Analysis and Ma hema ical Physics (2024) 14:119
h ps://doi.o g/10.1007/s13324-024-00981-4
On he Ha dy numbe o Koenigs domains
Manuel D. Con e as1·F ancisco J. C uz-Zamo ano1·Ma ia Kou ou2·
Luis Rod íguez-Piazza3
Recei ed: 25 June 2024 / Re ised: 22 Sep embe 2024 / Accep ed: 4 Oc obe 2024 /
Published online: 22 Oc obe 2024
© The Au ho (s) 2024
Abs ac
This wo k s udies he Ha dy numbe o hype bolic plana domains sa is ying Abel’s
inclusion p ope y, which a e usually known as Koenigs domains. Mo e explici ly, we
p o e ha he Ha dy numbe o a Koenings domains whose complemen is non-pola
is g ea e han o equal o 1/2, and his lowe bound is sha p. In con as o his esul ,
we p o ide examples o gene al domains whose Ha dy numbe s a e a bi a ily small.
Addi ionally, we ou line he connec ion o he a o emen ioned class o domains wi h
he disc e e dynamics o he uni disc and ob ain esul s on he ange o Ha dy numbe
o Koenigs maps, in he hype bolic and pa abolic case.
Keywo ds Ha dy spaces ·Abel’s equa ion ·Koenigs domain ·I e a ion in he uni
disc ·Koenigs map
M. D. Con e as, F. J. C uz-Zamo ano and L. Rod íguez-Piazza a e pa ially suppo ed by Minis e io de
Inno ación y Ciencia, Spain, p ojec PID2022-136320NB-I00, and Jun a de Andalucía, p ojec
P20_00664. F. J. C uz-Zamo ano is also pa ially suppo ed by Minis e io de Uni e sidades, Spain,
h ough he ac ion Ayuda del P og ama de Fo mación de P o eso ado Uni e si a io, e e ence
FPU21/00258. M. Kou ou is pa ially suppo ed by he Alexande on Humbold Founda ion.
BManuel D. Con e as
[email p o ec ed]
F ancisco J. C uz-Zamo ano
[email p o ec ed]
Ma ia Kou ou
ma ia.kou ou@uni-wue zbu g.de
Luis Rod íguez-Piazza
[email p o ec ed]
1Depa amen o de Ma emá ica Aplicada II and IMUS, Escuela Técnica Supe io de Ingenie ía,
Uni e sidad de Se illa, Camino de los Descub imien os, s/n, 41092 Se illa, Spain
2Julius-Maximilians-Uni e si ä Wü zbu g, Ins i u ü Ma hema ik, Emil Fische S aße 40,
97074 Wü zbu g, Ge many
3Depa men o de Análisis Ma emá ico and IMUS, Facul ad de Ma emá icas, Uni e sidad de Se illa,
Calle Ta ia, s/n, 41012 Se illa, Spain
119 Page 2 o 21 M. D. Con e as e al.
Ma hema ics Subjec Classi ica ion P ima y 30D05 ·30H10 ·30C85; Seconda y
39B32 ·37F99
1 In oduc ion
Th ough he Ha dy spaces on he uni disc, Hp(D), one can de ine he Ha dy numbe
o a holomo phic unc ion ∈Hol(D,C), gi en by
h( ):= sup {p>0: ∈Hp(D)}∪{0}∈[0,+∞],
which somehow measu es he “g ow h” o . A simila idea was in oduced by Hansen
[9] in o de o examine he ange o holomo phic unc ions aking alues in a domain
. The so-called Ha dy numbe o a domain is de ined as
h() := in {h( ): ∈Hol(D,)}.
The s udy o he Ha dy numbe o a domain  ocuses on he case when is
unbounded, since h() =+∞i is bounded.
