Ark. Mat., 3 6 (1998), 31 39
@ 1998 by I n s t i t u t Mittag-Leitter. All rights reserved
On the dynamics of composite entire functions
Walter Bergweiler and Yuefei Wang(1)
A b s t r a c t . Let f a n d g be n o n l i n e a r entire functions. T h e relations b e t w e e n t h e d y n a m i c s
of f~ a n d gof are discussed. D e n o t e by J ( . ) a n d ~-(- ) t h e Julia a n d F a t o u sets. It is proved
t h a t if z E C , t h e n zEJ(fog) if a n d only if g(z)CJ(g~f); if U is a c o m p o n e n t of -~(f~ a n d V is
t h e c o m p o n e n t of .T(gof) t h a t c o n t a i n s g(U), t h e n U is w a n d e r i n g if a n d only if V is wandering;
if U is periodic, t h e n so is V a n d moreover, V is of t h e s a m e t y p e according to t h e classification
of periodic c o m p o n e n t s as U. T h e s e r e s u l t s are used to show t h a t certain n e w classes of entire
f u n c t i o n s do n o t have w a n d e r i n g d o m a i n s .
1. I n t r o d u c t i o n a n d m a i n r e s u l t s
The Fatou set jz(f) of a nonlinear entire (or rational) function f is the subset
of the complex plane (or Riemann sphere) where the iterates fn of f form a normal
family. The complement of }-(f) is called the Julia set and denoted by J ( f ) .
The Fatou set is open and completely invariant; that is, zEJZ(f) if and only if
f(z)CJz(f). The Julia set is closed and also completely invariant. It is also known
to be the closure of the set of repelling periodic points. If U0 is a component of
~c(f), then f~(Uo) lies in some component Us of }-(f) and Un\fn(Uo) is either
empty or contains exactly one point by a result of Heins [22]. If Unr
for all
n e r o , then U0 is called a wanderin 9 domain of f. Otherwise U0 is called preperiodic
and if Un=Uo for some h E N , then U0 is called periodic. Sullivan [30] proved that
rational functions do not have wandering domains. Transcendental entire functions,
however, may have wandering domains, see [2], [31, [41, [16], [30], but various classes
of entire functions without wandering domains are known [3], [6], [7], [9], [12], [13],
[18], [21], [28].
Already before Sullivan's work a classification of periodic components of $c(f)
was known. Let f be an entire function and U0 a periodic component of ~c(f), say
(1) T h e second a u t h o r was s u p p o r t e d by M a x - P l a n c k - G e s s e l l s c h a f t Z F D W , a n d by T i a n Y u a n
Foundation, NSFC.
32
Walter Bergweiler and Yuefei Wang
U,~=Uo. Then one of the following possibilities holds:
9 There exists zoEUo such that f'*~nluo---~Zo as m--+~c, fn(zO)=zo and
[(fn)'(zo)l < 1. Then Uo is called an attracting domain and Zo is called an attracting
periodic point.
9 There exists ZoEOUo such that f'*[uo--~zo as m--+oz, fn(zo)=Zo and
( f ~ ) r ( Z o ) = l . Then Uo is called a parabolic domain and zo is called a parabolic
periodic point.
9 There exists a conformal m a p r {zEC:]z]<l}--+Uo and a E R \ Q such that
r162
for I z l < l . With Zo=r
we have f~(zo)=Zo and ( f ~ ) ' ( z o ) =
e 2~i~. Then U0 is called a Siegel disc.
9 The sequence fnmlV0-~OC as m---~,~c. T h e n U0 is called a Baker domain.
We note here that in the case of a Siegel disc U0 the limit functions of the family
{f~ Iu0 } are all non-constant, while in the other cases they are all constant. We also
note that if U0 is periodic of period n>2, then the components U1, ... , U ~ - t of the
periodic cycle which U0 belongs to are of the same type according to the above
classification.
There is a similar classification for rational functions. Here Baker domains do
not play a special role, but there is the additional possibility of a Her~nan ring.
As an introduction to iteration theory, we recommend Beardon's [8], Carleson and
Gamelin's [14], and Steinmetz's [29] books as well as Milnor's [25] lecture notes
for rational functions and the survey articles of Baker [5] and Er~menko and Lyubich [17] for rational and entire functions. The iteration theory of transcendental
meromorphic functions is surveyed in [10]. The classical references are Fatou [19]
and Julia [23] for rational and Fatou [20] for transcendental entire functions.
