Recei ed: 28 May 2024 Accep ed: 26 Janua y 2025
DOI: 10.1112/jlms.70077
Jou nal o he London
Ma hema ical Socie y
RESEARCH ARTICLE
C i e ia o ex ension o commu a i i y o
ac ional i e a es o holomo phic sel -maps in
he uni disc
Manuel D. Con e as1San iago Díaz-Mad igal1
Pa el Gumenyuk2
1Camino de los Descub imien os, s/n,
Depa amen o de Ma emá ica Aplicada II
and IMUS, Uni e sidad de Se illa, Se illa,
Spain
2Depa men o Ma hema ics, Poli ecnico
di Milano, Milan, I aly
Co espondence
Manuel D. Con e as, Camino de los
Descub imien os, s/n, Depa amen o de
Ma emá ica Aplicada II and IMUS,
Uni e sidad de Se illa, Se illa, 41092
Spain.
Email: con e [email p o ec ed]
Funding in o ma ion
Minis e io de Inno ación y Ciencia,
G an /Awa d Numbe :
PID2022-136320NB-I00; INdAM
(GNASAGA)
[Co ec ion added on 12 May, 2025, a e
i s online publica ion: The copy igh
line was changed.]
Abs ac
Le 𝜑be a uni alen non-ellip ic sel -map o he uni
disc 𝔻and le (𝜓𝑡)be a con inuous one-pa ame e semi-
g oup o holomo phic unc ions in 𝔻such ha 𝜓1≠𝗂𝖽𝔻
commu es wi h 𝜑. This assump ion does no imply ha
all elemen s o he semig oup (𝜓𝑡)commu e wi h 𝜑.In
his pape , we p o ide a numbe o su icien condi ions
ha gua an ee ha 𝜓𝑡◦𝜑=𝜑◦𝜓𝑡 o all 𝑡>0:This
holds, o example, i 𝜑and 𝜓1ha e a common bound-
a y ( egula o i egula ) ixed poin di e en om hei
common Denjoy–Wol poin 𝜏, o when 𝜓1has a bound-
a y egula ixed poin 𝜎≠𝜏a which 𝜑is isogonal, o
when (𝜑 − 𝗂𝖽𝔻)∕(𝜓1−𝗂𝖽
𝔻)has an un es ic ed limi a 𝜏.
In addi ion, we analyze how 𝜑beha es in he pe als o
he semig oup (𝜓𝑡).
MSC 2020
30C55, 37F44, 30D05 (p ima y)
Con en s
1. INTRODUCTION AND MAIN RESULTS ......................... 2
2. NOTATION AND PRELIMINARIES............................ 7
2.1. No a ion ....................................... 7
2.2. Holomo phic sel -maps o he uni disc ....................... 7
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p o ided he o iginal wo k is p ope ly ci ed.
J. London Ma h. Soc. (2) 2025;111:e70077. wileyonlinelib a y.com/jou nal/jlms 1o 31
h ps://doi.o g/10.1112/jlms.70077
2o 31 CONTRERAS e al.
2.3. Holomo phic models o uni alen sel -maps .................... 8
2.4. Commu ing holomo phic sel -maps ......................... 9
2.5. One-pa ame e semig oups in he uni disc ..................... 11
3. THE STARTING EXAMPLE AND A CHARACTERIZATION OF SELF-MAPS
COMMUTING WITH A SEMIGROUP .......................... 12
4. PROOF OF THEOREM 1.3................................. 13
5. PETALS AND COMMUTATIVITY I. AUXILIARY RESULTS ............... 15
6. PROOF OF THEOREM 1.4................................. 18
7. PETALS AND COMMUTATIVITY II. PROOF OF THEOREMS 1.8 AND 1.9 ....... 20
8. PETALS AND ISOGONALITY............................... 22
9. PROOF OF THEOREM 1.2................................. 28
10. COMMUTING ONE-PARAMETER SEMIGROUPS ................... 28
APPENDIX A: THE ELLIPTIC CASE ............................. 29
ACKNOWLEDGEMENTS................................... 30
REFERENCES......................................... 30
1 INTRODUCTION AND MAIN RESULTS
This pape is mo i a ed by he ollowing na u al and qui e old p oblem in disc e e holomo phic
i e a ion: Gi en a holomo phic sel -map o he uni disc, ha is, 𝜑∈𝖧𝗈𝗅(
𝔻), de e mine o a leas
analyze hose 𝜓∈𝖧𝗈𝗅(
𝔻) ha commu e wi h 𝜑,see[1, sec ion 4.10]. This kind o ques ions has
also been ea ed in he amewo k o ac ional i e a ion o holomo phic sel -maps, wi h he aim
o s udy all con inuous one-pa ame e semig oups in he uni disc (𝜓𝑡) ha commu e wi h a gi en
con inuous one-pa ame e semig oup (𝜑𝑡)in he sense ha 𝜓𝑡◦𝜑𝑠=𝜑
𝑠◦𝜓𝑡 o all 𝑡,𝑠 ⩾0, see,
o ins ance, [2]and[3].
In his pape , we ackle an in e media e si ua ion, which has in e es ing implica ions in he
disc e e as well as in he ac ional amewo k. Gi en 𝜑∈𝖴(
𝔻), a uni alen sel -map o he uni
disc, he cen alize (𝜑) o 𝜑is de ined as
(𝜑) ∶= {𝜓 ∈ (𝔻)∶𝜑◦𝜓=𝜓◦𝜑}, (𝔻)∶={𝜓∶𝔻→𝔻holomo phic injec i e}.
The p oblem we a e in e es ed in is he ollowing:
P oblem 1.1. Fix 𝜑∈(𝔻)and a con inuous one-pa ame e semig oup in he uni disc (𝜓𝑡).
Suppose 𝜓1∈(𝜑),wi h𝜓1≠𝗂𝖽𝔻. Does i necessa ily ollow ha 𝜓𝑡∈(𝜑) o all 𝑡>0? I no ,
p o ide condi ions on 𝜑and 𝜓1— and maybe on some ini e numbe o 𝜓𝑡s— unde which he
ela ion 𝜓1∈(𝜑) implies ha he whole semig oup (𝜓𝑡)is con ained in (𝜑).
Fi s o all, i is wo h men ioning ha he assump ion o 𝜑being uni alen is comple ely na u al
and no es ic i e. As we will see (P oposi ions 3.3 and A.1), uni alence is a necessa y condi ion
o 𝜑 o commu e wi h a non- i ial con inuous one-pa ame e semig oup. We would also like o
unde line ha he con ex o his p oblem is eally di e en om he o he wo men ioned abo e
because o i s lack o symme y. In pa icula , o wo con inuous one-pa ame e semig oups in he
uni disc, (𝜑𝑡)and (𝜓𝑡),wecanha e𝜓1∈(𝜑𝑡) o all 𝑡>0and a he same ime, 𝜑1∉(𝜓𝑡) o
some 𝑡>0; see Example 3.1. The same example shows ha , in gene al, he answe o he ques ion
in P oblem 1.1 is nega i e.
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EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 3o 31
When 𝜑is ellip ic, he abo e P oblem 1.1 was implici ly sol ed by Cowen [4]. Fo he sake
o comple eness, a he end o he pape , we include a b ie appendix wi h a e y simple, and
independen om Cowen’s wo k, solu ion o he ellip ic case; see P oposi ion A.1. In he es o
he pape , we es ic ou sel es o he non-ellip ic case.
In ha non-ellip ic case, i is no di icul o gi e a comple e answe o P oblem 1.1 in e ms
o he Koenigs unc ion 𝐻o he semig oup (𝜓𝑡). Indeed, (𝜓𝑡)⊂(𝜑) i and only he e exis s
𝑐∈ℂsuch ha 𝐻◦𝜑=𝐻+𝑐, see P oposi ion 3.3. The la e condi ion means exac ly ha 𝜑
is a ine wi h espec o 𝜓1(see De ini ion 2.6 and, in gene al, Sec ion 2.4 o u he de ails).
Mo eo e , as a qui e di ec consequence o ou esul s in [5], we can answe he ques ion in
P oblem 1.1 posi i ely i he semig oup (𝜓𝑡)is hype bolic o pa abolic o ze o hype bolic s ep.
