数理解析研究所講究録 1269 巻 2002 年 1-11

1

AHLFORS THEORY AND COMPLEX DYNAMICS:PERIODIC POINTS OF ENTIRE FUNCTIONSWALTER BERGWEILER
ABSTRACT. We give anew proof of the result that transcendental entire functions haveinfinitely many periodic points of all periods greater than one, and we discuss the main
tool used there: the Ahlfors theory of covering surfaces.

1. INTRODUCTION

This paper is an extended version of aseries of two talks given at the Research Instituteof Mathematical Sciences in Kyoto about the Ahlfors theory of covering surfaces, and theapplications it has found in complex dynamics. Some applications of one of the principal
results of the Ahlfors theory -the five islands theorem -to various questions in complex
dynamics have been surveyed in [11]. This includes topics such as the Hausdorff dimensionof Julia sets or the existence of singleton components of Julia sets. In the first part of thispaper (\S \S 2-3) we discuss Ahlfors’ “Scheibensatz,” which contains the five islands theorem
as aspecial case. Then we describe in some detail how the Ahlfors theory can be used
to prove the existence of periodic points of agiven period, atopic treated rather briefly
in [11]. Thus the present paper complements the survey [11] in some sense.
Based on ideas introduced by Essen and Wu $[18, 19]$ , and extended in [5], we present
areasonably self-contained proof of the result (Theorem in \S 5) that atranscendental
entire function has infinitely many repelling periodic points of all periods greater than$\mathrm{E}$one.
2. THE PRINCIPAL RESULTS OF THE AHLFORS THEORY
be ameromorphic function. Let
be adomain and let :is
be aJordan domain. Asimply-connected component of $f^{-1}(V)$ with
: $Uarrow V$ is aproper map. The degree
called an island of over . Note that thenof this proper map is called the multiplicity of the island . An island of multiplicity 1iscalled asimple island.
,
be Jordan domains with,, and let,Let nowand ,family of all functions
thedenoteweBypairwise disjoint closures.
, for allovermeromorphic in $D$ which have no island of multiplicity less thanLet

$D\subset\hat{\mathbb{C}}:=\mathbb{C}\cup\{\infty\}$$Darrow\hat{\mathbb{C}}$$f$$\overline{U}\subset D$$U$$V\subset\hat{\mathbb{C}}$

$V$

$f$

$f|_{U}$$U$$q\in \mathrm{N}$$\mu_{1}$$\ldots$$\mu_{q}\in \mathrm{N}$$D_{1}$$D_{q}\subset\hat{\mathbb{C}}$$\ldots$$\mathcal{F}(D, \{(D_{j}, \mu_{j})\}_{j=1}^{q})$$\mu_{j}$$D_{j}$

This paper is based on talks given at the RIMS Kyoto. My visit to Japan was made possible byProfessor Shunsuke Morosawa, with funding from the Japan Society for the Promotion of Science (grantin-aid 12640182). Ithank Professor Morosawa for the invitation to Japan, and aperfect organization of mystay there, and Ithank the Japan Society for the Promotion of Science for the funding provided. Supportof my research by G.I.F., G-643-117.6/1999 and by INTAS-99-00089 is also gratefully acknowledged

2
WALTER BERGWEILER$i\in\{1, \ldots, q\}$. We shall always suppose that(1)$\sum_{j=1}^{q}(1-\frac{1}{\mu_{j}})>2$.

One of the main results of the Ahlfors theory (called “Scheibensatz” by Ahlfors [1, p. 190])
can be stated as follows.
Theorem$\mathrm{A}.\mathrm{I}$.

$\mathcal{F}(D, \{(D_{j}, \mu_{j})\}_{j=1}^{q})$is normal

Aclosely related statement is as follows.

Theorem A.2.

$\mathcal{F}(\mathbb{C}, \{(D_{j}, \mu_{j})\}_{j=1}^{q})$contains only the constant functions.

In the above results, we may put, meaning that $1/\mu_{j}=0$ in (1) and that thefunctions in have no islands at all over .
We discuss some special cases of Theorem.
$\mu_{j}=\infty$$\mathcal{F}$$D_{j}$$\mathrm{A}.\mathrm{I}$(i) $q=5$ , $\mu_{1}=\mu_{2}=Is$$=\mu_{5}=2$ . Then Theorem A.I says that afamilyof meromorphic functions is normal, if the functions in the family do not have asimple
island over any of five given Jordan domains with disjoint closures. This is the celebrated
Ahlfors five islands theorem.
$=\mathrm{P}4$(ii) $q=4$ , $\mu_{1}=\mu_{2}=\mu_{3}=2$ ,. We note that if is holomorphic and,then has no island over
. Theorem A.I thus implies that afamily of holomorphic
functions is normal, if the functions in the family do not have asimple island over any ofthree given plane Jordan domains with pairwise disjoint closures.
$f$$\mu_{4}=\infty$$f$$\infty\in D_{4}$$D_{4}$(iii) $q=3$ , $\mu_{1}=\mu_{2}=3$ , $\mu_{3}=\infty$ . with
we now deduce from Theorem A.I
that afamily of holomorphic functions is normal, if the functions in the family do nothave an island of multiplicity less than three over any of two given plane Jordan domains$\infty\in D_{3}$with disjoint closures.

