(ml -9054/82| 0021
874$ | .50+ 0.20l0
@ 1982BirkhiuserVertag,Basel
AequationesMathematicaeA (l 982)I 87- I 94
Univenity of Waterloo
Analytic solutionsof Biittcher's functional equation
in the unit disk
Cenr C. CoweN
Suppose/ is analytic on the unit disk D, maps D into itself, and has the Taylor
seriesf (z): a&r * ar+tzr+r+'" where arl0 and k > 2. This papergivesnecessary and sufficientconditions for the existenceof single-valuedanalytic solutions
definedon all of,D to Bdttcher'sfunctionalequationa"f : qn.It is easilyseen[5]
that the only non-zero solutions ocrur when k : m- There is always a solution of
the equation a of : o' that is holomorphic and univalent in a neighborlnod of
zero: seeValiron [9, pp.124-127J.When o is a solution of Bottcher's equation,so
is q" for n:2,3,..., so our problem is to determinewhen one of thesehas a
continuationto all of the disk. We shallseethat the existenceof such
single-valued
solutionsdependson a multiplicity condition on the zeroesof iteratesof /, and we
deterrnineall solutionswhen the condition is met. The solutionsare computablein
the sensethat the Taylor coeftcients of o can be obtained recursively.
As usual, from the solutions of this fundamental equation one can obtain
information about solutions of the classicalfunctional equations of Abel and
Schroederand aboutfractionaliteratesof /. Sincethesenecessitate
considerationof
multiple-valuedfunctions,we confineour attention on thesequestionsto those/
that are real-valuedand increasingon [0,1). We obtain entirelyanalogousresultsto
those of Szekeres[8], Kuczma [4J, and Ger and Smajdor [3]. The additional
information obtained here is that the natural fractional iteration semigroup
l,(x)= F(:,t) is realanalyticin t aswell asr whenf'$)>0 for 0< r < I and that
if lim,-r-f(x):1, this semigroupis actuallyembeddedin a continuousgroup.
Referencesto the extensiveliterature on the subjectof iteration may be found in
[6]. I would like to thank the refereesfor several helpful suggestionsand for
pointing out some relevanl references.
AMS (1980)subject classification:Primary 30D05, Secondary39810.
Researchsupported in part by National ScienccFoundation Grant MCS-7902018.
Manuscript recivd Juru 2, 1981,ard, in final fon4 Decembcr 6,'1981.
187
lE8
AEQ. MATH.
CARL C, CI}WEN
We beginby examiningthe appropriatenotion of multiplicity.For zoin D, n a
positiveinteger,let m(zo,n) be the multiplicityof the zero of f, at zs, that is, if
*. - . then rn(zo,n)= minf, b,l 01. (Here f,
f^(z): bo* b'(z zo)+br(z zo)2
d e n o t e st h e n t h i t e r a t eo f / , t h a t i s , f r = f a n df , . t : f " f ^ I o r f , : 1 , 2 , . . . . )
Sincetn(zo,n) > 0 if and only if f.(zo): 0, we have rn(zo,n) = 0 for all n for all
but countablymany zo in D. Sincef (z)= atzr +. ", w€ see t?r(O,n): k' for
t = 1 , 2 , 3 , . . . . I f 1 @ o ) : 0 a n d m ( z o , l ) = i ( s ot h a tf ( z ) : b i Q - z o l * . . . ) t h e n
nr(zo,n): jk'-t (sincef UQ)): a. (biQ - zol +. - -)' + - . ., etc.).
For f analyticin D, mappingD into itself with /(z )= arz' + . . - where arl0
and k 22,we define the multiplicitysetof f to be the set Qr: lk-^m(z,n); z e D,
n'=lr2r3r...l.
As a consequenceof the above observationswe have the following.
PROPOSITION. Forf as abooe,Qr) {0,1}.Moreotser,
if {z eD: f^(z)=0 for
some n ) is finite, then Q1 is finite.
We now compute the multiplicity sets for some particular functions.