A classical p oblem has been o de e mine he ange o he Ha dy numbe o a
hype bolic plana domain (i.e., hose whose bounda y con ains a leas wo poin s)
in e ms o i s geome y and bounda y beha io . As a ma e o ac , se e al wo ks
ha e con ibu ed o es ima es on he Ha dy numbe o ce ain classes o hype bolic
domains. Hansen [9,10] p o ided a cha ac e iza ion o he Ha dy numbe o s a like
and spi allike wi h espec o he o igin domains. A ew yea s la e , Essén [7] ob ained
es ima es o he Ha dy numbe o a gene al hype bolic domain in e ms o ha monic
measu es and he loga i hmic capaci y. The ea e , Kim and Sugawa [14] examined
he Ha dy numbe o unbounded K-quasidisks. Qui e ecen ly, he ange o he Ha dy
numbe o comb domains was s udied by Ka a yllia [13].
The s epping s one, howe e , o he cu en wo k is he pape by Poggi-Co adini
[16], who examined he Ha dy numbe o hype bolic domains sa is ying Sch öde ’s
inclusion p ope y; i.e. λ ⊆, o someλ∈D. Inspi ed by he impac o Sch öde ’s
and Abel’s unc ional equa ions on he disc e e dynamics o he uni disc D, ou main
ocus lies on he ange o he Ha dy numbe o hype bolic domains sa is ying Abel’s
inclusion p ope y; namely +1⊆. Due o i s ele ance, hese domains will be
called Koenigs domains. One can easily no ice ha all domains in he a o emen ioned
class a e unbounded.
I u ns ou ha he Ha dy numbe o hese domains s ongly depends on he loga-
i hmic capaci y o hei complemen ; see Sec . 2.1. Recall ha , i ⊆Cis a domain
whose complemen is pola (i.e., i has ze o loga i hmic capaci y), hen he Ha dy
numbe o  anishes. This ollows, o ins ance, om [15, Theo ems 5.1.1 and 5.4.2,
p. 209 and 211], whe e i is p o ed ha e e y uni e sal co e ing map p:D→has
non- angen ial limi s almos nowhe e on he uni ci cle Ti he loga i hmic capaci y
o C is ze o. In con as , ou main esul p o ides a lowe bound o he Ha dy
numbe o a Koenigs domain whose complemen has posi i e loga i hmic capaci y:
On he Ha dy numbe o Koenigs domains Page 3 o 21 119
Theo em 1.1 Le ⊆Cbe a Koenigs domain. Then, h() ≥1/2i and only i he
complemen o is non-pola . In pa icula , o e e y Koenigs domain i holds ha
h() ∈{0}∪[1/2,+∞].
In Sec .5we emphasize he ole o he Koenigs domains in he la e esul . Namely,
o any p e ixed p∈(0,+∞)we build a domain ⊆Cwhose Ha dy numbe is
exac ly p.Fo p∈[1/2,+∞), easie examples a e known ( o ins ance, see he
commen s a e Lemma 2.3). In he case whe e p∈(0,1/2), as a as we know,
Theo em 5.2 is no el.
The p oo o Theo em 1.1 s ongly depends on po en ial heo y. Mo e p ecisely,
a sha p es ima e o he loga i hmic capaci y o compac se s ob ained as union o
in ege ansla ions o a compac non-pola subse o C. Recall ha a compac se in
he complex plane is
pola
i i s loga i hmic capaci y is ze o.
Theo em 1.2 Le E be a compac non-pola subse o D(0,1/4). Se
Kn=
n