As already mentioned, Theorems A.I and A.2 can be considered as the main resultsof the Ahlfors theory of covering surfaces. Besides Ahlfors’s original paper [1], we refer
to [21, Chapter 5], [28, Chapter XIII] or [34, Chapter ] for an account of the Ahlfors
theory. Anew proof of Theorems A.I and A.2 was given in [10]. (Actually [10] wasmainly concerned with the Ahlfors five islands theorem, but it was pointed out in [10,\S 5.1] that the method used also yields the more general “Scheibensatz.”) In the first partof the proof in [10] it was shown by afairly simple and elementary argument that theconclusion of Theorems A.I and A.2 holds if theare sufficiently small disks. In thesecond part of the proof quasiconformal mappings were used to reduce the case of general
Jordan domains
to the case of small disks.
Since the version where theare small disks suffices for the applications consideredin this paper (as well as for many other applications), and since its proof is considerablyeasier and more elementary than the proof of the general version, we state this simplified
version formally. We use the notation $D(a, r):=\{z\in \mathbb{C} : |z-a|<r\}$ forand$r>0$ . In the following, let ,,be distinct and let ,,, and supposethat (1) is satisfied$\mathrm{V}\mathrm{I}$$D_{j}$$D_{j}$$D_{j}$$a\in \mathbb{C}$$a_{1}$$\ldots$$a_{q}\in \mathbb{C}$$\mu_{1}$$\ldots$$\mu_{q}\in \mathrm{N}$3
AHLFORS THEORY AND COMPLEX DYNAMICS

Theorem$\mathrm{B}.\mathrm{I}$. There exists

Theorem B.2. There exists
constant functions.

$\epsilon$$>0$ such that $\mathcal{F}(D, \{(D(a_{j}, \epsilon), \mu_{j})\}_{j=1}^{q})$$\epsilon>0$such that$\mathcal{F}(\mathbb{C}, \{(D(a_{j}, \epsilon), \mu_{j})\}_{j=1}^{q})$is normal.

contains only the

For completeness we include aproof of Theorems B.I and B.2 in \S 3 below, following
the arguments of [10, 11].

3. Aproof OF THEOREMS B.1 AND B.2
We denote the spherical derivative of ameromorphic function f by$f^{\#}$.

be a domain and let be a family offunctions meromorphic in $D$ .
Lemma 1. Let
in $D$ , a sequenceof positive real
If is not normal, then there exist a sequence$z_{0}\inD$meromorphic
and a non-constantnumbers, a sequencein , a point
function$f_{k}(z_{k}+\rho_{k}z)arrowf(z)$uniformly
in C.
locally,and:such that.
Moreover, can be chosen such that $f^{\#}(z)\leq 1=f\#(0)$ for all$D\subset \mathbb{C}$$\mathcal{F}$$(f_{k})$$f$$(\rho_{k})$$(z_{k})$$\mathcal{F}$$\mathcal{F}$$\mathbb{C}arrow\hat{\mathbb{C}}$$\rho_{k}arrow 0$$z_{k}arrow z_{0}$$z\in \mathbb{C}$$f$This lemma is due to Zalcman [35]. For asurvey of various applications of this lemma
we refer to [36]. We shall also need the following result.
Lemma 2. Let
satisfied. Let :multiplicity at least. Suppose that (1) is,,distinct and ,be a meromorphic function. Suppose that the -points of have
, for all $j\in\{1, \ldots, q\}$ . Then is constant.

$q\in \mathrm{N}$$f$,$a_{1}$,$a_{q}\in\hat{\mathbb{C}}$$\mu_{1}$$\ldots$$\ldots$$\mu_{q}\in \mathrm{N}$$\mathbb{C}arrow\hat{\mathbb{C}}$$f$$a_{j}$$\mu_{j}$$f$This result was proved by Nevanlinna using his theory on the distribution of values,see [27, p. 102] or [28, \S X.3]. Adifferent proof was given by Robinson [29]. For aproof of
Lemma 2based on Lemma 1we refer to [10, \S 3].
It is clear that Lemma 2follows from Theorem B.2. Using Lemma 1, however, we will
see that Theorems B.I and B.2 can in turn be deduced from Lemma 1.