Howe e , when (𝜓𝑡)is pa abolic o posi i e hype bolic s ep, gi ing any signi ican answe o
P oblem 1.1 ha does no in ol e 𝐻o he in ini esimal gene a o 𝐺=1∕𝐻
′is eally no ha
easy.
In he ollowing heo em, we summa ize ou posi i e answe s o P oblem 1.1 including he wo
ones desc ibed in he o me pa ag aph (as i ems (a) and (b)). Fo he de ini ion o isogonali y
a a bounda y poin in ol ed in (e), see Sec ion 8. O he use ul de ini ions can be ound in he
p elimina ies, see Sec ion 2. No e ha in ou e minology e e y non-ellip ic sel -map is di e en
om he iden i y map 𝗂𝖽𝔻. Fu he mo e, by a epelling ixed poin , we mean a bounda y egula
ixed poin di e en om he Denjoy–Wol poin .
Theo em 1.2. Le 𝜑∈(𝔻)be non-ellip ic and (𝜓𝑡)be a con inuous one-pa ame e semi-
g oup in 𝔻such ha 𝜓1≠𝗂𝖽𝔻and 𝜑◦𝜓1=𝜓
1◦𝜑. Assume ha one o he ollowing condi ions
holds:
(a) 𝜑is a ine wi h espec o 𝜓1,
(b) 𝜓1is ei he hype bolic o pa abolic o ze o hype bolic s ep,
(c) he e exis 𝑟∈(0,+∞)⧵ℚsuch ha 𝜓𝑟∈(𝜑),
(d) he limi
lim
𝑧→𝜏
𝜑(𝑧) − 𝑧
𝜓1(𝑧)−𝑧,
whe e 𝜏is he Denjoy–Wol poin o 𝜓1(and hence also o 𝜑), exis s un es ic edly in 𝔻,
(e) 𝜓1has a epelling ixed poin a which 𝜑is isogonal,
o
( ) 𝜑and 𝜓1ha e a common bounda y ixed poin di e en om he Denjoy–Wol poin .
Then, 𝜓𝑡∈(𝜑) o all 𝑡>0.
In gene al, condi ions (b) and (d)–( ) in Theo em 1.2 a e only su icien ones. A he same ime,
i is wo h ema king ha hese condi ions in ol e 𝜑and 𝜓1, bu do no depend on he knowledge
o o he elemen s o he semig oup (𝜓𝑡), i s Koenigs map, o in ini esimal gene a o . A p io i, his
is no he case o he necessa y and su icien condi ion (a), see De ini ion 2.6. Howe e , in many
cases, ou nex esul allows o bypass his di icul y.
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4o 31 CONTRERAS e al.
Theo em 1.3. Le 𝜑∈(𝔻)be pa abolic o posi i e hype bolic s ep, and le 𝜓∈(𝜑) ⧵ {𝗂𝖽𝔻}.
Deno e by 𝜏 he Denjoy–Wol poin o 𝜑and le
𝑅𝜑,𝜓(𝑧) ∶= 𝜓(𝑧) − 𝑧
𝜑(𝑧) − 𝑧,𝑧∈
𝔻.
Then, he ollowing s a emen s hold.
(A) The sequence (𝑅𝜑,𝜓 ◦𝜑◦𝑛)con e ges locally uni o mly in 𝔻 o some unc ion 𝑓𝜑,𝜓 ∈𝖧𝗈𝗅(𝔻,ℂ).
(B) 𝜓is a ine wi h espec o 𝜑i and only i he unc ion 𝑓𝜑,𝜓 is cons an in 𝔻. In ac , in such a
case, 𝑓𝜑,𝜓(𝜁) = ∠ lim𝑧→𝜏 (𝜓(𝑧)−𝑧)∕(𝜑(𝑧)−𝑧) o all 𝜁∈𝔻.
(C) Le (𝑆, ℎ𝜑,𝑧↦𝑧+1),whe e𝑆=ℍ∶= {𝑧∶ Im 𝑧 > 0} o 𝑆=−ℍ, s and o he canonical holo-
mo phic model o 𝜑.Then𝑓𝜑,𝜓 ◦ℎ−1
𝜑ex ends holomo phically o a map 𝐺∶𝑆→𝑆∪ℝwi h
𝐺(𝑤 + 1) = 𝐺(𝑤) o all 𝑤∈𝑆.
(D) The sel -map g(𝑤) ∶= 𝑤 + 𝐺(𝑤) is uni alen in 𝑆,g(ℎ𝜑(𝔻)) ⊂ ℎ𝜑(𝔻),and
𝜓=ℎ
−1
𝜑◦g◦ℎ𝜑=ℎ
−1
𝜑◦(ℎ𝜑+𝑓
𝜑,𝜓).
Coming back o Theo em 1.2, condi ion ( ) is p obably he mos in e es ing and deepes esul
o he pape . In ac , acco ding o he ollowing heo em, his condi ion implies he s onge con-
clusion ha 𝜑is an elemen o (𝜓𝑡). We exclude om he s a emen hype bolic au omo phisms,
because in his case, he esul is essen ially known: I 𝜑is a hype bolic au omo phism, hen (𝜓𝑡)is
a hype bolic one-pa ame e g oup and all he h ee condi ions in Theo em 1.4 below hold, excep
ha in condi ion (a) he wo ds “ o some 𝑡0>0” ha e o be eplaced by “ o some 𝑡0∈ℝ”; see,
o example, [1, sec ion 4.10] (see also [5]).
Theo em 1.4. Le 𝜑∈(𝔻)be a non-ellip ic sel -map di e en om a hype bolic au omo phism,
and suppose ha i has a bounda y ixed poin 𝜎di e en om i s Denjoy–Wol poin . Le (𝜓𝑡)be
a con inuous one-pa ame e semig oup such ha 𝜓1∈(𝜑) ⧵ {𝗂𝖽𝔻}. Then he ollowing condi ions
a e equi alen :
(a) 𝜑=𝜓
𝑡0 o some 𝑡0>0;
(b) (𝜓𝑡)⊂(𝜑);
(c) 𝜎is a bounda y ixed poin also o (𝜓𝑡).
Rema k 1.5. I a leas one o he sel -maps 𝜑o 𝜓1— and hence bo h o hem†— a e hype bolic,
hen all he equi alen condi ions (a), (b), and (c) in he abo e heo em a e au oma ically sa is ied.
This ollows om he ac ha by [5, P oposi ions 6.6 and 6.9], in he hype bolic case, we ha e
(𝜑) = (𝜓1)={𝜓
𝑡∶𝑡⩾0}.
Rema k 1.6. Le 𝜑and 𝜓be wo commu ing holomo phic sel -maps o 𝔻and suppose ha 𝜑has
a bounda y ixed poin 𝜎. I is known (see, e.g., [6]; see also Rema k 3.2) ha in gene al, 𝜎does
no ha e o be a bounda y ixed poin o 𝜓, e en i we addi ionally assume ha 𝜎is egula ,
𝜑is uni alen , and 𝜓is embeddable in a con inuous one-pa ame e semig oup. The e o e, he
implica ions (b) ⇒(c) and (b) ⇒(a) in he heo em s a ed abo e can be ega ded as an illus a ion
†See [4, Co olla y 4.1].
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EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 5o 31
o a nea di e ence be ween commu a i i y wi h a sel -map (and hence wi h all i s na u al i e a es)
and commu a i i y wi h all he ac ional i e a es.
I a con inuous one-pa ame e semig oup (𝜓𝑡)is con ained in he cen alize o a non-ellip ic
sel -map 𝜑, hen by Theo em 1.4 e e y bounda y ixed poin o 𝜑has o be among bounda y ixed
poin s o (𝜓𝑡). The con e se is no ue: We may ha e a con inuous one-pa ame e semig oup
(𝜓𝑡)⊂(𝜑) wi h many bounda y ixed poin s, while 𝜑has no bounda y ixed poin o he han i s
Denjoy–Wol poin ; see, o example, Example 3.1 wi h (𝜑𝑡)and (𝜓𝑡)in e changed. This leads us
o he ollowing na u al ques ion (in which, in ac , we impose some weake assump ions).
P oblem 1.7. Le 𝜑∈(𝔻)be a non-ellip ic sel -map di e en om a hype bolic au omo phism
and le (𝜓𝑡)be a con inuous one-pa ame e semig oup such ha 𝜓1∈(𝜑) ⧵ {𝗂𝖽𝔻}. Fu he , sup-
pose 𝜑∉{𝜓
𝑡∶𝑡⩾0}. Is he e any ela ionship be ween 𝜑and he bounda y ixed poin s o (𝜓𝑡)
in his case?