Baker and Singh [7] proved t h a t if 9(z)=a+bexp(27riz/c) and if f is entire,
then fog has no wandering domains if gof has no wandering domains. They used
this to show that e x p ( e x p z ) - e x p z does not have wandering domains. Here we
compare the dynamics of fog and gof without assuming that g has the special
form above. Our main results are as follows.
T h e o r e m 1. Let f and g be nonlinear entire functions and zEC.
J ( f o g ) if and only if g(z)EJ(gof).
Then zC
It follows t h a t if U0 is a component of .T'(fog), then g(Uo) is contained in a
component V0 of .T'(gof). The result of Heins [22] already mentioned implies that
Vo\g(U0) contains at most one point.
T h e o r e m 2. Let f and g be nonlinear entire functions. Let Uo be a component
of JZ(fog) and let Vo be the component of ,T(g~f) that contains g(Uo). Then
(i) Uo is wandering if and only if Vo is wandering,
33
On the dynamics of composite entire functions
(ii) if Uo is periodic, then so is Vo, moreover, Vo is of the same type according
to the classification of periodic components as Uo.
In particular it follows that fog has wandering domains if and only if gof has
wandering domains. We use Theorem 2 to show that certain new classes of entire
functions do not have wandering domains.
T h e o r e m 3. Let F={eiZd=z, i(eZ~=z), s i n z •
cosz=t=z} and G={glog2 .....
gin; g j = s i n z or cosz, j = l , 2 , . . . ,m, m E N } . Then for any f E F and gcG, fog
has no wandering domains.
For an entire function f , we denote by A ( f ) the set of asymptotic values of f,
by C ( f ) the set of critical values of f, and by sing(f -1) the set of singularities of
the inverse function of f. Then sing(f -1) = A ( f ) U C ( f ) .
T h e o r e m 4. Let f be a real entire function satisfying If (x)] ~ [x[ for - 1 < x < 1.
Suppose that s i n g ( f - 1 ) c R . Then f(sinz) does not have wandering domains.
Here an entire function f is called real if f ( R ) c R . To give specific examples
of entire functions which Theorem 4 applies to we recall that the Pdlya-Laguerre
class LP consists of all entire functions f which have a representation
f(z)=exp(-az2+bz+e)z n n
(z)
e•
,
k~l
where a,b, cCR, a > 0 , h e N 0 , z k e R \ { 0 } for all k c N , and Ek~r 1 IZkl-2<oc. In
particular, real entire functions of order less than two with only real zeros are
in LP. P61ya [27] and Laguerre [24] proved that an entire function f is in LP if and
only if there is a sequence of real polynomials with only real zeros which converges
locally uniformly to f .
P r o p o s i t i o n 1. Let f = f l o f 2 ..... fn where f l , f 2 , . . . , f n E L P . Then we have
s i n g ( f - 1 ) c R . In particular, s i n g ( f - 1 ) c R / f f E L P .
We note that f ( z ) = z cos z E L P and obtain from Theorem 4 and Proposition 1
that sin z cos(sinz) does not have wandering domains. More generally, an odd
function f is in LP if and only if it has the form
~(
z2)
k=l
~k
f(z) =exp(-az2+c)z I I 1--
with nE{1,3, 5, ...} and a, c, zk as above. It is easy to see that the hypothesis of
Theorem 4 are satisfied if c_~0 and ]zkl~_l/v~ for all k c N .
34
Walter Bergweiler and Yuefei Wang
We remark that Theorems 3 and 4 are just examples of applications of Theorem 2 and t h a t we have not tried to state these results in their most generM forms.
Using T h e o r e m 2 one can find more classes of entire functions without wandering
domains.
Acknowledgement. We would like to t h a n k Professor G. Frank for hospitality
and helpful discussions.
2. P r o o f o f t h e t h e o r e m s
We need the following lemma.
L e m m a 1. Let f and g be nonlinear entire functions and zo GC. If Zo is a
periodic point of fog, then g(zo) is a periodic point of gof.
Proof. Let h-=fog and k=g~f. Suppose h'~(Zo)=zo where n E N . Then g(zo)=
g(hn(zo) ):kn(g(ZO) ).