To deduce Theorem B.I from Lemma 1we assume that Theorem B.I is false. Applywe obtain ameromorphic funcing Lemma 1to the family$\leq 1=f_{\epsilon}\#(0)$ for all $z\in$
C. It is easy to see that
:
with
tion
is normal. Thus
. By Marty’s theorem,
iffor some meromorphic
tending to zero such that
there exists asequence
we have $f^{\#}(0)=1$ so that isfor all. Sincefunction :have multiplicity at least ,of-points
Moreover,all
we see that
non-constant.
$j\in\{1$ ,, 5 , contradicting Lemma 2.
$\mathcal{F}(\mathbb{C}, \{(D(a_{j}, \epsilon), \mu_{j})\}_{j=1}^{5})$$\mathbb{C}arrow\hat{\mathbb{C}}$$f_{\epsilon}$$f_{\epsilon}\#(z)$$f_{\epsilon}\in \mathcal{F}(\mathbb{C}, \{(D(a_{j}, \epsilon’), \mu_{j})\}_{j=1}^{5})$$\epsilon’>\epsilon$$\{f_{\epsilon}\}_{\epsilon>0}$$f_{\epsilon_{k}}arrow f$$(\epsilon_{k})$$f$$\mathbb{C}arrow\hat{\mathbb{C}}$$\epsilon>0$$f_{\epsilon}\#(0)=1$

$f$

$f$

$a_{j}$$\mu_{j}$$\mathrm{f}\mathrm{o}\mathrm{r}\square$$\ldots$$\}$

is anon-constant meromorphic
To prove Theorem $B.2$ we note that if :function, thenis not normal at 0. Thus Theorem B.2 follows
from Theorem B.2.
We note that Lemma 1can in turn be used to deduce Theorem B.I from Theorem B.2.
$f$$\mathbb{C}arrow\hat{\mathbb{C}}$$\{f(nz)\}_{n\in \mathrm{N}}$$\mathrm{i}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{y}\square$

4. FIXED POINTS AND PERIODIC POINTS:
be afunction. We define the iterates
be sets and let :$n\geq2$ . Note$X_{n}:=f^{-1}(X_{n-1}\cap\mathrm{Y})$$f^{n}:=f^{n-1}\circf$,$f^{1}:=f$for,
by $X_{1}:=X$ ,and.
that $X_{2}=f^{-1}(X_{1}\cap \mathrm{Y})\subset X=X_{1}$ and thus $X_{n+1}\subset X_{n}\subset X$ for all
Let $X$ ,$\mathrm{Y}$$f$$Xarrow \mathrm{Y}$$f^{n}$$n\in \mathrm{N}$$n\in \mathrm{N}$$X_{n}arrow \mathrm{Y}$4
WALTER BERGWEILER

and $f^{n}(\xi)=\xi$ , but
is called aperiodic point of period of if$f^{m}(\xi)\neq\xi$ for $1\leq m\leq n-1$ . Aperiodic point of period 1is called
afixed point. The
which are not fixed points ofperiodic points of period are thus the fixed points of$m$for any less than .
We shall be concerned with periodic points of holomorphic functions. Let be aperiodic point of period of aholomorphic function . Then A $:=(f^{n})’(\xi)$ is called
the multiplier of 4. We say that aperiodic point is repelling, indifferent or attracting
depending on whether the modulus of its multiplier is greater than, equal to or lessthan 1. An indifferent periodic point is called rationally indifferent if the multiplier is aroot of unity and irrationally indifferent otherwise.
The following lemma indicates why the Ahlfors theory may be useful to prove theexistence of fixed points and periodic points.
Apoint$\xi$$\in X$$f^{n}$$n$$f^{m}$$\xi\in X_{n}$

$f$

$n$

$n$

$\xi$

$f$

$n$

be a domain, :Lemma 3. Let
Jordan domain.
(i) If has a simple island over such thatin .
(ii) If has an island
over such thatis repelling or has multiplier 1.
$D\subset\hat{\mathbb{C}}$

$f$

$U$

$f$

$V$

a meromorphic function and$Darrow\hat{\mathbb{C}}$$\overline{U}\subset V$, then$f$$V\subset \mathbb{C}a$

has a repelling fixed point

$U$

$U$

$f$

$\overline{U}\subset V$$V$, then$f$

has a fixed point in

$U$

which

Here we shall need only (i), but for completeness we have also included (ii). We will,
however, only sketch the proof of (ii).
To prove (i) we note that there exists abranch of the inverse function of such that$\phi(V)=U$ . It follows easily thatas $narrow\infty$ for some $u\in U$ , locally uniformly in$V$ .
This implies that is an attracting fixed point of , and thus arepelling fixed point
Cl
of .
$f$

$\phi$$\phi^{n}arrow u$$u$$\phi$$f$under iterates
To prove (ii) we consider the set $P$ of all images of critical points of
$P$
$c\in
U$ such that;that is, is the set of all $z\in V$ for which there existsandof$f’(c)=0$ , $f^{n}(c)=z$ and $f^{m}(c)\in U$ for $1\leq m<n$ . Thenis finite. Given
with $f(u)=v$ and we can connect and by apath
there exists.
. Moreover, there exists asimply-connected domain $W$ with
Let now :be the branch of the inverse function of which satisfies $\phi(v)=u$ .
Then we find that all iterates of are defined on , and it turns out thatdefines acurve which ends at apoint $a\in U$ . Moreover, is afixed point of which isrepelling or has multiplier 1; see, . , [24, p. 154] or [33, p. 57] for more details of this0argument.
$f|_{U}$$n\in \mathrm{N}$$f|_{U}$$P\cap V\backslash U$$u\in U\backslash P$$v\in V\backslash P$$u$$\phi$$v$$\gamma\subset W\subset V\backslash P$$7\subset V\backslash P$$Warrow \mathbb{C}$