A na u al way o a ack his p oblem is h ough he concep o pe al o a con inuous one-
pa ame e semig oup, concep in ima ely linked o he no ion o bounda y egula ixed poin o
such a semig oup. We e e he eade o Sec ion 5 o mo e de ails, o o he monog aph [7, sec ion
13] o a de ailed exposi ion, which includes a numbe o ela ed esul s. I is wo h men ioning
ha his heo y o pe als plays also an impo an ole in he p oo o Theo em 1.4.
We begin by showing ha , as a gene al ac , 𝜑always maps pe als in o pe als and in a a he spe-
ci ic way. The degene a e case o (𝜓𝑡)being a one-pa ame e g oup is excluded om he s a emen
o he heo em below; i is co e ed sepa a ely, see Rema k 7.1 in Sec ion 7.
Theo em 1.8. Le 𝜑∈(𝔻)be non-ellip ic and le (𝜓𝑡) ⊄ 𝖠𝗎𝗍(𝔻)be a con inuous one-pa ame e
semig oup in 𝔻wi h Denjoy–Wol poin 𝜏∈𝜕𝔻such ha 𝜓1∈(𝜑) ⧵ {𝗂𝖽𝔻}.
(A) Assume ha Δis a hype bolic pe al o (𝜓𝑡)wi h associa ed bounda y epelling ixed poin 𝜎∈
𝜕𝔻. Then, one and only one o he ollowing h ee s a emen s holds:
(i) 𝜑(Δ) = Δ. In his case, he e exis s 𝑡0>0such ha 𝜑=𝜓
𝑡0and, in pa icula , 𝜎is a
epelling ixed poin o 𝜑.
(ii) Δ∩𝜑(Δ)=∅and he e exis s a hype bolic pe al Δ′o (𝜓𝑡)wi h associa ed epelling ixed
poin 𝜎′such ha 𝜑(Δ) ⊂ Δ′and ∠lim
𝑧→𝜎 𝜑(𝑧)=𝜎
′.
(iii) Δ∩𝜑(Δ)=∅ and he e exis s a pa abolic pe al Δ′o (𝜓𝑡)such ha 𝜑(Δ) ⊂ Δ′and
∠lim
𝑧→𝜎 𝜑(𝑧) = 𝜏.In hiscase,𝜎is an i egula con ac poin o 𝜑.
(B) Assume ha Δis a pa abolic pe al o (𝜓𝑡). Then he ollowing wo s a emen s hold:
(a) 𝜑(Δ) ⊂ Δ;
(b) 𝜑(Δ) = Δ i and only i 𝜑=𝜓
𝑡0 o some 𝑡0>0.
In he hype bolic case, by he eason men ioned in Rema k 1.5, only al e na i e (i) may
occu , and mo eo e , (B) becomes i ial because a hype bolic semig oup canno ha e pa abolic
pe als.
In he pa abolic case, based on Theo em 1.8, we a e hen able o gi e a qui e comple e answe
o P oblem 1.7 as ollows. We again exclude om conside a ion he case (𝜓𝑡) ⊂ 𝖠𝗎𝗍(𝔻), in which
he esul , wi h some ob ious modi ica ions, holds i ially.
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6o 31 CONTRERAS e al.
Theo em 1.9. Le 𝜑∈(𝔻)be a non-ellip ic sel -map wi h Denjoy–Wol poin 𝜏∈𝜕𝔻and le
(𝜓𝑡) ⊄ 𝖠𝗎𝗍(𝔻)be a con inuous one-pa ame e semig oup in 𝔻such ha 𝜓1∈(𝜑) ⧵ {𝗂𝖽𝔻}. Suppose
ha 𝜑is no an elemen o (𝜓𝑡). Then he ollowing s a emen s hold.
(I) Bo h 𝜑and (𝜓𝑡)a e pa abolic. Mo eo e , (𝜓𝑡)has a mos one pa abolic pe al.
(II) Suppose (𝜓𝑡)has a pa abolic pe al Δ∗. Then o each hype bolic pe al Δo (𝜓𝑡), he eexis a
ini e collec ion Δ1,Δ
2,…, Δ
𝑛o pai wise disjoin hype bolic pe als o (𝜓𝑡)such ha
Δ1=Δ, 𝜑(Δ
𝑘)⊂Δ
𝑘+1, 𝑘=1,…,𝑛−1, 𝜑(Δ
𝑛)⊂Δ
∗.
Mo eo e ,
𝜑(𝜎𝑘)=𝜎
𝑘+1, 𝑘=1,…,𝑛−1, 𝜑(𝜎
𝑛)=𝜏,
whe e 𝜎𝑘(𝑘 =1,…,𝑛)s ands o he epelling ixed poin associa ed wi h Δ𝑘.
(III) Suppose ha (𝜓𝑡)has no pa abolic pe al. Then o each hype bolic pe al Δo (𝜓𝑡), he e exis s
asequence(Δ𝑛)o pai wise disjoin hype bolic pe als o (𝜓𝑡)such ha
Δ1=Δ, 𝜑(Δ
𝑛)⊂Δ
𝑛+1,𝑛∈ℕ.
Mo eo e ,
𝜑(𝜎𝑛)=𝜎
𝑛+1,𝑛∈ℕ,and lim
𝑛→∞ 𝜎𝑛=𝜏,
whe e 𝜎𝑛(𝑛 ∈ ℕ)s ands o he epelling ixed poin associa ed wi h Δ𝑛.
Rema k 1.10. The abo e wo heo ems imply ha e e y epelling ixed poin 𝜎o 𝜓∶=𝜓
1is a
con ac poin o 𝜑and ha 𝜑(𝜎) is also a bounda y egula ixed poin o 𝜓.Mo eo e ,i 𝜑(𝜎) ≠𝜎,
hen he ( o wa d) o bi o 𝜎wi h espec o 𝜑(ex ended o a.e. poin o 𝜕𝔻by angula limi s)
ei he hi s he Denjoy–Wol poin wi hin ini e ime, o i consis s o pai wise dis inc epelling
ixed poin s o 𝜓, bu ends o he Denjoy–Wol poin in he limi . In pa , his esembles he
si ua ion wi h wo gene al commu ing holomo phic sel -maps 𝜑,𝜓 ∈ 𝖧𝗈𝗅(𝔻)s udiedbyB acci
[6]. In his gene al case, 𝜑can ha e a epelling cycle consis ing o epelling ixed poin s o 𝜓,bu
his possibili y is uled ou i 𝜑is uni alen and non-ellip ic, see [6, P oposi ion 5.2]. A he same
ime, he esul s es ablished in [6] do no seem o exclude some o he si ua ions no occu ing in
ou case, such as a possibili y ha he o bi o 𝜎hi s wi hin ini e ime a common ixed poin o 𝜑
and 𝜓di e en om hei Denjoy–Wol poin . Mo eo e , in con as o he con ex o Theo em 1.9,
i is no clea in gene al whe he any wo o bi s s a ing om di e en epelling ixed poin s o 𝜓
necessa ily all in o he same ca ego y.
Theo ems 1.8 and 1.9 a e p o ed in Sec ion 7. Fu he , in Sec ion 8, we conside he las wo
al e na i es gi en in Theo em 1.8(A) and analyze in which case equali y 𝜑(Δ) = Δ′occu s. This
will be used, along wi h se e al o he esul s, in he p oo o Theo em 1.2 gi en in Sec ion 9. Finally,
in Sec ion 10, based on ou indings, we subs an ially imp o e a esul by Elin e al. [2, sec ion 5]
on su icien condi ions o wo pa abolic con inuous one-pa ame e semig oups (𝜑𝑡)and (𝜓𝑡) o
commu e, gi en ha 𝜑1◦𝜓1=𝜓
1◦𝜑1.