j=1
(E+j).
Then,
lim
n→+∞
cap(Kn)
cap([0,n])=1.
Mo eo e , log(cap(Kn)) =log(n/4)+O(1/√n).
A e a p elimina y in oduc ion on loga i hmic capaci y, he Ha dy numbe o
domains and ha monic measu e in Sec .2, we p o e Theo em 1.2 in Sec .3and The-
o em 1.1 in Sec .4. A e his, he a o emen ioned amily o examples appea s in
Sec .5.
The main mo i a ion o he la e esul s is ha Abel’s inclusion p ope y is di ec ly
connec ed o non-ellip ic disc e e dynamics o he uni disc, mainly h ough he use
o Koenigs maps. To his ex en , i is o in e es o s udy any implica ions o Theo em
1.1 in he con ex o disc e e i e a ion heo y. To mo i a e his, se e al examples a e
p esen ed in Sec .6 o ela e he Ha dy numbe o Koenigs maps wi h he p ope ies
o i s associa ed sel -map.
No a ion 1.3 We w i e an=O(bn)i he e exis C >0and N ∈Nsuch ha
|an|≤Cbn o all n ≥N. Mo eo e , we w i e xn≥yn+O(bn)i he e exis s
an=O(bn)such ha xn≥yn+an o all n ∈N.
2 P elimina ies
2.1 Loga i hmic capaci y
Le us in oduce some opics abou he undamen als o po en ial heo y, which can
be ound in [18, Chap e s 3-5].
119 Page 4 o 21 M. D. Con e as e al.
De ini ion 2.1 Le μbe a ini e posi i e measu e on Cwi h compac suppo . I s
(loga i hmic) po en ial is he unc ion pμ:C→[−∞,+∞)gi en by
pμ(z)=C
log |z−w|dμ(w), z∈C.
I s (loga i hmic) ene gy I(μ) ∈[−∞,+∞)is gi en by
I(μ) =C
pμ(z)dμ(z)=CC
log |z−w|dμ(w)dμ(z).
F om hese concep s one can de ine he (loga i hmic) capaci y o a se X⊆C, ha is
cap(X)=sup
μ
exp(I(μ)) (2.1)
whe e he sup emum is aken among e e y p obabili y measu e μwhose suppo is a
compac subse o X.
In hese de ini ions, we unde s and ha I(μ) =−∞i he o me in eg al is no
con e gen . Indeed, i cap(X)=0, hen I(μ) =−∞ o e e y p obabili y measu e
μwi h compac suppo in X. I his happens, Xis said o be a pola se . A p ope y
is said o be sa is ied nea ly e e ywhe e in X⊂C, i i is sa is ied o all poin s in X,
excep maybe o a Bo el pola subse . In gene al, pola se s a e negligible om he
po en ial- heo e ic poin o iew. Mo eo e , he ollowing lemma holds:
Lemma 2.2 [18, Co olla y 3.2.5] A coun able union o Bo el pola se s is pola .
When i comes o compac a, he heo y o he po en ials is iche : i X⊂Cis a
compac se , hen he e exis s a p obabili y measu e νwi h suppo in Xa aining he
sup emum in (2.1). In ac , his measu e is unique i Xis non-pola and in such a case
νis said o be he equilib ium measu e o X;see[18, Theo em 3.7.6]. The po en ial
associa ed o he equilib ium measu e sa is ies he ollowing p ope y:
Theo em A (F os man’s Theo em) [18, Theo em 3.3.4] Le X ⊆Cbe a non-pola
compac se , and le νbe i s equilib ium measu e. Then, pν(z)≥I(ν) o all z ∈C.
Mo eo e , pν≡I(ν) nea ly e e ywhe e on X.
2.2 Ha dy numbe s
Gi en 0 <p<+∞, he Ha dy space Hp(D)is de ined as he se o all holomo phic
maps :D→Csuch ha
sup
0< <1T| ( ξ)|pdm(ξ) < +∞,
whe e mis he no malized leng h measu e on he bounda y o he uni disc T.In
case p=+∞, he Ha dy space H∞(D)s ands o he se o all holomo phic maps
On he Ha dy numbe o Koenigs domains Page 5 o 21 119
:D→C ha a e bounded, ha is,
sup
z∈D| (z)|<+∞.
We e e o [6] o a comple e in oduc ion o his opic. One o he p ope ies o
hese spaces is ha hey o m a dec easing amily, ha is, Hp(D)⊇Hq(D)i 0 <
p≤q≤+∞. These ela ions sugges he ollowing idea: gi en a holomo phic map
:D→C, i s Ha dy numbe h( )is de ined as
h( )=sup({0}∪{p>0: ∈Hp(D)})∈[0,+∞].
No e ha ∈Hp(D) o e e y 0 <p<h( ), and /∈Hp(D)i p>h( ).
In a simila manne , his idea can be ansla ed o domains. Gi en a domain ⊆C,
i s Ha dy numbe is de ined as
h() =in {h( ): ∈Hol(D,)}.
He e, Hol(D,) deno es he se o all holomo phic maps :D→Csuch ha
(D)⊆. Le us s a e some well-known p ope ies o he Ha dy numbe o a domain:
Lemma 2.3 [14, Lemmas 2.1 and 2.3] Le ,⊆Cbe wo domains. Then
(a) h() =+∞,i is bounded.
(b) h()≤h(),i ⊆.
(c) h(ϕ()) =h() o a complex a ine map ϕ(z)=az +b, a = 0.
(d) h() =0,i C is bounded.
(e) h() ≥1
2,i = Cis simply connec ed.
( ) h() =h(p), whe e p :D→is a uni e sal co e ing map o .
The Ha dy numbe o ce ain simply connec ed domains is al eady known. Fo
ins ance, h(H)=1 o a hal -plane H. In he case o a s ip e.g. S(a,b)={z∈
C:a<Im z<b}, wi h a,b∈R, i is known ha h(S(a,b)) =+∞.TheHa dy
numbe o sec o s has also been examined in [9]: suppose θ∈(0,2π]and Sθ:=
 eiφ: >0,|φ|<θ
2, hen h(Sθ)=π
θ.
A cha ac e iza ion o he Ha dy numbe o ce ain domains can be ob ained h ough
he ollowing classical esul :
Theo em B [15, Theo ems 5.1.1 and 5.4.2, p. 209 and 211] A uni e sal co e ing map
p:D→has non- angen ial limi s almos e e ywhe e on Ti and only i he
complemen o is a non-pola .
Recall ha any unc ion whose Ha dy numbe is posi i e ( ha is, i belongs o some
Ha dy space) has non- angen ial limi s almos e e ywhe e on T. Then, om he abo e
esul and he p ope ies o he Ha dy numbe o a domain, i ollows ha he Ha dy
numbe o e e y domain wi h pola complemen is ze o.