Proof of Theorem 1. Let zoCJ(fog). Since the Julia set is the closure of the
set of repelling periodic points, there are periodic points zj of fog such t h a t zj---~z0.
By L e m m a 1, g(zy) are periodic points of gof and hence g(zo) is a limit of periodic
points of gof because g(zy)-*g(zo). It follows that g(zo)E,7(gof).
Interchanging the role of f and g we see that if WoEJ(gof), then f(wo)e
J ( g o f ) . Suppose now t h a t z 0 e C and g(zo)Cfl(g~f). T h e n f(g(zo))efl(fog).
Because of the complete invariance of the Julia set we conclude that zo Gfl(fog).
The proof is complete.
Proof of Theorem 2. Let h=f~g and k--g~f. For h E N , let Un be the component of J(h) containing h~(Uo) and let V,~ be the component of fl(k) containing kn(VO). Since g(h"(Uo))=kn(g(Uo)) for all n G N we see that g(U,~)cVn and
analogously f (Vn) C Un+1. We conclude that if Um = U~, then V,,, = V,~ and if Vm = V,~
then Um+l = U n + l . In particular, if Uo=Un, then Vo=V,.
Let now Uo=Un. Suppose that hn~]vo--~O as j--,oc where r
Take a
domain V* in V0 such that a branch g*: V* --~ U* C U0 of the inverse function of g is
defined. Then k~lv . =goh~og*[v . and hence k ~j Iv*--*~P:=g~176 If U0 is a Siegel
disc, then r is nonconstant, hence r is also nonconstant and thus V0 is a Siegel
disc. If U0 is an attracting domain, then r is a constant lying in ~ ( h ) , hence ~p is
a constant in Y(k) and thus V0 is an attracting domain. The case of a parabolic
domain is analogous, except that r and ~ are in J ( h ) and J ( k ) now.
The arguments show that if V0 is an attracting domain, parabolic domain or
Siegel disc, then so is U1 and hence U0. It follows that if U0 is a Baker domain,
then so is V0. This completes the proof.
On the dynamics of composite entire functions
35
Remark. The above proof also shows that if V0 is periodic, then U1 is periodic.
We note that U0 need not be periodic. To see this simply take f = g such that 9v(f)
has an invariant component V0 which is not completely invariant. Then take U0 as
a component of f-i(Vo)\Vo.
To prove Theorems 3 and 4, we also need the following results.
L e m m a 2. Let f and g be two entire functions. Then
C(f og) C C(f)U f(C(g) ),
A ( f og) c A( f )U f (A(g) ),
and
s i n g ( ( f o g ) - l ) C sing(f-1)Uf(sing(9-1)).
Proof. We have (fog)'=f'(g)g' and thus C(fog)cC(f)Of(C(g)). If fog tends
to a E C along a path 7 tending to oc, then along ~/either g tends to oc or g tends
to a point/3 satisfying f ( / 3 ) = a (see [7] for details). We have s E A ( f ) in the first
case and a~f(A(g)) in the second case. Now the second and the last conclusion
follow.
L e m m a 3. (Denjoy-Carleman-Ahlfors theorem [26, w
/ f the inverse
function of a meromorphic function f has n direct singularities, n>_2, then
lim inf T(r, f__~)> O.
r~oc
rn/2
Consequently, the inverse function to a meromorphic function of finite order 0 has
at most max{20, 1} direct singularities. Moreover, an entire function of finite order
0 has at most 20 finite asymptotic values.
Proof of Theorem 3. The functions sin z and cos z have the critical values
• and no asymptotic values. And any f c F has at most finitely many asymptotic values by Lemma 3. Thus g(f) has only finitely many asymptotic values by
Lemma 2. (In fact, it is not difficult to see that functions in F and hence the
function g(f) have no asymptotic values at all.) Since all the critical values of g(f)
are among the finitely many values •
gl(•
glog2(:kl),..., glog2 ..... g , ~ ( i l ) ,
glOg2 . . . . . gm(:~(17~-~i)), .qlo~2 . . . . . ~ m - - l ( - { - g , , ( i ) ) , .q1:~2 ..... C g m _ l ( Q - g m ( 0 ) ) , and
glog2 ..... gm-l(•189
again by Lemma 2, g(f) has only finite many critical
values. Hence g(f) is of finite type (i.e. the inverse function to g(f) has only a finite
number of singularities) and thus g(f) has no wandering domains by [18] or [21].