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EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 7o 31
The pa o he pape p eceding Sec ions 7–10, he con en o which we ha e al eady desc ibed,
is o ganized as ollows. In Sec ion 2, we ecall some p elimina ies om holomo phic dynamics
ha will be needed o ollow he pape . In Sec ion 3, we p esen in de ail an example ela ed o
P oblem 1.1 and al eady men ioned abo e. In he same sec ion, we es ablish a cha ac e iza ion o
holomo phic sel -maps 𝜑commu ing wi h a non-ellip ic semig oup (𝜓𝑡)in e ms o he Koenigs
unc ion o (𝜓𝑡). In he nex sec ion, Sec ion 4, we p o e Theo em 1.3, which yields ano he cha ac-
e iza ion in e ms o he bounda y beha iou o 𝜓1and 𝜑along o wa d o bi s unde he disc e e
dynamics o 𝜓1. Sec ion 5con ains a b ie su ey on he heo y o pe als and some lemma a ela -
ing commu a i i y o pe als, which we use in he p oo o Theo em 1.4 gi en in Sec ion 6,aswell
as in he p oo o Theo ems 1.8 and 1.9.
2 NOTATION AND PRELIMINARIES
Below we in oduce some no a ion and basic heo y used u he in he pape . Fo mo e de ails
and o he p oo s o he esul s p esen ed in his sec ion, we e e he in e es ed eade s o he
ecen monog aphs [1, 7].
2.1 No a ion
As usual, we deno e he uni disc by 𝔻∶= {𝑧 ∈ ℂ∶|𝑧|<1},and𝔻∗∶= 𝔻⧵{0}.Wew i eℍ∶=
{𝑤 ∈ ℂ∶Im𝑤>0} o he uppe hal -plane and ℍRe ∶= {𝑧 ∈ ℂ∶Re𝑤>0} o he igh hal -
plane.
Fu he mo e, deno e by 𝖧𝗈𝗅(𝐷, 𝐸) he class o all holomo phic mappings o a domain 𝐷⊂ℂ
in o a se 𝐸⊂ℂ,andle (𝐷, 𝐸) s and o he class o all uni alen (i.e., injec i e holomo phic)
mappings om 𝐷 o 𝐸.Asusual,weendow𝖧𝗈𝗅(𝐷, 𝐸) and (𝐷, 𝐸) wi h he opology o locally
uni o m con e gence. In case 𝐸=𝐷,wewillw i e𝖧𝗈𝗅(𝐷) and (𝐷) ins ead o 𝖧𝗈𝗅(𝐷,𝐷) and
(𝐷, 𝐷), espec i ely.
Fo a sel -map 𝜑∶𝐷→𝐷o a domain 𝐷⊂ℂand 𝑛∈ℕ,we deno e by 𝜑◦𝑛 he 𝑛 h i e a e o 𝜑,
and le 𝜑◦0∶= 𝗂𝖽𝐷.Mo eo e ,i 𝜑is an au omo phism o 𝐷, hen o e e y 𝑛∈ℕ, we deno e by
𝜑◦−𝑛 he 𝑛 h i e a e o 𝜑−1.
2.2 Holomo phic sel -maps o he uni disc
The s udy o he dynamics o an a bi a y holomo phic sel -map 𝜑o he uni disc 𝔻is a classical
and well-es ablished b anch o Complex Analysis. An impo an ole is played by he ixed poin s,
all o which — excep o a mos one — lie on he bounda y and hence should be unde s ood in he
sense o angula limi s: 𝜎∈𝜕
𝔻is a bounda y ixed poin o 𝜑∈𝖧𝗈𝗅(𝔻)i 𝜑(𝜎) ∶= ∠lim𝑧→𝜎 𝜑(𝑧)
exis s and coincides wi h 𝜎. I is known ha he angula de i a i e 𝜑′(𝜎) exis s a e e y bounda y
ixed poin 𝜎, bu i can be in ini e. In his la e case, 𝜎is e e ed o as a supe - epelling ixed
poin ; o he wise, ha is, when 𝜑′(𝜎) is ini e, i is in ac a posi i e eal numbe and he bounda y
ixed poin 𝜎is said o be egula (BRFP o sho ).
The cen al esul in he a ea is he Denjoy–Wol Theo em, which s a es ha i 𝜑is di e en
om an ellip ic au omo phism (i.e., no an au omo phism o 𝔻possessing a ixed poin in 𝔻), hen
he sequence o he i e a es (𝜑◦𝑛)con e ges locally uni o mly in 𝔻 o a ce ain poin 𝜏∈𝔻.This
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8o 31 CONTRERAS e al.
poin is called he Denjoy–Wol poin o 𝜑.Mo eo e ,i 𝜏∈𝜕𝔻, i is he unique bounda y ixed
poin a which he angula de i a i e 𝜑′(𝜏) is ini e and belongs o (0,1]. In pa icula , o e e y
BRFP 𝜎di e en om he Denjoy–Wol poin 𝜏,weha e𝜑′(𝜎) ∈ (1, +∞). By his eason such
poin s a e e e ed o as epelling ixed poin s o 𝜑.
Acco ding o he posi ion o he Denjoy–Wol poin 𝜏and o he alue o he mul iplie 𝜑′(𝜏),
holomo phic sel -maps 𝜑∈𝖧𝗈𝗅(
𝔻)di e en om ellip ic au omo phisms a e di ided in o h ee
ca ego ies. Namely, 𝜑is called:
(a) ellip ic i 𝜏∈𝔻,
(b) hype bolic i 𝜏∈𝜕
𝔻and 𝜑′(𝜏) < 1,and
(c) pa abolic i 𝜏∈𝜕
𝔻such ha 𝜑′(𝜏) = 1.
The iden i y mapping 𝗂𝖽𝔻and all ellip ic au omo phisms o 𝔻a e con en ionally included in he
ca ego y (a) o ellip ic sel -maps. Simila ly, o an ellip ic au omo phism di e en om 𝗂𝖽𝔻,i s
Denjoy–Wol poin is de ined o be i s unique ixed poin in 𝔻.
Pa abolic sel -maps can ha e e y di e en p ope ies depending on he so-called hype bolic
s ep. Deno e by 𝜌𝔻 he hype bolic dis ance in 𝔻,andle 𝜑∈𝖧𝗈𝗅(𝔻)be non-ellip ic. Thanks o he
Schwa z–Pick Lemma, o he o bi (𝑧𝑛)∶= (𝜑◦𝑛(𝑧0))o any poin 𝑧0∈𝔻, he e exis s a ini e
limi 𝑞(𝑧0)∶=lim
𝑛→+∞ 𝜌𝔻(𝑧𝑛,𝑧
𝑛+1).I isknown,see, o example,[1, Co olla y 4.6.9], ha ei he
𝑞(𝑧0)>0 o all 𝑧0∈𝔻,o 𝑞≡0in 𝔻. The sel -map 𝜑is said o be o posi i e o o ze o hype bolic
s ep depending on whe he he o me o he la e al e na i e occu s. I 𝜑is hype bolic, hen i is
always o posi i e hype bolic s ep. Howe e , he e exis pa abolic sel -maps o ze o as well as o
posi i e hype bolic s ep.
2.3 Holomo phic models o uni alen sel -maps
An indispensable ole in ou s udy is played by he concep o a holomo phic model, which goes
back o Pomme enke [8], Bake and Pomme enke [9], and Cowen [10] and which is discussed
below o he special case o a uni alen sel -map. The e minology we use is mainly bo owed
om [11].
De ini ion 2.1. Aholomo phic model o 𝜑∈(𝔻)is any iple ∶=(𝑆,ℎ,𝛼), whe e 𝑆is a
Riemann su ace, 𝛼is an au omo phism o 𝑆,andℎis a uni alen map om 𝔻in o 𝑆sa is ying
he ollowing wo condi ions:
(HM1) ℎ◦𝜑=𝛼◦ℎ,and
(HM2) 𝑆=⋃𝑛⩾0𝛼◦−𝑛(ℎ(𝔻)).
The Riemann su ace 𝑆is called he base space, and he map ℎis called he in e wining map o
he holomo phic model .
E e y 𝜑∈(𝔻)⧵{𝗂𝖽
𝔻}admi s an essen ially unique holomo phic model. Mo e p ecisely, he
ollowing undamen al heo em holds.
Theo em 2.2 [11, Theo em 1.1]. E e y 𝜑∈(𝔻)admi s a holomo phic model. Mo eo e such a
model is unique up o a model isomo phism; ha is, i (𝑆1,ℎ
1,𝛼
1)and (𝑆2,ℎ
2,𝛼
2)a e holomo phic
models o 𝜑, hen he e exis s a biholomo phic map 𝜂o 𝑆1on o 𝑆2such ha
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EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 9o 31
ℎ2=𝜂◦ℎ1,𝛼
2=𝜂◦𝛼1◦𝜂−1.