$f$

$V$

$\phi$$\bigcup_{n=1}^{\infty}\phi^{n}(\gamma)$

$f$

$a$

$\mathrm{e}$$\mathrm{g}.$An alternative, less elementary proof of Lemma 3, (ii) is described after Lemma 4in\S 6 below.
The periodic points play an important role in complex dynamics. Let be an entire
or rational function. The basic objects studied in complex dynamics are the Julia setof which, by definition, is the set where the iterates of fail to be normal, and its
complement, the set of normality or Fatou set of . One of the fundamental results of thetheory is that the Julia set is equal to the closure of the set of repelling periodic points.
While this result was obtained for rational functions already by Fatou [20, \S 30, p. 69]
and Julia [23, p. 99, p. 118] in their memoirs that founded the theory, it was proved for
$f$

$f$

$f$

$f$

5
AHLFORS THEORY AND COMPLEX DYNAMICS

entire functions much later by Baker [3]. Baker’s proof is based on Theorem A.I andLemma 3, (i). Meanwhile, however, simpler proofs based on Lemma 1are available [4,12, 31]; see also [11, \S 6.2] for further discussion.

5. EXISTENCE OF pER1OD1C pOINTS OF A G1VEN pERIOD
We consider the polynomial $p(z):=-z+z^{2}$ . We note that $p(z)-z=z(z-2)$ and. Thus and have the same fixed points so that has no periodic
point of period 2. The following result of Baker [2] shows that is essentially the only
polynomial of degree greater than one where periodic points of some period are missing.

$p^{2}(z)-z=z^{3}(z-2)$$p$$p^{2}$$p$$p$. Suppose that has no
Theorem C. Let be a polynomial of degree $d\geq 2$ and let
periodic point of period . Then $d=n=2$ . Moreover, there exists a linear transformationsuch that $f(z)=L^{-1}(p(L(z)))$ , with $p(z)=-z+z^{2}$ .
$n\in \mathrm{N}$

$f$

$f$

$n$

$L$

The following result was conjectured in [22, Problem 2.20] and proved in [7, Theorem 1]and [8, \S 1.6, Satz 2].
Theorem D. Let be a transcendental entire function and let
has infinitely many periodic points of period .
$f$, $n\geq 2$ . Then$f$

, $n\geq 2$ . Then$f$$n\in \mathrm{N}$$n$Actually the following stronger result was proved in [7, 8].
Theorem E. Let be a transcendental entire function and let
has infinitely many repelling periodic points of period .
$f$$n\in \mathrm{N}$$n$Similarly to Theorem C one can also describe the cases where apolynomial fails tohave repelling periodic points of some period [8, \S 1.4, Satz 1].

. Suppose that
Theorem F. Let be a polynomial of degree $d\geq 2$ and let
no repelling periodic point of period . Then one of the following cases holds:$n\in \mathrm{N}$

$f$

$f$

has

$n$

(i) $n=1$ , $d\geq 2$ ,(ii) $n=2$ , $d=2$ ,(iii) $n=2$ , $d=3$ ,(iv) $n=2$ , $d=4$ ,(v) $n=3$ , $d–2$ .

Examples in [8, \S 1.4] show that each of the five exceptional cases listed in Theorem
does occur.
The proof of Theorem actually gives the following result.

$\mathrm{F}$$\mathrm{F}$

Theorem G. Let be a polynomial of degree $d\geq 2$ and letof repelling periodic points of period N. Then
$f$

(2)

$n\in \mathrm{N}$. Let $N$ be the number

N $\geq d^{n}-\sum_{k<n,k|n}d^{k}-2n(d-1)$ .

Anew proof of Theorem
was given in [5, \S 4]. Here we show that the arguments
developed there also lead to anew proof of Theorem E. As we will use Theorem there,
we will also sketch its proof$\mathrm{D}$$\mathrm{G}$6
WALTER BERGWEILER

6. POLYNOMIAL-L1KE MAPS
Besides the results from the Ahlfors theory discussed in \S 2, we will need the conceptof apolynomial-like map to prove Theorem D. By definition, if ,are bounded,
simply-connected domains with, and if : $Uarrow V$ is aproper holomorphic map(of degree ), then the triple $(f, U, V)$ is called apolynomial-like map (of degree ). We
note that if , and are as in Lemma 3, then $(f, U, V)$ is apolynomial-like map. The
fundamental result about polynomial-like maps is the following one (see [13, Theoremor [16, Theorem 1] .
$U$$\overline{U}\subset V$$V\subset \mathbb{C}$

