example Hille [8], p. 351), t h a t {]~(z)} is normal i n D. E x t r a c t a subsequence {]mr(z)} which tends uniformly to q(z) in D. Obviously,
q(z) is univalent and ~ constant in D. B y (9.7)
R_~,(z) - ~ =/~,(q(z) + e~(z)),
where the c o n s t a n t s / x ~ - + 0, v - + ~ , and sz~(z)--> O uniformly in D.
If D N F = 7 then take z E 7 such t h a t z # $*. Consider
lim arg (R_,n,(Z)- ~). (9.8)
v - - ~ Oo
If lim~_,~ arg #mr = 0 and arg q(z) = ~ , t h e n 0 + ~ is one of the limits of (9.8). Hence, for (9.8) to have a unique limit, it is necessary t h a t arg q(z) be constant when z moves along 7. Thus, the image 7" =q(7) m u s t be a straight line. Since ~(z) is univalent, 7 m u s t be an analytic arc. B y Theorem 4.3, there exists an integer N such t h a t F = RN(7). Thus if 7 is analytic, then F is analytic and it follows from L e m m a 9.1 t h a t F is a circle or a straight line.
Since ~ was arbitrary, no points of F have tangents, except when F is a circle or a straight line. Thus the theorem is proved.
1O. The structure of F when the number of attractive fixpoints equals 1 As usual a set is said to be totally disconnected if all its components are single points. The following theorem, stated b y Fatou, shows t h a t t h e set F can have this structure under quite general conditions. F a t o u only outlined the proof, b u t we shall give a detailed proof here, since both the theorem a n d some details of the proof will be of great importance in our investigations.
120
ARKIV FOR MATEMATIK. Bd 6 nr 6 Theorem 10.1. I / :r is a /irst order attractive /ixpoint o/ a rational /unction R(z) and i/ C c A*(a) then the set F is totally disconnected.
Proo/. F r o m t h e a s s u m p t i o n we conclude t h a t t h e r e exists no a t t r a c t i v e or r a t i o n a l l y i n d i f f e r e n t f i x p o i n t o t h e r t h a n ~ a n d t h a t A*(cc)=A(:r T h u s A ( : r C F a n d C F is a connected set. F o r s i m p l i c i t y we m o v e ~ t o z = oo b y a M6bius t r a n s f o r m a t i o n . According t o t h e a s s u m p t i o n , it is possible t o cover F b y a s i m p l y connected closed set E o such t h a t
E o n C = r ~ E o n F = r
I f C E 0 = B t h e n B c A(co). Since Rn(B) t e n d s u n i f o r m l y to z = co t h e r e exists a n integer p such t h a t
R n ( B ) c B if n>~p.
N o w set RAz) = p ( z )
a n d consider t h e i t e r a t e s {p~(z)}. Since e v e r y inverse b r a n c h is h o l o m o r p h i c in E0, we can use t h e same a r g u m e n t s as in t h e proof of T h e o r e m 6.5. Thus, {p_n(z)} is n o r m a l in E 0 a n d e v e r y c o n v e r g e n t subsequence t e n d s t o a c o n s t a n t . F u r t h e r m o r e , t h e functions {p'~(z)} t e n d u n i f o r m l y t o zero in E 0 a n d t h u s t h e r e exists a n i n t e g e r h such t h a t
< k < l if z 0 E E 0 a n d n ~> h. Set ph(z) =h(z),
where t h e degree of h(z) is m = d € if d is t h e degree of R(z). B y m a p p i n g E o b y t h e inverse function h l(z ) we o b t a i n m s i m p l y c o n n e c t e d sets (EI~}. Because of t h e choices of E 0 a n d h(z) these sets s a t i s f y
m
F ~ ( J E I ~ = E 1CE0; E I ~ N E I ~ = r if v~=/~.
V = I
Map E 1 b y h_l(z ) a n d t h e n E~ =h_l(E1) b y h l(Z ) a n d so on. A f t e r n such m a p p i n g s we o b t a i n m n s i m p l y c o n n e c t e d closed sets {En,} satisfying
m n
F c U E , ~ = E n c E , , _ 1 c . . . c E 0 ; En~ N E ~ = ~ b , if v=~/~.
I f t h e b o u n d a r i e s (~E,:} h a v e t h e l e n g t h s (l,v} t h e c o n d i t i o n ]h'__l(z)l < k < l , z E E0, implies
ln: < kl<n-1)~, < ..~. < k"lo (10.1)
a n d lim l~+ = 0 for e v e r y v.
Thus, e v e r y c o m p o n e n t of F is a single p o i n t a n d t h e p r o p o s i t i o n is established.