The ype o a uni alen sel -map (ellip ic, hype bolic, o pa abolic) is e lec ed in, and ac ually
can be ully de e mined om he kind o holomo phic model 𝜑admi s. Fo an open in e al 𝐼⊂ℝ,
we de ine
𝑆𝐼∶= ℝ×𝐼={𝑥+𝑖𝑦∶𝑥∈ℝ,𝑦∈𝐼}.
Theo em 2.3 ([10], see also [11]). Le 𝜑∈(𝔻)⧵{𝗂𝖽
𝔻}. The ollowing s a emen s hold.
(1) 𝜑is an ellip ic au omo phism wi h mul iplie 𝜆∈𝜕
𝔻⧵{1}i and only i 𝜑admi s a holomo phic
modelo he o m𝜑∶= (𝔻,ℎ,𝑧↦𝜆𝑧),whe eℎ ∈ 𝖠𝗎𝗍(𝔻).
(2) 𝜑is an ellip ic sel -map wi h mul iplie 𝜆∈𝔻∗(and hence i is no an au omo phism) i and
only i 𝜑admi s a holomo phic model o he o m 𝜑∶= (ℂ,ℎ,𝑧↦𝜆𝑧).
(3) 𝜑is a hype bolic sel -map wi h mul iplie 𝜆∈(0,1)i and only i 𝜑admi s a holomo phic model
o he o m 𝜑∶= (𝑆𝐼,ℎ,𝑧↦𝑧+1),whe e𝐼=(𝑎,𝑏)is a bounded open in e al o leng h
𝑏−𝑎=𝜋∕|log 𝜆|.
(4) 𝜑is a pa abolic sel -map o posi i e hype bolic s ep i and only i 𝜑admi s a holomo phic model
o he o m 𝜑∶= (𝑆𝐼,ℎ,𝑧↦𝑧+1),whe e𝐼is an open unbounded in e al di e en om he
whole ℝ.
(5) 𝜑is a pa abolic sel -map o ze o hype bolic s ep i and only i 𝜑admi s a holomo phic model o
he o m 𝜑∶= (ℂ,ℎ,𝑧↦𝑧+1).
Rema k 2.4. In he abo e heo em, we may assume he ollowing:
- incase(1),ℎ′(𝜏) > 0, whe e 𝜏is he Denjoy–Wol poin o 𝜑;
- incase(2),ℎ′(𝜏)=1, whe e 𝜏is he Denjoy–Wol poin o 𝜑;
- in cases (3) and (5), ℎ(0) = 0;
- incase(4),Reℎ(0) = 0 and 𝑆𝐼=𝑆
(0,+∞) =ℍo 𝑆𝐼=𝑆
(−∞,0) =−ℍ.
Using he uniqueness pa o Theo em 2.2, one can show (see, e.g., [1, Co olla y 4.6.12] o de ails)
ha he abo e assump ions play he ole o a no maliza ion unde which he holomo phic model
𝜑 o a gi en 𝜑∈(𝔻)⧵{𝗂𝖽
𝔻}is unique. No e ha he no maliza ion o cases (3) and (5) would
also wo k in case (4), bu we p e e o use ano he no maliza ion, so ha o pa abolic sel -maps
o posi i e hype bolic s ep, he base space 𝑆𝐼o 𝜑coincides wi h ℍo −ℍ. Mo eo e , eplacing,
i necessa y, 𝜑wi h 𝑧↦𝜑(
𝑧) we may assume 𝑆𝐼=ℍ.
De ini ion 2.5. The unique holomo phic model 𝜑o a sel -map 𝜑∈(𝔻)⧵{𝗂𝖽
𝔻}de ined in
Theo em 2.3 and no malized as in Rema k 2.4 is called he canonical (holomo phic) model o 𝜑.
The in e wining map ℎo he canonical model 𝜑is called he Koenigs unc ion,andΩ∶=ℎ(𝔻)
is called he Koenigs domain o 𝜑.
2.4 Commu ing holomo phic sel -maps
I is clea ha i wo holomo phic sel -maps 𝜑,𝜓 ∈ 𝖧𝗈𝗅(𝔻)⧵{𝗂𝖽
𝔻}commu e, ha is, 𝜑◦𝜓=𝜓◦𝜑,
and i one o hem is ellip ic, hen he o he is also ellip ic and hey sha e he Denjoy–Wol poin .
The si ua ion is no so e iden when we conside non-ellip ic sel -maps. In 1973, Behan [12], see
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16 o 31 CONTRERAS e al.
Following he con en ional e minology in dynamical sys ems, we call he abo e BRFP 𝜎 he 𝛼-
poin o he co esponding hype bolic pe al Δ. In case o a pa abolic pe al, by i s 𝛼-poin we mean
he Denjoy–Wol poin 𝜏.
In he es o his sec ion, we suppose 𝜑∈(𝔻)is a non-ellip ic sel -map wi h Denjoy–Wol
poin 𝜏∈𝜕
𝔻, ha 𝜑is di e en om a hype bolic au omo phism, and ha (𝜓𝑡)is a con inuous
one-pa ame e semig oup o 𝔻sa is ying 𝗂𝖽𝔻≠𝜓1∈(𝜑).
Lemma 5.2. In he abo e assump ions, 𝜑()⊂,whe es ands o he backwa d in a ian se
o (𝜓𝑡). In pa icula , he image 𝜑(Δ) o any pe al Δo (𝜓𝑡)is con ained again in some pe al o (𝜓𝑡)
(which may coincide wi h Δ).
P oo . Deno e by ℎ he Koenigs unc ion o (𝜓𝑡). Using Abel’s equa ion (2.2), and aking in o
accoun ha ℎis uni alen , i is easy o see ha
ℎ()=⋂
𝑡⩾0
ℎ(𝔻)+𝑡 = ⋂
𝑛∈ℕ
ℎ(𝔻)+𝑛. (5.1)
The second equali y holds because, again by (2.2), ℎ(𝔻)+𝑡
2⊂ℎ(𝔻)+𝑡
1whene e 𝑡2⩾𝑡1⩾0.
By he hypo hesis 𝜑∈(𝜓1). The e o e, by [5, Theo em 3.4], he e exis s a uni alen holo-
mo phic map g∶ℎ(
𝔻)→ℎ(𝔻)such ha ℎ◦𝜑=g◦𝜑and g(𝑤 + 1) = g(𝑤) + 1 o any 𝑤∈ℎ(𝔻).
The e o e,
ℎ(𝜑())=g(ℎ())=g(⋂
𝑛∈ℕ
ℎ(𝔻)+𝑛)=⋂
𝑛∈ℕ
g(ℎ(𝔻)+𝑛
)=⋂
𝑛∈ℕ
g(ℎ(𝔻))+𝑛. (5.2)
Combining he ac ha g(ℎ(𝔻))⊂ℎ(𝔻)wi h he equali ies (5.1)and(5.2), we see ha
ℎ(𝜑())⊂ℎ().Sinceℎis uni alen , his p o es he inclusion 𝜑()⊂. Finally, he s a e-
men ega ding he pe als ollows immedia ely om his inclusion because he map 𝜑is
con inuous and open. □
No e ha he pe als o (𝜓𝑡)a e pai wise disjoin . The e o e, in Lemma 5.2 p o ed abo e, he
pe al Δ′ ha con ains 𝜑(Δ) is unique.
Lemma 5.3. Le Δ′be he pe al o (𝜓𝑡)de ined as abo e, ha is, 𝜑(Δ) ⊂ Δ′.Then
𝜑(𝜎) ∶= ∠ lim
𝑧→𝜎 𝜑(𝑧) = 𝜎′,
whe e 𝜎and 𝜎′s and o he 𝛼-poin s o he pe als Δand Δ′, espec i ely.