$f$

$d$

$d$

$f$

$U$

$V$

$\mathrm{V}\mathrm{I}.1.1]$$)$

Lemma 4. Let $(f, U, V)$ be a polynomial-like map of degree . Then there exists a polynomial of degree and a quasiconformal map
such that $f(z)=\phi^{-1}(p(\phi(z)))$
$z\in
U$
all
.
Moreover,contains
the
Julia
set
for
filledof and thus, in particular, all
$d$

$d$

$p$

$\phi:\mathbb{C}arrow \mathbb{C}$$\phi(U)$$p$periodic points of .
$p$

Here the filled Julia set of apolynomial is defined as the set of all points that do nottend tounder iteration of .
We remark that it easy to see that polynomials have afixed point which is repelling or
has multiplier 1. Therefore Lemma 3follows from Lemma 4. However Lemma 4can beconsidered as afairly advanced result, and thus the proof of Lemma 3sketched in \S 4 isconsiderably more elementary than the one via Lemma 4.
Using Lemma 4one can generalize many results about the dynamics of polynomials
to polynomial-like mappings. In particular this applies to Theorems ,and , aswell as to Lemmas 8-10 in \S 9 below. Alternatively, we can prove these Theorems andLemmas directly for polynomial-like maps, with essentially the same proofs that work forpolynomials, and thus avoid the use of Lemma 4.
$p$$\infty$$p$$\mathrm{C}$7. PRELIMINARIES FOR THE PROOF OF THEOREM$\mathrm{F}$$\mathrm{G}$$\mathrm{E}$As mentioned, we shall use arguments similar to those used in [5], which in turn werevery much inspired by papers by Ess\’en and Wu $[18, 19]$ .
Let :be holomorphic and let ,be Jordan domains. Similarly as in [5]we will use the notationif
has an island of multiplicity at most $m$ over .
$U\sim Vf$
We writeif
has an island over ;that is, if
for some. Note$V\wedge Wg,n$$W$
that ifandthen. Lemma 3now takes the following form.
Lemma 5.
(i) If V, then f has a repelling fixed point in V.
(ii) If V, then f has a fixed point in V which is repelling or has multiplier 1.
As in [5], we shall use some elementary concepts from graph theory. For aset andaset $E\subset V\cross V$ we call the pair $G=(V, E)$ adigraph. The elements of are calledvertices and those of $E$ are called edges. In contrast to usual terminology we allow edgesof the form $e=(v, v)$ with $v\in V$ . (Such edges are called loops.)
Let
and $w=$. Then is called aclosed walk of length
if $(v_{k-1}, v_{k})\in E$ for $k\in\{1, \ldots, n\}$ and $v_{0}=v_{n}$ . Note that we have not excluded thecase that $v_{j}=v_{k}$ for , $k\in\{1, \ldots, n\}$ , $j\neq k$ . We call aclosed walk $w=(v_{0}, v_{1}, \ldots, v_{n})$primitive if there does not exist, $1\leq p<n$ , such that $p|n$ and $v_{j}=v_{k}$ for all$f$$Darrow \mathbb{C}$$U$$U^{f}\sim^{m}’ V$$V\subset \mathbb{C}$$f|_{D\cap U}$$U^{f}\sim^{m}’ V$$V$$f|_{D\cap U}$$U^{f}\sim^{m}’ V$$V$$m\in \mathrm{N}$’$U^{g\circ f}\sim^{mn}$$\sim Vf,1$$\sim Vf$

$V$

$V$

$e$

$n\in \mathrm{N}$$(v_{0}, v_{1}, \ldots, v_{n})\in V^{n+1}$$j$$p\in \mathrm{N}$

$w$

$n$

7
AHLFORS THEORY AND COMPLEX DYNAMICS, $k\in\{1, \ldots, n\}$ satisfying $p|(j-k)$ . Aprimitive closed walk is thus aclosed walk which
is not obtained by running through aclosed walk of smaller length several times.
and aholomorphic function we consider
Given pairwise disjoint disks , ,the digraphs $G(f, \{D_{j}\}_{j=1}^{q})=(V, E)$ and $G_{1}(f, \{D_{j}\}_{j=1}^{q})=(V, E_{1})$ , with vertex set $V:=$$j$$D_{1}$$\{D_{1}, \ldots, D_{q}\}$$\ldots$$f$$D_{q}\subset \mathbb{C}$, and edge sets given by $E:=\{(D_{j}, D_{k})\in V\cross V : D_{j}\sim f D_{k}\}$ and $E_{1}:=$$\{(D_{j}, D_{k})\in V\cross V : D_{j}\sim^{1}’ D_{k}\}f$.