119 Page 6 o 21 M. D. Con e as e al.
2.3 Ha monic measu e
Fo a gi en domain ⊆Cwhose complemen is non-pola , le B⊆∂ be a Bo el
se . The ha monic measu e ω(z,B,) o Ba a poin z∈is he solu ion o he
gene alized Di ichle p oblem in wi h bounda y alues 1 on Band 0 on ∂ B.
Fo a ixed Bo el se B⊆∂,z→ ω(z,B,)is a ha monic and bounded
unc ion. In addi ion, o a ixed poin z∈, hemapB→ ω(z,B,)is a Bo el p ob-
abili y measu e on ∂. We e e o [18, Sec ion 4.3] o an in oduc ion o ha monic
measu e.
In [7, Lemma 1], Essén p oposed a ela ion be ween Ha dy numbe and ha monic
measu e, which was la e imp o ed by Kim and Sugawa in he ollowing esul , whe e
we use he no a ion D(z,R):= {w∈C:|w−z|<R}.
Theo em C [14, Lemma 3.2] Le ⊆Cbe a domain wi h 0∈. Then,
h() =lim in
R→+∞−log ω(0,FR,R)
log R,
whe e Ris he connec ed componen o ∩D(0,R)con aining he o igin and
FR=∂R∩{|z|=R}.
This esul will be use ul in Sec .4 o deduce Theo em 1.1 om Theo em 1.2, and
in Sec .5 o p o ide examples o domains sa is ying h() =p o e e y gi en
p∈(0,+∞).
3 P oo o Theo em 1.2
Be o e mo ing on o he p oo o Theo em 1.2, we p o e wo auxilia y lemmas ela ed
o he cons uc ion o he equilib ium measu e o he compac se s Kn,n∈N.
In he sequel we will use he equilib ium measu e o a compac in e al, which can
easily be de i ed om [19, Eq. (1.7), p. 25]. Fo any n∈N, he equilib ium measu e
μo he in e al [0,n]is absolu ely con inuous (wi h espec o Lebesgue’s measu e
m) and i is gi en by
dμ
dm( )=χ[0,n]( )
π√ (n− ).(3.1)
In pa icula , i ollows (c . [19, Eq. (1.8), p. 25])
pμ( )=I(μ) =log(cap([0,n])) =log(n/4), o all ∈[0,n]; (3.2)
he abo e esul is a combina ion o Theo em Aand [18, Theo ems 4.2.2 and 4.2.4].
Using his equilib ium measu e, we de ine he ollowing coe icien s:
αj:= j
j−1
dμ( )=1
πj
j−1
d
√ (n− ),j∈N,1≤j≤n.
On he Ha dy numbe o Koenigs domains Page 7 o 21 119
No ice ha αjalso depends on n. Howe e , in he seek o clea ance, his i no explici ly
w i en in he no a ion. Some p ope ies o hese numbe s can be de i ed di ec ly
om he de ini ion. Fo example, i ollows ha αj>0 o all j=1, ..., nand
ha n
j=1αj=1. I is also possible o no ice ha hese numbe s a e endowed
wi h some symme y, namely αj=αn−j+1, and ha αjis non-inc easing o j=
1, ..., (n+1)/2, whe e xdeno es he in ege pa o he eal numbe x.
We p o e he ollowing es ima ions:
Lemma 3.1 (a) The e exis s C1>0such ha αj≤C1/√n, o all n ∈Nand all
j=1,...,n.
(b) The e exis s C2>0such ha
n