P oo . Fix any poin 𝑧0∈Δ. Then acco ding [7, P oposi ion 13.4.2], he e is a unique egula back-
wa d o bi (𝑧𝑡)⊂Δo (𝜓𝑡)s a ing om 𝑧0and con e ging o 𝜎as 𝑡→+∞. Mo e p ecisely, o
each 𝑡⩾0 he e is a unique poin 𝑧𝑡∈Δsuch ha 𝜓𝑡(𝑧𝑡)=𝑧
0and he map [0, +∞) ∋ 𝑡 ↦ 𝑧𝑡is
con inuous and injec i e, wi h lim𝑡→+∞ 𝑧𝑡=𝜎and
sup
𝑡⩾0
𝜌𝔻(𝑧𝑡,𝑧
𝑡+1) < +∞. (5.3)
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EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 17 o 31
Since 𝜑commu es wi h 𝜓1and hence wi h 𝜓𝑛 o any 𝑛∈ℕ,weha e
𝜓𝑛(𝜑(𝑧𝑛))=𝜑
(𝜓𝑛(𝑧𝑛))=𝜑(𝑧
0).
Taking in o accoun ha 𝑧𝑛∈Δand hence 𝜑(𝑧𝑛)⊂Δ
′ o all 𝑛∈ℕ∪{0}, we see ha he poin s
𝑤𝑛lie on he backwa d o bi o 𝑤0∶= 𝜑(𝑧0)∈Δ
′. The e o e, 𝑤𝑛→𝜎
′as 𝑛→+∞.
Thus, we see ha 𝜑(𝑧𝑛)→𝜎
′ o a sequence (𝑧𝑛)con e ging o 𝜎and such ha he hype bolic
dis ance 𝜌𝔻(𝑧𝑛,𝑧
𝑛+1), acco ding o (5.3), is bounded. This leads, see, o example, [11, Theo em
1.4], o he desi ed conclusion ha 𝜑has angula limi a 𝜎equal o 𝜎′.□
Lemma 5.4. I Δis a pa abolic pe al o (𝜓𝑡), hen𝜑(Δ) ⊂ Δ.
P oo . By he de ini ion o a pa abolic pe al, he 𝛼-poin 𝜎o he pe al Δcoincides wi h he Denjoy–
Wol poin 𝜏o he semig oup (𝜓𝑡), which in u n coincides wi h he Denjoy–Wol poin o 𝜑.
The e o e, 𝜑(𝜎) = 𝜎. Acco ding o Lemma 5.3, i ollows ha he pe al Δ′con aining 𝜑(Δ) mus
be also pa abolic.
By [7, Theo em 13.5.7], (𝜓𝑡)has a mos wo pa abolic pe als. I i has exac ly one pa abolic
pe al, hen we a e done. I he e a e wo dis inc pa abolic pe als, hen (again acco ding [7,The-
o em 13.5.7]) he images 𝑃𝑘,𝑘=1,2, o he wo pa abolic pe als wi h espec o he Koenigs
map ℎo (𝜓𝑡)a e o he o m 𝑃1={𝑤∶Im𝑤<𝑎}and 𝑃2={𝑤∶Im𝑤>𝑏} o some 𝑏⩾𝑎.As
a consequence, he semig oup (𝜓𝑡), and hence he sel -map 𝜓1, a e o ze o hype bolic s ep. Since
𝜑∈(𝜓1),by[5, P oposi ion 4.3], 𝜑is a ine wi h espec o 𝜓1, ha is, ℎ◦𝜑=ℎ+𝑐 o a sui able
𝑐∈ℂ. I ollows ha i Δand Δ′we e wo di e en pa abolic pe als, hen he ansla ion by 𝑐o
−𝑐 would map 𝑃1in o 𝑃2, which is clea ly impossible. Thus, 𝜑(Δ) ⊂ Δ′=Δ.□
Co olla y 5.5. In he no a ion o Lemma 5.3,𝜑(Δ) ⊂ Δ i and only i 𝜑(𝜎)=𝜎.
P oo . I Δis a hype bolic pe al, hen he co olla y ollows om Lemma 5.2 and he ac ha any
wo dis inc pe als ha e dis inc 𝛼-poin s unless bo h pe als a e pa abolic.
So suppose Δis a pa abolic pe al. Then 𝜑(Δ) ⊂ Δ by Lemma 5.4. Mo eo e , as we ha e seen in
he p oo o Lemma 5.4,𝜑(𝜎) = 𝜎 because in his case, 𝜎=𝜏is he Denjoy–Wol poin o (𝜓𝑡)
and hence also o 𝜑. These wo obse a ions comple e he p oo . □
Lemma 5.6. Suppose ha 𝜑is no an elemen o (𝜓𝑡).Then(𝜓𝑡)can ha e a mos one pa abolic pe al.
P oo . Suppose ha (𝜓𝑡)has wo dis inc pa abolic pe als. Then, as in he p oo o Lemma 5.4,we
see ha ℎ◦𝜑=ℎ+𝑐, whe e he cons an 𝑐mus be eal because each o he hal -planes 𝑃1and 𝑃2
a e mapped by he ansla ion 𝑤↦𝑤+𝑐in o i sel . Mo eo e , ℎ(𝔻)+𝑐=ℎ
(𝜑(𝔻))⊂ℎ(𝔻)and
hence, by [5, Theo em 3.1 (B)], 𝑐⩾0. I ollows ha 𝜑=𝜓
𝑐, which con adic s he hypo hesis. □
We conclude his sec ion wi h a lemma, which is de ini ely known o specialis s and holds also
o non-uni alen sel -maps, bu lacking a e e ence we p e e o include a sho p oo making use
o commu a i i y.
Lemma 5.7. Le 𝜑1,𝜑
2∈(𝔻)be such ha 𝜑◦𝑘
1=𝜑
◦𝑘
2 o some 𝑘∈ℕ.I 𝜑1is non-ellip ic,
hen 𝜑2=𝜑
1.
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18 o 31 CONTRERAS e al.
P oo . Since 𝜑1is uni alen and non-ellip ic, so is 𝜑3∶= 𝜑◦𝑘
1.Le (𝑆,ℎ,𝑧↦𝑧+1) s and o
he canonical holomo phic model o 𝜑3. Clea ly, bo h 𝜑1and 𝜑2commu e wi h 𝜑3. The e o e,
see, o example, [5, Theo em 3.4], 𝜑𝑚=ℎ
−1◦g𝑚◦ℎ,𝑚=1,2, wi h ce ain g𝑚∈(𝑆) sa is ying
g𝑚(𝑤 + 1) = g𝑚(𝑤)+1 o all 𝑤∈𝑆. The equali y 𝜑◦𝑘
2=𝜑
◦𝑘
1=𝜑
3implies
g◦𝑘
𝑚(𝑤) = 𝑤 + 1, 𝑚 = 1, 2, o all 𝑤∈𝑆.
In pa icula , g1,g2∈ 𝖠𝗎𝗍(𝑆). Using he one- o-one co espondence be ween he au omo phisms
o 𝑆and hose o 𝔻induced by a con o mal mapping be ween 𝑆and 𝔻and aking in o accoun ha
he i e a es o any 𝑓 ∈ 𝖠𝗎𝗍(𝔻)ha e he same ixed poin s as 𝑓, i is no di icul o see ha bo h g1
and g2a e ansla ions along ℝ.Sinceg◦𝑘
1=g◦𝑘
2, i ollows ha g1=g2.Thus,𝜑2=𝜑
1.□
6PROOF OF THEOREM 1.4
Recall ha by he hypo hesis 𝜑∈(𝔻)is non-ellip ic and di e en om a hype bolic au omo -
phism and ha 𝜓1∈(𝜑) ⧵ {𝗂𝖽𝔻}. In pa icula , i ollows ha he semig oup (𝜓𝑡)is non- i ial
and non-ellip ic. Mo eo e , by Behan’s Theo em, 𝜑and (𝜓𝑡)ha e he same Denjoy–Wol poin 𝜏.
Deno e by (𝑆,𝐻,(𝑧↦𝑧+𝑡)
𝑡⩾0) he canonical holomo phic model o (𝜓𝑡).
𝑃𝑟𝑜𝑜𝑓 𝑜𝑓 (a) ⇒(b) is comple ely i ial.
𝑃𝑟𝑜𝑜𝑓 𝑜𝑓 (b) ⇒(c). Since (𝜓𝑡)⊂(𝜑),byP oposi ion3.3 we ha e 𝐻◦𝜑=𝐻+𝑐 o some 𝑐∈ℂ.