Combining Lemma 5with the remarks preceding it we obtain the following result.
Lemma 6.
(i) If $G_{1}(f, \{D_{j}\}_{j=1}^{q})=(V, E_{1})$ contains a primitive closed walk of length , then has
belonging to the walk.
a repelling periodic point of period in each(ii) If $G(f, \{D_{j}\}_{j=1}^{q})=(V, E)$ contains a primitive closed walk of length , then has
belonging to the walk.
periodic point of period in each
$f$

$n$

$n$

$D_{j}$

$f$

$n$

$n$

$a$

$D_{j}$

To give conditions where the hypothesis of Lemma 6are satisfied, we recall that theoutdegree of avertex in agraph $(V, E)$ is defined to be the cardinality of the set of
all $u\in V$ for which $(v, u)\in E$ . We have the following elementary results [5, Lemmas 6
and 9].
$v$, $n\geq 2$ , and let $G=(V, E)$ be a digraph with vertices.
contains(i) If $q\geq 6$ and if the outdegree of each vertex is a least $q-2$ , thenprimitive closed walk of length .
contains(ii) If $q\geq 4$ and if the outdegree of each vertex is a least $q-1$ , thenprimitive closed walk of length .

Lemma 7. Let ,$q$$n\in \mathrm{N}$

$q$

$G$

$a$

$G$

$a$

$n$

$n$

Finally we recall that afamilyof functions holomorphic in adomain $D$ is called
quasinormal (cf. [14, 25, 30]) if for each sequenceinthere exists asubsequence$E\subset
D$. If theconverges locally uniformly inand afinite set
such that,cardinality of the exceptional set $E$ can be bounded independently of the sequenceand if is the smallest such bound, then we say that is quasinorrmal of order .
of functions holomorNote that the maximum principle implies that if asequence$D$phic in adomain
for some finite subset $E$ of $D$ ,converges locally uniformly in.
but not in $D$ , thenin$\mathcal{F}$$(f_{k})$$(f_{k_{j}})$$\mathcal{F}$$D\backslash E$$(f_{k_{\mathrm{j}}})$$(f_{k})$$\mathcal{F}$

$q$

$q$

$(f_{k})$$D\backslash E$$f_{k}arrow\infty$$D\backslash E$

8. Aproof OF THEORY$\mathrm{E}$by $f_{k}(z)=$:and define
in which tends to
We choose asequence$f(c_{k}z)/c_{k}$ . It is easy to see that no subsequence ofis normal at 0. We note that if
is aperiodic point of , with the same period and4is aperiodic point of , thenmultiplier. Let. We consider two cases.
andis not quasinormal of order 6. Then there exists asubsequence
Case 1:,
is normal at one ofsix distinct points ,such that no subsequence of
these six points. Without loss of generality we may assume that $a_{1}=0$ and that for, 6 , no subsequence ofis normal at . We choose $>0$ as ineach $\ell\in\{1$ ,, 6 and is large enough,,. It follows from Theorem B.I that if
Theorem,for at least four values of $m\in\{1$ ,, 6 . Thus each vertex ofthen$G_{1}(f, \{D(a_{j}, \epsilon)\}_{j=1}^{6})=(V, E_{1})$ has outdegree at least 4. Lemmas 6and 7now imply that$(c_{k})$$\mathbb{C}$$f_{k}$$\infty$$\mathbb{C}arrow \mathbb{C}$$(f_{k})$$f$$c_{k}\xi$$f_{k}$$\mathcal{F}:=\{f_{k}\}_{k\in \mathrm{N}}$$(f_{k_{j}})$$\mathcal{F}$$a_{1}$$\ldots$$\ldots$$\}$$\mathrm{B}.\mathrm{I}$$a_{6}\in \mathbb{C}$$(f_{k_{j}})$$(f_{k})$$\epsilon$$a_{\ell}$$\ell\in\{1$$\ldots$$D(a_{\ell}, \epsilon)\sim^{1}D(a_{m},\epsilon)f_{k}$$\ldots$

$k$

$\}$

$\}$

8
WALTER BERGWEILER

is large enough, thenhas arepelling periodic pointof period . Moreover, we
may assume that
. This implies thatwith
. Sinceis arepelling periodic point of , we conclude that has infinitely many repelling periodic
points.
if$k$$f_{k}$$\ell_{k}\neq 1$$\xi_{k}\in D(a_{\ell_{k}},\epsilon)$$\zeta_{k}:=c_{k}\xi_{k}arrow\infty$$\zeta_{k}$