EXAMPLE l. Ict f be the inner function f(z)= zL exp((z+lXr -1f'),
wheret *2. Since hQ)=0 if and only rt z :0, we havem(z,n):0 for zl0,so
Qr= {0'U-
EXAMPLE 2. L-etf(z):0.32'-0.62t,
rhen Qr = {0,1,t,1}. To see this we
o b s e r vteh a t f ( z ) : 0 i f a n d o n l yr t z : 0 o r z : l ' , f r ( r ) : 0 i f a n d o n l y i fz : 0 , 2 : l
-0.8. Since
or z:z'l h ? ) l = ( 0 . 9 ) ' < 0 . 5 f o r k = 3 a n d s i n c ef r ( z ) f z ' f o r
f r l a l , w e s e et h a tm ( z , n ) : 0 e x c e p tz = 0 , z : l a n d z : z ' . I t i s e a s y t o s eteh a t
ttt(0,n):2' for all n, fr(l,n; :2^-t for all n, and fr(Z',n) = 2^-2for n >2 with
Dt(z',1): 0.
EXAMPLE 3. For f(z)=2-"2'(z -r)', since lf (z)l<t
Qr : {0, 1,3}.
for lrl<1,
we see
EXAMPLE 4. SupposeB(z): arzt +'' . is a finite Blaschkeproductof order
M, at I 0 with k >z,and B (z)/ arzl.SinceB hasorderM, B'(z):p for at most
M - 1 points in D. Choose u in D, af 0 such that B(ot):g and let r =
m a x { l z l : B ' ( z ) - - 0 } . B y S c h w a r z ' l e m ml an ,e ) f < l z l t o r A < 1 , l < I a n dr h eo n l y
fixed point in D is 0, so we can find art in D, with f r' I > r, and B,-r(a ') : ,. Let
m : m(r',l)> l. Now if B^(z): ar' then m(z,n * l):7n and mk-"-t€ Q".
S i n c eB " m a p s D o n t oD f , o re v e r y n , w e s e teh a t m k - " - t€ Q " f o r n = 1 , 2 , 3 , . . . .
Carrying the ideasslightly further we see that.Qs is a bounded infinite set.
Analytic solutionsof B<ittcher'sfunctionalequation
Yol.24, 1982
189
E X A M P L E 5 . L e t t ; - 1 - 2 - i t o r i = 1 , 2 , 3 , . . . , l e t k > 2 b e g i v e na n d l e t
f(z)=it,fr(tr)'J'
(the Blaschke condition guarantees the convergence of the product.) Since
f(D)CrD and ra(4,1):ki, we haveQr:t0,1,2,3,...1.
We now state the main theorem.
THEOREM 1. I-etf (z)= arz' +'' ' , wherek >2 atd at/0, be analyticin D
withf (D)C D, and letI*0 be gioen.Thereis o analyticin D, with o(D)C D, such
that o(f (z))= Io(z)^ if and only if m = k and thereis an integerI suchthat IQl is a
subsetof the integers.Moreooer, in this case, there is a unique solution o(z) =
9z' +. -. fo, each suchI and solutiong of the equation
gr-t : I-taL.
The first stefl in the proof is the constructionof a solution near zero for A : I
where o(z)=Pz+..'.
This step does not involve the multiplicity set. Tbe
solutions of the general equation are of the form (o,o)t where a is a complex
numberand I a positiveinteger.Theselocal solutionshavesingle-valuedextensions
to all of the disk exactly when lQt is a subsetof the integers.
Proof.I. Find e >0 such that ltl<,
and f(z):0
+
>
Re
&,
F
0}.
Define
:
H
H
to
be
any
branch of
{ar:
imply z:0.
Let H -
F(r) : -logf (ee-' ) + log e.
All branchesare single-valuedsince H is simply connectedand F is arbitrarily
continuablein H. For fixed ar,since ee-o-2'a winds around zero once for 0 = , = 1,
the imagef (ee-'-'n)winds around zero k times,so F(or +2d):
F(r.r)+Zzrki.(lt
follows that there are k branchesof logf (ee--) on ^FL)
Now
,
- log l/(ee-')l + loe a
F(r ) =
lim Re
lim
r-o
r€e
X
_ rr t l:t t_
-
=
X
- l o g f a r e * e - - + a . * r e e ' t e - ( - * t t+s. . . I
+ o,lr"'r-' + "'t - o.
-toelo,r'
;; o
AEo' MATH
cARL c. cowEN
190
L e t t i n g4 ( l ) = r " * i y " , f r o m 1 7 , p . M A l w e h a v eS @ ) = l i m " - - [ ( F " ( r . r ) - i y " Y x " 1
existsand g(F(r)) = p(g(r)) whereE@)= ou" + ip,for somea ) 0 and p real.