j=1
j=k
αj
|j−k|≤C2
√n,
o all n ∈Nand all k =1,...,n.
P oo (a) Fix a na u al numbe n≥2. Then
α1=1
π1
0
d
√ (n− )≤1
π√n−11
0
d
√ =O1
√n.
Due o he mono onici y and symme ic p ope ies o he coe icien s αj, we ha e ha
b∗
1≤1, b∗
2≤1, b∗
3≤1/2, b∗
4≤1/2, and hence (a) holds.
(b) Fix n∈Nand de ine
S(k)=
n

j=1
j=k
αj
|j−k|,k=1,...,n.
By symme y, no ice ha S(k)=S(n−k+1). Thus, i is enough o wo k wi h
k∈Nsuch ha k≤(n+1)/2. I his is he case, no ice ha o any j∈Nwi h
1≤j≤(n+1)/2 i is possible o check ha αj=αn−j+1bu |j−k|≤|n−j+1−k|.
The e o e,
S(k)≤2(n+1)/2

j=1
j=k
αj
|j−k|.
Gi en wo ini e sequences {a1,a2, ..., am}and {b1,b2, ..., bm}o non-nega i e eal
numbe s, he Ha dy-Li lewood inequali y asse s ha
m

j=1
ajbj≤
m

j=1
a∗
jb∗
j,
119 Page 8 o 21 M. D. Con e as e al.
whe e {a∗
j}deno e he sequence o elemen s aja anged in dec easing o de and
simila ly o {b∗
j}(see [11, §10.2] and also [3, p. 43]). No ice ha he map {1,...,(n+
1)/2}  j→ αjis dec easing. Mo eo e , aking bj=1/|j−k|, o j=1, ..., (n+
1)/2,j= k, and bk=0, we ha e ha b∗
1≤1,b∗
2≤1,b∗
3≤1/2,b∗
4≤1/2,...,
b∗
(n+1)/2=0. In pa icula , b∗
j≤2/j. Thus
(n+1)/2

j=1
j=k
αj
|j−k|=(n+1)/2

j=1
αjbj≤2(n+1)/2

j=1
αj
j.
Bu now, no ice ha , o n≥2,
(n+1)/2

j=1
αj
j≤2
π(n+1)/2
0
d
( +1)√ (n− )
≤2
π√(n−1)/2+∞
0
d
( +1)√ =O1
√n,
whe e, in he i s inequali y, we ha e used ha 1/j≤2/( +1)whene e j−1≤
≤j. Thus, he esul ollows. 
To p oceed wi h he p oo o Theo em 1.2,le E⊆D(0,1/4)be a compac non-
pola se wi h 0 ∈Eand equilib ium measu e ν.Fixn∈Nand se
Kn=
n

j=1
(E+j).
Le us de ine he posi i e measu e σgi en by
σ(A)=
n

j=1
αjν(A−j),
whe e A⊆Cis a Bo el se . No ice ha σis a p obabili y measu e whose suppo is
a compac se lying on Kn. Once mo e, obse e ha σdepends on n.
Recalling he De ini ion 2.1, le us p o e he ollowing lemma:
Lemma 3.2 Unde he abo e no a ion, he ollowing s a emen s hold:
(a) pσ(x)≥log(n/4)+O(1/√n) o e e y x ∈Kn.
(b) pσ(x)=log(n/4)+O(1/√n) o nea ly e e y x ∈Kn.
(c) |pσ(x)−pσ(y)|=O(1/√n) o nea ly e e y x,y∈Kn.
(d) pσ(x)=log(n/4)+O(1/√n) o e e y x ∈E.
(e) |pσ(x)−pσ(y)|=O(1/√n) o e e y x ∈E and nea ly e e y y ∈Kn.
In ac , he unde lying cons an s do no depend on x and y.
On he Ha dy numbe o Koenigs domains Page 9 o 21 119
P oo In his p oo we will always assume ha n≥2. No e ha , by he de ini ion o
he p obabili y measu e σ, i s po en ial pσcan be w i en as
pσ(z)=
n

j=1
αjpν(z−j), z∈C.
Le k∈{1, ..., n}. No ice ha , by Theo em A,pν(x−k)≥I(ν) o all x∈E+k
and pν(x−k)=I(ν) nea ly e e ywhe e in x∈E+k. The e o e
pσ(x)=
n

j=1
αjpν(x−j)≥αkI(ν) +
n

j=1
j=k
αjE
log |x−j−y|dν(y)
o all x∈E+kand he inequali y u ns o be an equali y nea ly e e ywhe e. We
claim ha , gi en x∈E+k,
αkI(ν) +
n

j=1
j=k
αjE
log |x−j−y|dν(y)=log n
4+O1
√n.(3.3)
Thus, assuming his claim, we clea ly conclude (a) and (b).
Le us p o e he claim. Take x∈E+kand w i e x=x−k∈E. Then
log |x−j−y|=log |x−y+k−j|=log |k−j|+log 
1+x−y
k−j
=log |k−j|+O1
|k−j|,
o all y∈E, since |x−y|≤1/2. Due o Lemma 3.1.(b), we ob ain
αkI(ν) +
n

j=1
j=k
αjE
log |x−j−y|dν(y)
=αkI(ν) +
n

j=1
j=k
αjlog |k−j|+O1
√n.(3.4)
119 Page 16 o 21 M. D. Con e as e al.
de ine he domain
=C ∞