We now apply Theorem 2 to conclude that f(g) has no wandering domains. This
completes the proof.
36
Walter Bergweiler and Yuefei W a n g
Remark. It is not hard to see that such fog has infinitely many different critical
values and is not of finite type.
Proof of Theorem 4. We define h ( z ) = s i n f(z). Then
sing(h -1) C { - 1 , 1}Usin(sing(f-1)) C [-1, 1]
by Lemma 2. It now follows from a result of Er~menko and Lyubich [18] that there is
no component U0 of ~'(h) such that hn]vo-~oc as n--*~c. Thus if h has a wandering
domain U0, then there is a sequence (nk) of positive integers and a E C such that
h nk ]Uo--~a as k--~cc. Clearly we have aEff(h). Let P(h)=[.Jn~__0 hn(sing(h-1)). It
follows from a result of Baker [1] that aEP(h). (Actually we even have that a is
a limit point of P(h), but we do not need this result proved in [12] here.) Our
hypotheses imply that ]h(x)]<lx ] for 0<]x[_<l. We conclude that P ( h ) c [ - 1 , 1],
that hn][_l,1]--~0 as n - - - ~ and that 0 is an attracting or parabolic fixed point of h.
If 0 is attracting, then [-1,1] C~'(h) and thus P(h)NJ(h)=O, contradicting a c
P(h)QT(h). If 0 is parabolic, then [-1, 0) and (0, 1] are contained in the parabolic
domains associated to 0. We conclude that [-1, 1 ] n J ( h ) = { 0 } , so that a = 0 . The
dynamics near parabolic fixed points are well understood. In particular, it is known
and not difficult to see that a parabolic fixed point cannot be a limit function of a
sequence of iterates in a wandering domain. Thus we again have a contradiction.
Hence h has no wandering domains. Theorem 2 now implies that f(sin z) does not
have wandering domains.
3. P r o o f o f P r o p o s i t i o n 1
L e m m a 4. ([11]) Let f be a meromorphie function of finite order. Ira is an
asymptotic value of f, then a is a limit of critical values ak ~ a or all singularities
of f - 1 over a are logarithmic.
Proof of Proposition 1. Let f E L P . It follows from the characterization of LP
mentioned in the introduction that f~E LP. Hence all critical values of f are real.
We now assume that f has an asymptotic value ~ c C \ R and seek a contradiction. Clearly c~ is also an asymptotic value of f . It follows from Lemma 4 that
f - 1 has logarithmic (and hence direct) singularities over ~ and 6. From a theorem
of Lindel5f [26, w
we deduce that between the paths where f tends to a and
5, there must be paths where f tends to oc and thus there are also two direct
singularities over oc. Thus f - 1 has at least four direct singularities. By Lemma 3
we thus have
(1)
liminf log M(r, f) > O.
r--~oc
7"2
O n t h e d y n a m i c s of c o m p o s i t e entire f u n c t i o n s
37
We may write f in the form f(z)~-e--aZ2p(z), where p is an entire function of genus
0 or 1 and akO. It follows that logM(r,p)=o(r 2) as r--*oo and hence a>O by (1).
This implies that f(z)---~O as z-~oo along the positive or negative real axis. Using
Lindel6f's theorem again we conclude that between the real axis and the paths
where f tends to a and a there must be paths where f tends to oc. This leads to
four direct singularities of f - 1 over oo and thus altogether to six direct singularities
of f - 1 . Hence
liminf log M(r, f) > 0
r--~
~3
by Lemma 3. On the other hand, we have log M(r, f ) = O ( r 2) as r--+~ by the form
of f. This is a contradiction. Thus all asymptotic values of f are real.
Altogether we see that s i n g ( f - 1 ) c R if f E L P . The case that f has the form
f = f l o f 2 ..... f~ with fl, f2,... , f.~ELP now follows from Lemma 2.
38
Walter Bergweiler and Yuefei Wang
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39
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Received February 28, 1997
Walter Bergweiler
Mathematisches Seminar
Christian-Albrechts-Universit/it zu Kiel
Ludewig-Meyn-Str. 4
D-24098 Kiel
Germany
email: bergweiler~math, uni-kiel, de
Yuefei Wang
Institute of Mathematics
Academia Sinica
Beijing 100080
China
email: [email protected]