Now
B(r.,+zni)=ligry
,r^F,(co\*Zrk^i
;-:;
f"
- iy^:
lim
nd
E, (r.r)- iy"+ZriE
X;
fi
: s(a)+Zilgg
#
Thus 7 : lim"*(k'lx^'1: (Zz{.f'[8(al +2zri)- g(r)l exists.
Sinceg is univalentfor ar in largarf< rl4,lorl> p for p sufficientlylarge [7,
Theorem 3, p. 4411,we find that ll0.
We also have
g(F(ar +27ri))= e(7ko +2zli))= o;g(a+2ri)+
ip = ag(a)+ a"vlni + ip
and on tlrt other hand
s(F(a, +zfli)): g(F(co)+}f:ki)= g(F(or))+2rk7i:
ag(ct)+2rk"yi + ip-
Since0< y 4o, this meansa : k.
Now define A by A(t)= T-'[g(ar)+ p(k - lrrtJ. Thus A(t *?ni):
a n d A ( F ( r ) ) = k e ( o r ) . D e f i n i n gd b y 6 ( z ) = e x P ? A ( - l o g z ) ) f o r
A(t\+2d
0<lrl< t, we obtain from theserelations
e x P (- A ( - l o gz + Z l r : i ) ) =e x P? A ( - l o gz ) - Z t i ) :
e x P? A ( - l o gz ) )
so that 6 is single-valuedand
oUQ)) = exP? A( - loe(/(z )))
= €xPe A(F( - log z))) : exP? kA( - log z))
: [exP? A( - log z ))J' = 6(z)r.
B y T h e o r e m3 o f [ 7 , p . M l l , w e h a v e l a l a r ; f ' * o a n d a r g @ - ' A ( ' ) - 0 a s
a s & r - + c ow i t h f I m r . ' l = z r .I t
o r + @ w i t h f l m , l = z r , w h i c hm e a n sR e A ( r ) - c
follows(by choosingfIm(logz)l= z) that lim,<6(z)=0, so 6 has a removable
singularityat 0.
Vol. 24, 1982
Analytic solutionsof B6rtcher'sfunctionaleguation
l9l
and 6(z):0 if and only if Z =0, so 6(z)=
Thus d is analyticfor lrl<t
-.
where i=1, 5i10. Under thesecircumstances,
each branchof 6rl, is
6,tt +.
and analyticinlzf <e and (6tti)'(0)10. Chooseone branch and
single-valued
denote it 6o. We have lo"U|))l'= 6(f(zy=lo(z)J'=l6o(z)tJ', which means
6 o(f (z)\: eio6 s 1z)kfor somer ealg . Defining ooQ) - eiq(k I)-' 6o(z),we haverhat as is
analytic,single-valued
for lzl<t,60(0)-0, o'o(0)10,and oo(fQ\):ooQ)k. That
is, oe is a local solution to our functional equation.
II. SupposeI is an integersuchthat lQ is a subsetof the integers.For I z 11 e,
we defineo(z): (ao(z))tro thato(z): bef + ' . . and o(f (z))= o(z)'. We want to
show that o has a single-valuedextensionon all of D.
For eachinteger n, one of the branchesof,lo(f,(r))J"t' Ior lz f ce is o(z).
Considerthe analyticcontinuationof this branchto lz: lf"(z)l< r). The possible
branch points o(this function are the points zo such that f.(zo)=0. For 6 small
positive, y(t): Lo* 6e'-', 0 s I - 1, winds around Zo onc€ and f"(7(t)) winds
around zero m(zo,n) times,so c(f" (l(t))) winds around zero Im(znn) times. By
the choiceof I, k" divideslm(zo,n) so continuinglofi,(z))|"" along 7 givesthe
same function element lor t=0 as for t=l. In other words IrU,G\)J"" is
single-valuednear eachbranch point, so it is a single-valuedanalytic continuation
of.o(z) to {z :lf"(z)l< r}. In particular,this meansthat, if rn < n and lf^(z)l< t
then IoU^G))l''" :lo(f^(z))J"". Thus,defining<r on D by o(z):ls(f.(z))l"t',
where n is an integersuchthat f/"(z)lce, makeso into a single-valued
analytic
function on D with o(/(z)): o(z)'.