n=1
(CRn n),
n=z=Rneiθ:|θ|<π
R2p
n.(5.1)
Le us also de ine
R,N=D(0,R) N

n=1
(CRn n),ω
R,N=ω0,CR,R,N,R>RN.
We can now p o e he ollowing:
Theo em 5.2 Fo any p ∈(0,+∞) he e exis s a domain ⊆C o which h() =p.
P oo Fix p>0. The domain is cons uc ed as in (5.1), whe e he sequence {Rn}
is induc i ely cons uc ed depending on p. To do his, se R1=2. Suppose ha Rnis
cons uc ed o all 1 ≤n≤N.Le R>RN, and no ice ha he maximum p inciple
implies ha
ωR,N≤ω(0,N,D(0,RN)) =1
R2p
N
,(5.2)
as ω(0,·,D(0,R)) is he no malized Lebesgue measu e on CR. In pa icula ,
limR→R+
NωR,N≤1/R2p
N, whe e he limi con e ges because he maximum p inciple
implies ha he unc ion R→ ωR,Nis dec easing o R>RN.
We now claim ha he e mus exis s R>RNsuch ha ωR,N=1/Rp.Tosee
his, a guing by con adic ion, using he con inui y gi en in Lemma 5.1 and he abo e
inequali y, we ha e ha ωR,N<1/Rp o all R>RN. Then, Theo em Cimplies ha
h(C (∪N
n=1(CRn n)) ≥p. Bu his con adic s Lemma 2.3.(d). Thus, i is possible o
de ine RN+1=min{R>RN:ωR,N=1/Rp}.F om(5.2), we see ha RN+1≥R2
N,
which yields ha RN→+∞as N→+∞.
I is clea ha o e e y R>R1 he e exis s N∈Nwi h RN<R≤RN+1.F om
he cons uc ion, i ollows
ω(0,CR,∩D(0,R)) =ωR,N≤1/Rp,
whe e equali y is a ained o R=RN+1. The e o e, applying Theo em C,
h() =p.
Rema k 5.3 In he se ing o Lemma 5.1, i is possible o p o e ha limR→+∞ ωRlog(R)
>0. This ac could be used ins ead o Theo em C o show he claim we made in
o de o p o e he exis ence o RN+1.

On he Ha dy numbe o Koenigs domains Page 17 o 21 119
6 Applica ions o disc e e i e a ion heo y
Theo em 1.1 can be o in e es in he non-ellip ic disc e e dynamics o he uni disc.
To in oduce his, le φbe a holomo phic sel -map o D. Le us deno e by (φn)n∈N,
whe e φn=φ◦... ◦φcomposing n imes, he sequence o i e a es o φ.
As usual, i φhas a ixed poin in D, hen φis called ellip ic. In he case whe e φis no
an ellip ic au omo phism o D, he Denjoy–Wol Theo em asse s ha (φn)con e ges
locally uni o mly in D o a poin τφ∈D. The poin τφis called he Denjoy–Wol poin
o φ.
The esea ch spec um o he cu en wo k ocuses on he case whe e φhas no ixed
poin s in Dand hus, τφ∈∂D. Fo his case, i is known ha 0 <φ
(τφ)≤1(in
he angula limi sense). Mo e speci ically, i φ(τφ)<1, hen φis called hype bolic,
whe eas, i φ(τφ)=1, φis called pa abolic.
Pa abolic sel -maps a e u he di ided in o wo subca ego ies depending on i s
hype bolic s ep. The pa abolic sel -map φ:D→Dis o ze o hype bolic s ep i o
some –and hence, o all– z∈D, i is ue ha ρD(φn(z), φn+1(z)) →0, n→+∞,
whe e ρDdeno es he pseudo-hype bolic dis ance in D. O he wise, φis pa abolic o
posi i e hype bolic s ep. Fo a u he in oduc ion on his opic, we e e o a ecen
book by Aba e [1, Chap e 4].
A p ominen ool in examining p ope ies o he i e a es (φn)a e he solu ions o
he so-called Abel’s equa ion, ha is, holomo phic maps σ:D→Csa is ying
σ◦φ=σ+1.
In he cou se o he pas cen u y, he exis ence o solu ions o his equa ion in he case
whe e φis non-ellip ic was shown. Indeed, his was achie ed h ough he indi idual
wo ks by se e al ma hema icians: Vali on examined he hype bolic case [20], while
he wo di e en pa abolic cases we e co e ed by Pomme enke [2,17], he second
one in collabo a ion wi h Bake . The main idea o hese h ee wo ks was o explici ly
cons uc a solu ion σ o he Abel’s equa ion o a gi en sel -map φin e ms o i s
“no malized” i e a es.
Some ime a e wa ds, Cowen [5] p oposed o ind solu ions o a gene al unc ional
equa ion ela ed o he sel -map φ. This e en ually led o he ollowing concep : a
iple (0,,) is said o be a model o φi 0⊆Cis a domain, :D→0
is a holomo phic unc ion, and is an au omo phism o 0, sa is ying he ollowing
h ee condi ions:
(i)
◦φ=◦, (6.1)
(ii)