The Koenigs unc ion 𝐻has angula limi s a all poin s on 𝜕𝔻, possibly excep o he Denjoy–
Wol poin , see, o example, [7, Co olla y 11.1.7]. Le us check ha 𝐻(𝜎) ∶= ∠lim𝑧→𝜎 𝐻(𝑧) =
∞. Suppose 𝐻(𝜎) ≠∞. Then 𝐻(𝜑(𝑟𝜎))=𝐻(𝑟𝜎)+𝑐→𝐻(𝜎)+𝑐as 𝑟→1
−. By he hypo hesis, 𝜎
is a bounda y ixed poin o 𝜑, ha is, 𝜑(𝑟𝜎) → 𝜎 as 𝑟→1
−. Acco ding o he Leh o–Vi anen’s
Theo em (see, e.g., [7, Theo em 3.3.1]) i ollows ha
𝐻(𝜎) = lim
𝑟→1−𝐻(𝜑(𝑟𝜎))=𝐻(𝜎)+𝑐
and hence 𝑐=0, ha is, 𝜑=𝗂𝖽
𝔻. The la e con adic s he hypo hesis. Thus, 𝐻(𝜎) = ∞ and, as a
consequence, see, o example, [7, P oposi ion 13.6.1], 𝜎is a bounda y ixed poin o (𝜓𝑡).
𝑃𝑟𝑜𝑜𝑓 𝑜𝑓 (c) ⇒(a). By an old esul o Heins [20, Lemma 2.1], 𝜓1canno be a hype bolic au o-
mo phism because i commu es wi h 𝜑, which is no a hype bolic au omo phism. Mo eo e , 𝜓1
is no a pa abolic au omo phism because by (c), i has a bounda y ixed poin 𝜎di e en om i s
Denjoy–Wol poin . I ollows ha he non-ellip ic semig oup (𝜓𝑡)canno be ex ended o a g oup.
Acco ding o he egula i y o 𝜎, we dis inguish wo cases.
Case I: 𝜎is a epelling ixed poin o (𝜓𝑡).Le Δbe he hype bolic pe al o (𝜓𝑡)associa ed wi h
𝜎, as explained in Sec ion 5.Since(𝜓𝑡)is no a g oup, acco ding o [7, Theo ems 13.2.7, 13.4.12],
he e exis s a uni alen map g om 𝔻on o Δsuch ha he o mula ˆ
𝜓𝑡∶= g−1◦𝜓𝑡◦g,𝑡⩾0, de ines
a hype bolic g oup (ˆ
𝜓𝑡)in 𝔻wi h Denjoy–Wol poin a −𝜎. By Co olla y 5.5,𝜑(Δ) ⊂ Δ. The e-
o e, ˆ
𝜑∶=g−1◦𝜑◦gis an elemen o (𝔻).Mo eo e ,𝜓1∈(𝜑) implies ˆ
𝜑∈(ˆ
𝜓1).Sinceˆ
𝜓1is
a hype bolic au omo phism and by [5, Rema k 6.4], we conclude ha he e exis s 𝑡0>0such
ha ei he ˆ
𝜑=ˆ
𝜓𝑡0o ˆ
𝜑=ˆ
𝜓−1
𝑡0. I he la e al e na i e would occu , hen (𝜓𝑡0◦𝜑)|Δ=𝗂𝖽
Δand
hence, by he iden i y p inciple, 𝜓𝑡0◦𝜑=𝗂𝖽
𝔻, which is impossible because 𝜓1∉ 𝖠𝗎𝗍(𝔻). The e-
o e, ˆ
𝜑=ˆ
𝜓𝑡0. I immedia ely ollows ha 𝜓𝑡0|Δ=𝜑|Δand hence, again by he iden i y p inciple,
𝜓𝑡0=𝜑as desi ed.
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EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 19 o 31
Case II: 𝜎is a supe - epelling ixed poin o (𝜓𝑡). Acco ding o [7, Co olla y 13.6.7], we know
ha he e is 𝑎∈ℝsuch ha
lim
𝑧→𝜎 Im𝐻(𝑧)=𝑎 and lim
𝑧→𝜎 Re 𝐻(𝑧) = −∞. (6.1)
No e ha bo h limi s a e un es ic ed. Taking in o accoun ha 𝜎is a bounda y ixed poin o 𝜑,
we he e o e ha e
𝑎= lim
𝑟→1−Im 𝐻(𝜑(𝑟𝜎)). (6.2)
I 𝜓1is hype bolic o pa abolic o ze o hype bolic s ep, hen by [5, P oposi ion 4.3], 𝜑∈(𝜓1)
implies 𝐻◦𝜑=𝐻+𝑐 o a sui able 𝑐∈ℂand, in iew o (6.1)and(6.2), i ollows ha
Im 𝑐 = lim
𝑟→1−(Im 𝐻(𝜑(𝑟𝜎)) − Im 𝐻(𝑟𝜎))=𝑎−𝑎=0.
The e o e, 𝑐is a eal numbe . Ce ainly 𝑐≠0because 𝜑≠𝗂𝖽𝔻.Mo eo e ,i 𝑐we e nega i e, we
would ob ain ha 𝜓−𝑐◦𝜑=𝗂𝖽
𝔻, which is impossible because 𝜓𝑡∉ 𝖠𝗎𝗍(𝔻) o any 𝑡>0. Hence,
𝑐>0andi ollows ha 𝜑=𝜓
𝑡0wi h 𝑡0∶= 𝑐. This comple es he p oo in he case when (𝜓𝑡)is
hype bolic o pa abolic o ze o hype bolic s ep.
F omnowonwesuppose ha 𝜓1is pa abolic o posi i e hype bolic s ep. Fo he sake o cla i y,
we will also assume ha he base space o (𝜓𝑡)is 𝑆=ℍ. The p oo in he o he case, ha is, o
𝑆=−
ℍ, is comple ely simila . Then he ac ha 𝜑∈(𝜓1)implies, by [5, P oposi ion 7.2], ha
he e exis s 𝐹∈𝖧𝗈𝗅(
𝔻,ℍ∪ℝ) such ha
𝐻◦𝜑(𝑧)=𝐻(𝑧)+𝐹
(𝑒2𝜋𝑖𝐻(𝑧)) o all 𝑧∈𝔻.
We no ice ha he unc ion 𝑤↦𝑤+𝐹(𝑒
2𝜋𝑖𝑤)is uni alen in ℍ, a ac ha will be used la e .
By (6.1)and(6.2), we ha e
lim
𝑟→1−Im 𝐹(𝑒2𝜋𝑖𝐻(𝑟𝜎))= lim
𝑟→1−(Im 𝐻(𝜑(𝑟𝜎)) − Im 𝐻(𝑟𝜎))=𝑎−𝑎=0. (6.3)
By (6.1), o any 𝑝=𝑒
2𝜋𝑖𝜃 ∈𝜕𝔻 he e exis a na u al 𝑁=𝑁(𝑝)and a sequence (𝑟𝑛)=(𝑟
𝑛(𝑝)) in
(0,1) con e gen o 1 such ha Re𝐻(𝑟𝑛𝜎)=𝜃−𝑛 o all 𝑛⩾𝑁. Fo his sequence (𝑟𝑛),weha e
lim
𝑛→∞ 𝑒2𝜋𝑖𝐻(𝑟𝑛𝜎) =lim
𝑛→∞ 𝑒2𝜋𝑖(𝜃−𝑛)𝑒−2𝜋 Im 𝐻(𝑟𝑛𝜎) =𝑝𝑒
−2𝜋𝑎.(6.4)
Since 𝐻(𝔻)⊂ℍ,weha e𝑎⩾0. Below we conside sepa a ely he cases 𝑎>0and 𝑎=0.
Subcase II.a: 𝑎>0. In his case, 𝑒−2𝜋𝑎 ∈ (0, 1).Le 𝐶𝑎be he ci cle o adius 𝑒−2𝜋𝑎 cen ed a
he o igin. Taking in o accoun ha 𝐶𝑎⊂𝔻and using (6.3)and(6.4), we deduce ha Im 𝐹(𝑧) = 0
o all 𝑧∈𝐶
𝑎. By he maximum p inciple o ha monic unc ions and he iden i y p inciple o
holomo phic unc ions, i ollows ha 𝐹≡𝑟 o some cons an 𝑟∈ℝ. Hence, 𝐻◦𝜑=𝐻+𝑟.By
essen ially he same a gumen as in he case o ze o hype bolic case, we see ha 𝑟>0and
he e o e, 𝜑=𝜓
𝑡0wi h 𝑡0∶= 𝑟, as desi ed.