III. Now, if A t'O andrt Br-r: A-t, then h : Fo satisfiesthe equationh o/:
lh t. This means there are at least k - I distinct solutions of the equation
h " f : A l r r w i t h h ( z ) = c F ' + . . . w h e r e c t f 0 . O n t h e o t h e rh a n d ,i f t r i s s u c ha
solution,equatingthe coefficients
of, z't yields cfiL= Aciso that ci-'= A-tal and
we see that there are at most k - | possibleleading coefficients,c1. Equating
coefficients
of za*t lor i :1,2,.. . we obtain
g 1f,crrct+t,., . ,ct*i): Ict*Fl-t + R, (1, c,,.. . , cr*i-r)
where ; =ljlkJ and $ and R1 are functionsdepending
on the variablesindicated.
Sincek>2, this meansthat cr*; is determinedby the choiceof cr. Sincethereare
k - | formal power seriessolutions,and we have k - 1 analyticsolutions,we find
that each formal solution actuallyconvergesin D.
This completesthe proof of existenceand uniquenessif we are givenan integerI
such that tQy is a subsetof the integers.We now turn to the necessityof the
multiplicity condition.
IV. As we sawin III, existenceof a solutionof h " f =lhr is equivalentto the
existenceof a solutionof o of = ct. Supposeo is single-valued,
non-constant,
and
t92
clnl
c GowEll
AEO. MATH.
analyticin D, and satisfies
o of = q'with c(z)=b,zt +..' whereb,f A. We will
show lQ; is a subsetof the integers.
We have o(0): o(/(0)):q(0)' and a(z)L' :q$.(z))-g.(0)
for all z in D.
Sinceo is non-constant,this meansthat o(0)=0 and o(D)CD. Choose€')0 so
that I z l< e' and c(z): 0 impliesz :0. Given zoinD suchthatf^(zo)= 0, choose
6>0 such that 0<lt-zol<26
i m p l i e sh Q ) l O
and, if y(r)= zs*6e2il,
0st<1, then f/"(y(t))l Ie'. Now as y windsaround zoonce,c(fn(y(r))) winds
around zero lm(zs,n) times, and a(l(t)) winds around zero j times where
o(z): ci(z - zol +. - . . Since a(f.(z)): o(z)t", *e obtain lm(zs,n): jk" so
k - ' l m ( z r , n ) = . j a n i n t e g e rS
. i n c eQ r : { 0 } U l k - ' m ( z , n ) : h Q ) : 0 , t r = 1 , 2 , . . . } ,
we see lO1 is a subset of the integers. tl
COROLLARY l. If Io is the hast integer such that loQt is a suDsetof the
integers,then euery solutionof h " I = lh' analytic in tlv unit disk is of theform ao^
w h e r eo " f = o ' a n d o ( z ) = b u z b a . .. .
,t
COROLLARY 2. If lz: f.(z)=A forsomenl is afinitesetthenh"f : Ih'has
nn-constant solutionsanalytic in tlu unit disk for eoeryA*0.
Proof. q is a finite set of rational numbers.Let I be the leastcommon multiple
of the denominators. tr
COROLLARY 3. If ll/" ll- <l for some n, then h"f : Ih' has non-constant
solutions analytic in the unit disk for eaery Il0.
Proof. Let e >0 be small enoughthat f zl<c and f(z):0 imply z:0. The
hypothesisguaranteesthat, for mo large enough,ll/^ll< € so that, if rn > no and
f^(z):O, then f^(z):0. It follows that ktQr is a subsetof the integers. fl
We now turn to the specialcasein whichf is real-valuedand increasingon [0,1).
(Ihis case has been studied more extensivelyin tbe literature, see for example,
Szekeres[8J.) We pay particular attention to the case in which lim,-r- f(r):\,
which includesprobability generatingfunctions.Tbeorems2 and 3 are restatements
of results of Kuczma [4J and Ger and Smajdor [3J for the analytic case,
THEOREM 2. Suppose
f is analyticin D, f(D)C D, f(z): arz' * - -., k 22,
with ar)A, and f'(r)>O for 0<r(1.