$f$

$f$

Case 2:$n$$\xi_{k}$is quasinormal of order 6. Then there exists asubsequenceand six
points ,,converges locally uniformly insuch that.
On the other hand, no subsequence ofis normal at 0and thus we deduce that$0\in\{a_{1}, \ldots, a_{6}\}$ and thatin
. Without loss of generality weshall assume that $f_{k}arrow\infty$ in. Since $f_{k}(0)=f(0)/c_{k}arrow 0$ we findthat if is large enough, then there exists acomponentof $f_{k}^{-1}(D(0,1))$ such that$(f_{k}, U_{k}, D(0,1))$ is apolynomial-like map, say of degree. It is easy to see that$karrow\infty$as. Lemma 4and Theorem now imply thatand hence have at least$N_{k}:=d_{k}^{n}- \sum_{\ell<n,\ell|n}d_{k}^{\ell}-2n(d_{k}-1)$ repelling periodic points of period. Since$karrow\infty$
we conclude that.
as$a_{1}$$\mathcal{F}$$(f_{k_{j}})$$a_{6}\in \mathbb{C}$$\ldots$$(f_{k_{j}})$$\mathbb{C}\backslash \{a_{1}, \ldots, a_{6}\}$$(f_{k})$$f_{k_{j}}arrow\infty$$\mathbb{C}$$\langle$$\{a_{1}, \ldots, a_{6}\}$$\mathbb{C}\backslash \{a_{1}, \ldots, a_{6}\}$$k$$U_{k}$$d_{k}arrow\infty$$d_{k}$$\mathrm{G}$$f$$f_{k}$$n$$d_{k}arrow \mathrm{o}\mathrm{o}$$N_{k}arrow\infty$$\square$Remark. If we only want to prove Theorem , then we can do without Lemma 4and
Theorem G. Instead we choosesuch that $f(c_{k})$ remains bounded so that $f_{k}(1)=$$f(c_{k})/c_{k}arrow 0$ . In Case 2we then find that $1\in\{a_{1}, \ldots, a_{6}\}$ and that if
is sufficiently
large, thenis the complete digraph, and thus contains primitive
closed walks of any length. This, together with Lemma 6, (ii) yields the conclusion.
$\mathrm{D}$$c_{k}$$k$$G(f_{k}, \{D(0, \frac{1}{4}), D(1, \frac{1}{4})\})$

9. SOME RESULTS FROM COMPLEX DYNAMICS
To prove Theorem we shall need some classical results from complex dynamics; see,. , [6, 13, 24, 26, 33] for an introduction to the subject. Let be apolynomial of degree
at least two and let
be aperiodic point of period of . For $1\leq j\leq p-1$ we define$z_{j}:=f^{j}(z_{0})$ . We callacycle of periodic points.
Suppose first that is attracting. Then the other are also attracting. We also saythat the cycle of periodic points is attracting. For $0\leq j\leq p-1$ we denote bythecomponent of the Fatou set that contains . Thenis called a cycle of immediate
attracting basins.
$\mathrm{G}$$f$$\mathrm{e}$$\mathrm{g}.$

$f$

$p$

$z_{0}$$\{z_{0}, z_{1}, \ldots, z_{\mathrm{p}-1}\}$$z_{0}$$z_{j}$$U_{j}$$Zj$$\bigcup_{j=0}^{p-1}U_{j}$Lemma 8. Each cycle of immediate attracting basins contains a critical point.
Suppose now thatis rationally indifferent and let be the smallest positive integer$(f^{p})’(z_{0})^{t}=1$such that. Thenhas the form(3)$f^{pt}(z)=z+a_{m+1}(z-z_{0})^{m+1}+O((z-\eta)^{m+2})$$t$$z_{0}$$f^{pt}$, with $a_{m+1}\neq 0$ . It turns out thatasMoreover for
we have$zarrow z_{0}$$m$is of the form $m=\ell t$ for some$\ell\in \mathrm{N}$.

$k\in \mathrm{N}$$f^{kpt}(z)=z+ka_{m+1}(z-z_{0})^{m+1}+O((z-z_{0})^{m+2})$as. Next, for $0\leq j\leq p-1$ there arecomponents $U_{\dot{|}j}(1\leq i\leq m)$ of theFatou set of such that
and
as
. The
are called Lean
$pm=p\ellt$domains. The set of theLeau domains
falls into disjoint subsets called
cycles of Leau domains, each of $pt$ domains, the domains of each subset being permuted
cyclically by .
$zarrow z_{0}$$f$$z_{j}\in\partial U_{\dot{|}j}$$m$$f^{\nu \mathrm{p}}|_{U_{\mathrm{j}}}.\cdotarrow z_{j}$$D_{j}$$\nuarrow\infty$$U_{\mathrm{j}}.\cdot$$\ell$$f$

Lemma 9. Each cycle of Leau domains contains a critical point.

9
AHLFORS THEORY AND COMPLEX DYNAMICS

Finally we mention the following result of Douady [15].
Lemma 10. A polynomial of degree
points.

$d$has at most $d-1$ non-repelling cycles of periodic

The idea in the proof of Lemma 10 is to perturb slightly so that the indifferent
periodic cycles become attracting. More specifically, aperturbation of the form }$f_{\epsilon}(z):=f(z)+\epsilon P(z)$ with asuitable polynomial(of high degree) and sufficiently small
degreeof as apolynomial may be larger
that
theyields the desired result. Noteandthan . The point is that if is sufficiently small, then there exist domains
mapofdegreeapolynomial-like
iscontaining the filled Julia set of such that
. The conclusion then follows from Lemmas 4and 8.
However, as remarked earlier, one can also prove Lemma 10 without making reference
to Lemma 4, by proving Lemma 8directly for polynomial-like maps. Thus Lemma 10 has
afairly elementary proof. The proof of the corresponding result for rational functions,due to Shishikura [32], is much more involved; see also [17].
Finally we note that for our purposes aweaker bound for the number of non-repelling
cycles of apolynomial of degree would suffice, . , the bound $2d-2$ obtained with
Fatou’s method.
$f$