n∈N
−n((D)) =0,
119 Page 18 o 21 M. D. Con e as e al.
(iii) he e exis s a domain A⊆Dsuch ha is injec i e on A,φ(A)⊆A, and o
e e y z∈D he e exis s n∈Nwi h φn(z)∈A.
The domain 0is called a base space, he holomo phic unc ion is called an
in e wining map, he au omo phism is called a no mal o m, and Ais called an
abso bing se .
Cowen showed ha e e y non-ellip ic sel -map φadmi s a model, which migh
be chosen so ha i ollows a ce ain canonical o m. This can be o mula ed in he
ollowing way, whe e he no a ion H={z∈C:Im(z)>0}is used:
Theo em D Le φbe a non-ellip ic holomo phic sel -map o D. The e exis s a model
(0,σ,) o φ, whe e (z)=z+1, so ha :
(i) φis hype bolic i and only i he e exis s λ>1such ha 0=S(λ) := {z∈
C:0<Im(z)<π/log(λ)}.
(ii) φis pa abolic o posi i e hype bolic s ep i and only i 0=Ho 0=−H.
(iii) φis pa abolic o ze o hype bolic s ep i and only i 0=C.
Rema k 6.1 This heo em is s a ed in [1, Theo em 4.6.8], al hough i has been e o -
mula ed in such a way ha (6.1) always co esponds o he Abel’s equa ion o φ.
E e y non-ellip ic holomo phic sel -map admi s an essen ially unique holomo phic
model (see [1, Co olla y 3.5.9]). As a byp oduc , he unc ion σin Theo em Dis unique
i we assume ha Re(σ(0)) =0 in cases (i) and (ii), and σ(0)=0 in case (iii). F om
now on, such no malized solu ion o he Abel’s equa ion σwill be called he Koenigs
map o φ. The image =σ(D)plays a p ominen ole in he p ope ies o φ. No e ha ,
om Abel’s equa ion, needs o sa is y ha +1⊆. The e o e, is a Koenigs
domain in he sense o he p e ious sec ions. Indeed, he conclusion in Theo em 1.1
migh be s eng hened i is assumed o be a Koenigs domain o a ce ain sel -map
φo D, depending on he p ope ies o φ.
Fo example, i φis hype bolic, i s Koenigs domain is con ained in a ho izon al
s ip. This means ha his Koenigs domain and he associa ed Koenigs map ha e an
in ini e Ha dy numbe , see Sec . 2.2. The e o e, in he hype bolic case, he Koenigs
map is in Hp o all 0 <p<+∞. A li le mo e can be ob ained h ough he space
BMOA o analy ic unc ions wi h bounded mean oscilla ion, ha is, he se o analy ic
unc ions :D→Cwi h
sup
w∈Dsup
∈[0,1)2π
0| w( eiθ)|2dθ<+∞,
whe e w(z)= z+w
1+wz− (w), z∈D.
We e e o [8] o a comple e in oduc ion o BMOA. I holds H∞⊆BMOA ⊆Hp,
o 0 <p<+∞. The key poin is ha Riemann mappings om Don o any s ip
S(λ) a e known o be in BMOA. The e o e, by a subo dina ion a gumen (see [8,
Co olla y 10.1]), he Koenigs map σco esponding o a hype bolic sel -map φis also
in BMOA. This can also be no iced geome ically using a deepe esul o Hayman
and Pomme enke, see [12, Theo em 1].
On he Ha dy numbe o Koenigs domains Page 19 o 21 119
In he pa abolic case, cha ac e izes he hype bolic s ep o φ, as no iced in Theo em
D. The e o e, i φis a pa abolic sel -map o posi i e hype bolic s ep, he conclusion
on Theo em 1.1 can be imp o ed: i s Koenigs domain is con ained in a hal -plane,
and so he Ha dy numbe o he Koenigs domain and he associa ed Koenigs map is
always g ea e han o equal o one.
Fu he mo e, o e e y p∈[1,+∞], i is possible o ind a pa abolic sel -map
φo posi i e hype bolic s ep such ha h() =h(σ) =p o some Koenigs map
σ. In he case whe e pis ini e, his cons uc ion is achie ed by means o sec o s:
ix some θ∈(0,π]and se he domain S(θ) := {z∈C:a g(z)∈(0,θ)}.Le
σ:D→S(θ) be a Riemann mapping, and de ine a sel -map φ:D→Dgi en by
φ(z)=σ−1(σ (z)+1). By Theo em D,φis a pa abolic sel -map o posi i e hype bolic
s ep and, up o a ansla ion, σis he Koenigs map o φ. The e o e, he p ope ies
o Ha dy numbe s show ha h(σ) =h(σ(D)) =h(S(θ)) =π/θ; see Sec . 2.2.By
choosing θ∈(0,π]app op ia ely, i is clea ha o any p∈[1,+∞)one can ind
explici examples wi h h(σ ) =p.
Simila ly, o he in ini e alue o Ha dy numbe , we conside he domain =
{z=x+iy ∈C:x>0,0<y<√x}. Wi h he de ini ions gi en abo e, no e
ha o e e y θ∈(0,π/2) he e exis s ≥0 such ha + ⊆S(θ). Due o he
p ope ies o he Ha dy numbe s, i ollows h() =h( + )≥h(S(θ)) =π/θ.
Le ing θ→0+leads o h() =+∞. Once mo e, conside a Riemann mapping
σ:D→and de ine a sel -map φ:D→Dgi en by φ(z)=σ−1(σ (z)+1).By
Theo em D,φis a pa abolic sel -map o Do posi i e hype bolic s ep and, up o a
ansla ion, σis a Koenigs map o φ. Once mo e, he p ope ies o Ha dy numbe s
lead o h(σ) =h(σ(D)) =h() =+∞.
While conside ing pa abolic sel -maps o ze o hype bolic s ep, Koenigs domains
a e no included in some ho izon al hal -plane o C. The e o e he main di e ence
is ha now i may ac ually occu ha C has ze o loga i hmic capaci y and hus,
h(σ) can be ze o. Fo example, se =C {n∈Z:n<0}and le p:D→be
a uni e sal co e ing map wi h p(0)=0. Using [4, Theo ems 8.1 and 8.2], one can
see ha he e exis s a pa abolic inne unc ion φ:D→Do ze o hype bolic s ep o
which pis i s Koenigs map in he sense o Theo em D,bu h(p)=h() =0.
I he complemen o he Koenigs domain is non-pola , he la e si ua ion can no
happen since Theo em 1.1 can be applied. I can also be shown ha no uni o m uppe
bound o h(σ) no h() exis s. To be mo e p ecise: o e e y p∈[1/2,+∞] he e
exis s a pa abolic sel -map φo ze o hype bolic s ep such ha h(σ) =h() =p
o he Koenigs map o φ. Simila ly as be o e, in he ini e cases, his is done by
means o he sec o s S(θ) := {z∈C:|a g(z)|<θ/2}wi h h(S(θ)) =π/θ and a
sui able Riemann mapping. The a gumen is also simila o he in ini e alue, whe e
={z=x+iy ∈C:x>0,|y|<√x}wi h h() =+∞is used.
To sum up, he o egoing conside a ions can be abb e ia ed as ollows:
P oposi ion 6.2 Le φbe a holomo phic non-ellip ic sel -map o Dand conside i s
associa ed Koenigs map σ.
(i) I φis hype bolic, hen σis in BMOA.
119 Page 20 o 21 M. D. Con e as e al.
(ii) I φis pa abolic o posi i e hype bolic s ep, hen h(σ ) ≥1. Indeed, o e e y
p∈[1,+∞] he e exis s a pa abolic sel -map φo posi i e hype bolic s ep o
which h(σ) =p.
(iii) I φis pa abolic o ze o hype bolic s ep and he complemen o σ(D)is non-
pola , hen h(σ ) ∈[1/2,+∞]. Indeed, o e e y p ∈[1/2,+∞] he e exis s a
pa abolic sel -map φo ze o hype bolic s ep o which h(σ) =p. Mo eo e , he e
exis s an inne pa abolic sel -map o ze o hype bolic s ep, o which h(σ) =0.
Acknowledgemen s The au ho s would like o exp ess hei g a i ude o he anonymous e e ee o he
ho ough eading o he a icle and he help ul commen s.
Au ho Con ibu ions Au ho s ha e been discussing and wo king oge he on he manusc ip , con ibu ing
equally o he con en , p esen a ion, and e iewing he manusc ip .
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