W e shall d e n o t e b y ml a n d m S Lebesgue m e a s u r e on a line a n d in t h e p l a n e respec- t i v e l y . I t is now n a t u r a l t o ask if i t is possible t o o b t a i n results concerning m I F a n d m 2 F u n d e r t h e conditions of T h e o r e m I 1.1. F a t o u g a v e some results b u t we can now give a more complete solution.
9* 121
H. BROLIN, Invariant sets under iteration o f rational functions
Theorem 10.2.
I/
~ i8 afirst
order attractive /ixpoint o/ a rational /unction R(z) and i/ C c A*(o~), then F is totally disconnected and we also have(i) m 2 F = O ,
(ii) m l F = O i / F = L , where L is a straight line.
Proo I (i). W e use t h e s a m e coverings {E~}[ r of F as in t h e proof of T h e o r e m 10.1 a n d i n t r o d u c e a new sequence of sets {O~} defined b y
O n = E~ - En+ 1 (10.2)
To p r o v e t h e t h e o r e m i t is sufficient to show t h a t t h e r e exists a fixed n u m b e r ~ > 0 i n d e p e n d e n t of n, such t h a t
As in (10.2) we i n t r o d u c e O n , = E n , - IJ m E (n+l)u,
p = I
a n d
m2 On > 4.
m2 En
where E(n+l)g =Env, ~ = 1, 2 . . . m,
~ . . _ m ~ 0 . .
m~ E nv "
I n forming t h e sets {E,} we used a f u n c t i o n h(z) satisfying 0 < K x ~ < [ h ' ( z ) [ ~ < K ~ < c r zfiEo, where K 1 a n d K2 are constants, a n d also
h(E.~) = E(.-1)v, h(O..) = O ( n _ l ) v ,
(10.3)
(10.4) where for s i m p l i c i t y , we k e e p t h e i n d e x v. I f
max Ih'(~)l=lh'(r and
m i nIh'(z)[=[h'(zn~)[
z 6 E n v zEEnv
we g e t from (10.4) t h a t
m 2 0 . , Ih'(znd[ ~
m,O(.-1)~
4 , , - - - >1
m.,E,, i ~ m,E~,_l);
M t e r r e p e a t i n g t h i s p r o c e d u r e n t i m e s we o b t a i n
a. >.m, ~ ~ I~'(~,)1 ~- ,~. h I h',(~) I ~
To v e r i f y t h e existence of a 4 > 0 , such t h a t 4 n , > 4 for all n a n d v, i t is sufficient to p r o v e t h a t t h e p r o d u c t
1~ Ih'(z.,)l'
n-1 I h'(r (1o.5)
122
A R K I V F O R M A T E M A T I K . B d 6 n r 6 is u n i f o r m l y b o u n d e d f o r all v. B u t a s u f f i c i e n t c o n d i t i o n f o r (10.5) t o be uni- f o r m l y b o u n d e d is t h a t t h e series
I h ' ( : . : ) ~ - h'(~.~): [ S n : ~ ~ fh'(~.~)[ ~
is u n i f o r m l y b o u n d e d f o r all v. O n a c c o u n t of (10.3) we g e t
S<~K, ~ [h'(z.~.)-h'($n.)l
n = l
a n d since h'(z) is a r a t i o n a l f u n c t i o n
A c c o r d i n g to t h e p r o o f of T h e o r e m 11.1 t h e l e n g t h ln. of 0En, satisfies In. < l 0 9 U , w h e r e 0 < k < l (see (10.1)). H e n c e we g e t
n ~ l n = l n - 1
W e h a v e t h u s p r o v e d t h a t t h e r e e x i s t s a ~t > 0 s u c h t h a t
m 20nv m 2 End,
f o r all n a n d v. B u t t h e n w e o b t a i n
m n
m 2 On ~ 1 m 2 On~
- - > ~ . E~ m2En"
m 2
S i n c e 0n = E ~ - En+~ w e h a v e
m 2 E~ = m 20~ + m 2 En+ 1 > 2. m z E~ + m 2 E~+ 1 a n d m 2 En+ 1 < (1 - 2)m 2 E . < . . . < (1 - ~)n+lm 2 Eo.
T h u s m 2 F ~ lira m~ En ~< lira (1 - 2)~ m 2 E 0 = 0
n --> oo n - - ~
a n d t h e first p a r t of t h e t h e o r e m is p r o v e d .
(ii) S u p p o s e n o w t h a t F c L w h e r e L is a s t r a i g h t l i n e a n d l e t E o N L = Lo; En~ N L = eon~, On~ fl L = ~n~.
W e t h e n d e f i n e in a n a l o g y t o t h e p r o o f of (i)
~ n v - m l gnv m I ~On~
123