$z$

$arrow$$P$$\epsilon>0$$f_{\epsilon}$

$d$

$V$

$U$

$\epsilon$$(f_{\epsilon}, U, V)$

$f$

$d$

$d$

$\mathrm{e}$$\mathrm{g}.$10. Aproof OF THEOREMS F AND

Proof of Theorem . For$\mathrm{G}$the number of fixed points of counted
the corresponding number where multiplicities areaccording to multiplicity, and byignored. Similarly, the number of periodic points of of period is denoted by , ifandmultiplicities are counted, and byotherwise. Clearly we have$G$

$k\in \mathrm{N}$

we denote by$f^{k}$$F_{k}$$\overline{F}_{k}$

$k$

$f$

$\overline{P}_{k}\leq\overline{F}_{k}\leq F_{k}=d^{k}$$\overline{P}_{k}$$\overline{F}_{n}-\overline{P}_{n}\leq\sum_{k<n,k|n}\overline{P}_{k}\leq\sum_{k<n,k|n}d^{k}$

We write$\overline{F}_{n}=F_{n}-(F_{n}-\overline{F}_{n})=d^{n}-(F_{n}-\overline{F}_{n})$(4)$P_{k}$.

and obtain$\overline{P}_{n}\geq\overline{F}_{n}-\mathrm{I}d^{k}=d^{n}-\sum_{kk|n<n,k|n}d^{k}-(F_{n}-\overline{F}_{n})$.

that contributes to it. Let, let
be afixed point of
To estimate the termof$m$ be the contribution ofis afixed point of;that is,to the term,
multiplicity $m+1$ . Let be the period of . Then the periodic cycle$z_{j}=f^{j}(z_{0})$cyclesof. Let be the number of, contributes $pm$ to the termwith. We shall show that
Leau domains associated to the periodic cycle$F_{n}-\overline{F}_{n}$$f^{n}$$z_{0}$$F_{n}-\overline{F}_{n}$$z_{0}$$p$$z_{0}$$f^{n}$$\{z_{0}, z_{1}, \ldots, z_{p-1}\}$$z_{0}$$F_{n}-\overline{F}_{n}$$\ell$$\{z_{0}, z_{1}, \ldots, z_{p-1}\}$

pm

(5)

$\leq n\ell$.

In fact, let be the smallest positive integer such that $(f^{p})’(z_{0})^{t}=1$ . Then $pt|n$ , andhas the form (3), with $m=\ell t$ . Now (5) follows since $pm=p\ell t$ and $pt\leq n$ .
, where is the number of cycles of Leau domains
It follows from (5) thatof . Since $L\leq d-1$ by Lemma 9, we obtain$f^{pt}$$t$$F_{n}-\overline{F}_{n}\leq nL$

$L$

$f$

(6)

$F_{n}-\overline{F}_{n}\leq n(d-1)$

10
WALTER BERGWEILER

Finally we note that if denotes the number of non-repelling periodic points of pe. Since the number of non-repelling periodic cycles of period
riod , thenis at most $d-1$ by Lemma 10, we obtain $Q\leq n(d-1)$ and thus
$Q$

$n$

$N.=\overline{P}_{n}$

-

$Q$

$n$

(7)

$N\geq\overline{P}_{n}-n(d-1)$.

Combining (4), (6) and (7) we obtain (2).
Proof of Theorem . As already mentioned, Theorem follows easily from Theorem G.
In fact, if $n=2$ , then $N\geq d^{2}-d-4(d-1)=(d-1)(d-4)$ by (2), and thus $N>0$ if$\square$$F$$\mathrm{F}$$d>4$ .
If $n=3$ , then (2) yields $N\geq d^{3}-d-6(d-1)=(d-1)(d-2)(d+3)$ so that $N>0$
if $d>2$ .

Finally, if $n\geq 4$ , let $m$ be the largest integer less thansince $n\geq 4$ . From (2) we obtain
$N$

$n$

that divides . Then $m\leq n-2$$n$$\geq$$d^{n}- \sum_{k\leq m}d^{k}-2n(d-1)$$=d^{n}- \frac{\Psi^{+1}-d}{d-1}-2n(d-1)$$\geq$$d^{n}-d^{m+1}+d-2n(d-1)$$\geq d^{n}-d^{n-1}+d-2n(d-1)$$=$$\geq$$(d^{n-1}-2n)(d-1)+d$$(2^{n-1}-2n)+d$$\geq d$,and this completes the proof of Theorem F.

口Remark. It follows from (4) and (6) that$\overline{P}_{n}\geq d^{n}-\sum_{k<n,k|n}d^{k}-n(d-1)$.

, except possibly if $n=d=2$ . Afurther investigation of theThis implies that
proof of Theorem given
case $n=d=2$ then leads to Theorem C. This is essentiallyby Baker [2].
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