Thenfor cach c>0 thereis a unique
o,
analytic
complex
ncar
0,
andreal
arialyticon l0,l) with s'(0)>0, such
furction
q
(
f
(
:
)
)
:
c
a
(
x
)
'
f
o
r
tlrat
0s:<1.
Moreooer,0<o(r)<1 ond o'(x)>O for
0<r <l and, if f satisfies
then s-r is real analytic(on
f'(x)>O for 0(.r(1,
(0,a(l -))) os well.
Yol.24,1982
Analytic rclutions o( B6ttcher'sfunctionalcquation
t93
hoof. We consider the case c = I first. I*t oo be the function constructedin
step I of the proof of Theorem 1. If we choose appropriate branches of the
functions in the construction, oo is non-negative on the interval [0, r). Since
f'(x)>0 for 0=r ( 1, we seethat hG)=0 if and only if r :0, so we define.' on
[0, 1) by cr(r) =Joo(f"(r))Jt"', where n is large enough,that /. (r) < e, and we take
the branch of, arlr' that is non-negativeon [0,1). It is easily checked that o is
welldefined, o(f (x)) = cr(r)r, and cr has tbe analyticity properties asserted.
From the functional equation we see that
o' U.(: ))/l(r ) : k^ lolo (r)J'-'o'(r ),
so that
a'(f" (r)) fl({)
v,-,tt\r', - k"
[4(r)l^-t
Thus,it f'(x)> 0 for 0 < r ( 1, then f'"(x)> 0 for all n and,sincer'A^QD *0 for n
. large enough,we have o'(x)> 0 for all .r in (0, 1). This meansa-r is welldefined
and real analytic on (0, s(l- ). An easy computation shows lr(.r) - c$-Lno'(x)
satisfiesthe equation ft "f = ch'.
As in Theorem 1, the uniquenessis a consequence
of the uniquenessof a formal
power seriessolution with o'(0) > 0. tr
As would be expected,the solutionof thisfunctionalequationleadsto solutions
of the classical functional equations on ' (0,1). The function A(x):
(logk)-'logllogo(r)f is a real analyticsolutionof Abel's equationA "f : A +l
for 0( r ( l.If a ) 0, al I, S1r; = [logo(x)l' wherep = (logc)(logk)-'is arcal
analyticsolutionof Schroeder's
equationS"/: cS for 0<x < 1.
We now turn our attention to fractional iteration.
THEOREM 3. Suppose
f is analyticin D, f(D)C D, f(z): atzr*-.- k>2
with ar>A and f'(x)>0 for 0< r <1. TIun thereis a functionF(x,t) definedfor
0s.r <l and 0= t <rr, real analyticin eachoariablesuchthatF(x,l): f (x) and
F ( F ( r , s ) , t ) : F ( r , s + t ) l o r 0 s x ( | a n d 0 = s , r ( o c . M o r e o u eir/, l i r n , - r -f ( x ) :
l, then F is defined (and has the sameproperties)for. -6< t1a.
Proof.FromTheorem2wehavethatthesolutionof oof = o'with o'(0)>0is
real analyticon (0,1) and has real analyticinverseon (0, o(1 - )). We defineF by
F(x,t)= s-'(Ia(r)J''). This functionis easilyseento haveappropriateproperties
for t>0.
194
.AEQ. MATI{.
CARL C. COWEN
Now, sinceo is increasing
and c(r)<l for 0<r<1, lirn,-r-a(x) exists.If
l i m , - r -f ( x ) = l , w e s e e f r o mo U G D : o ( x ) * t h a t o ( 1- ) = s ( l - ) ' s o g r ( 1 - ) = 1 .
It follows that o-r([o(r)J*') is well definedfor t(0 as wett. D
This theorem saysthat the discretesemigroupof iteratesof / (on [0, U) can be
embeddedin a continuoussemigroupand, il lim,*r- f (x):1, it can be embeddedin
a continuousgroup. In particular this is true if / is a probability generatingfunction
(ai>0 and Zi-rai: l). It is not dfficult to seethat this family of iteratesis regular
in the senseof Szekeres[8, p.216l.
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[6J KuczuA, M., Functional cquationsin a singb oariabb. Polish ScicntificPubtishers,Warszawa,1968.
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[8t SzerrnES, G., Regular itcration of rcal and complcr functbns. Acta Math. lm 0958),20T?J,8.
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