Subcase II.b: 𝑎=0.Since𝐹∈𝖧𝗈𝗅(
𝔻,ℍ∪ℝ), we ha e Im 𝐹 is a non-nega i e ha monic unc-
ion in he uni disc. By [21, Co olla y on p. 38], he e exis s 𝐸⊂𝔻o measu e ze o such ha
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20 o 31 CONTRERAS e al.
he angula limi ∠lim
𝑧→𝑝 Im 𝐹(𝑧) exis s o e e y 𝑝∈𝜕𝔻⧵𝐸.By(6.4), o e e y 𝑝∈𝜕𝔻, he
sequence (𝑒2𝜋𝑖𝐻(𝑟𝑛(𝑝)𝜎))con e ges non- angen ially o 𝑝. The e o e, by (6.3), we deduce ha
∠lim
𝑧→𝑝 Im 𝐹(𝑧) = 0, o all 𝑝 ∈ 𝜕𝔻⧵𝐸. (6.5)
Now, le us conside he unc ion in 𝖧𝗈𝗅(ℍ,ℂ)gi en by 𝐺(𝑤) ∶= 𝐹(𝑒2𝜋𝑖𝑤), 𝑤 ∈ ℍ. Simila ly o he
case 𝑎>0, i su ices o show ha 𝐺is a eal cons an . Conside g(𝑤) ∶= 𝑤 + 𝐺(𝑤). This unc-
ion is a uni alen holomo phic sel -map o ℍwi h g′(∞) ∶= ∠ lim𝑤→∞ g(𝑤)∕𝑤 = 1. The e o e,
gadmi s he ollowing ep esen a ion, see, o example, [22, Chap e V.4: (V.42), (V.44) (V.45)],
g(𝑤) = 𝑟 + 𝑤 + ∫ℝ(1
𝜉−𝑤−𝜉
1+𝜉
2)d𝜇(𝜉), 𝑤 ∈ ℍ,wi h a cons an 𝑟∈ℝand
a non-nega i e Bo el measu e 𝜇on ℝgi en by 𝜇([𝑎, 𝑏]) = lim
𝜂→0+
1
𝜋∫𝑏
𝑎
Im g(𝜉 + 𝑖𝜂) d𝜉
o any 𝑎,𝑏 ∈ ℝ,𝑎<𝑏, possibly excep o a coun ably many poin s a which 𝜇has a oms. Taking
in o accoun ha by (6.5), lim𝜂→0+Im g(𝜉 + 𝑖𝜂) = 0 o a.e. 𝜉∈ℝ, i emains o see ha Im gis
bounded in ℍ⧵ℍ1, whe e o 𝑏∈ℝwe deno e by ℍ𝑏 he hal -plane {𝑤∶ Im 𝑤 > 𝑏}.
No e ha g′(∞) ≠0and hence gis con o mal a ∞wi h g(∞) = ∞. Recall ha gis
uni alen in ℍ. Then by a s anda d a gumen , see, o example, [23, pp. 303–304], we ge
ha g(ℍ1)⊃𝑈
𝑅∶= {𝑤 ∶ a g 𝑤 ∈ (𝜋∕2, 3𝜋∕2), |𝑤|>𝑅} o some 𝑅>0.Le 𝑅0∶= max{1, 𝑅}.
Since g(𝑤 + 1) = g(𝑤)+1 o all 𝑤∈ℍ, i ac ually ollows ha ℍ𝑅0⊂⋃𝑘∈ℤ𝑈𝑅0+𝑘⊂g(ℍ1).
The e o e, g(ℍ⧵ℍ1)=g(ℍ)⧵g(ℍ1)⊂ℍ⧵ℍ𝑅0,andwea edone. □
7 PETALS AND COMMUTATIVITY II. PROOF OF THEOREMS 1.8
AND 1.9
Be o e we s a p o ing Theo em 1.8, le us analyze (in he ema k below) he case (𝜓𝑡) ⊂ 𝖠𝗎𝗍(𝔻),
which in he s a emen o he heo em is excluded om conside a ion. As i has been al eady
men ioned in Sec ion 2.5, in his case, we can (and do) ex end (𝜓𝑡) o a g oup by se ing 𝜓𝑡∶=
(𝜓−𝑡)−1 o all 𝑡<0.
Rema k 7.1. Suppose (𝜓𝑡) ⊂ 𝖠𝗎𝗍(𝔻)is a con inuous one-pa ame e g oup in 𝔻wi h 𝜓1≠𝗂𝖽𝔻,
and le 𝜑∈(𝔻)be a non-ellip ic sel -map commu ing wi h 𝜓1. Clea ly, (𝜓𝑡)cannonbeellip-
ic. The whole uni disc 𝔻is he unique pe al o (𝜓𝑡), which is hype bolic o pa abolic depending
on whe he (𝜓𝑡)is a hype bolic o a pa abolic g oup. Mo eo e , i (𝜓𝑡)is hype bolic, hen by [20,
Lemma 2.1], 𝜑is a hype bolic au omo phism ha ing he same ixed poin s as (𝜓𝑡), and hence
𝜑=𝜓
𝑡0 o some 𝑡0∈ℝ.I (𝜓𝑡)is pa abolic, hen 𝜑is no necessa ily an au omo phism o 𝔻,bu
i has o be a pa abolic sel -map wi h he same Denjoy–Wol poin as 𝜓1; see, o example, [1,
P oposi ion 2.6.11]. In his case, 𝜑 ∈ 𝖠𝗎𝗍(𝔻)i andonlyi 𝜑=𝜓
𝑡0 o some 𝑡0∈ℝ.
P oo o Theo em 1.8. Fi s o all, since 𝜓1is no a hype bolic au omo phism (o he wise, we
would ha e (𝜓𝑡) ⊂ 𝖠𝗎𝗍(𝔻)), by [20, Lemma 2.1], 𝜑canno be a hype bolic au omo phism ei he .
The e o e, by Behan’s Theo em, see [12]o [1, Theo em 4.10.3], he Denjoy–Wol poin o 𝜑
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EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 21 o 31
coincides wi h he Denjoy–Wol poin 𝜏o he semig oup (𝜓𝑡). In pa icula , i ollows ha (𝜓𝑡)
is non-ellip ic.
P oo o (A). By Lemma 5.2, he e exis s a pe al Δ′o (𝜓𝑡)such ha 𝜑(Δ) ⊂ Δ′.Being
connec ed componen s o ◦, pe als a e pai wise disjoin . The e o e, ei he 𝜑(Δ) ⊂ Δ,o
𝜑(Δ) ∩ Δ = ∅. In he o me case, by Co olla y 5.5,𝜑(𝜎) = 𝜎 in he sense o angula limi s.
Mo eo e , aking in o accoun ha 𝜎≠𝜏, by Theo em 1.4 we ha e ha in his case, 𝜑=𝜓
𝑡0 o
some 𝑡0>0and as a consequence 𝜑(Δ) = 𝜓𝑡0(Δ) = Δ.Thus,i 𝜑(Δ) ⊂ Δ, hen al e na i e (i) in
Theo em 1.8(A) holds.
Now suppose 𝜑(Δ) ∩ Δ = ∅.I Δ′is a hype bolic pe al o (𝜓𝑡), hen acco ding o Lemma 5.3,
al e na i e (ii) in Theo em 1.8(A) holds. I Δ′is a pa abolic pe al o (𝜓𝑡), hen again by Lemma 5.3,
𝜑(𝜎) = 𝜏 in he sense o angula limi s. In his case, aking in o accoun ha 𝜎≠𝜏and ha 𝜑′(𝜏) ≠
∞,by[24, Lemma 8.2], he angula de i a i e 𝜑′(𝜎) is ∞, and hus, al e na i e (iii) holds. This
comple es he p oo o Theo em 1.8(A).
P oo o (B). By hypo hesis, Δis a pa abolic pe al o (𝜓𝑡). Hence, s a emen (a) is jus
Lemma 5.4. Since one o he implica ions in s a emen (b) is clea , see Sec ion 5, i emains o p o e
ha i 𝜑(Δ) = Δ, hen 𝜑is con ained in (𝜓𝑡). To his end, conside he canonical holomo phic
model (𝑆,𝐻,(𝑧↦𝑧+𝑡)
𝑡⩾0) o he semig oup (𝜓𝑡). I is known, see, o example, [7, Theo-
em 13.5.7], ha Π∶=𝐻(Δ)is a hal -plane o he o m Π={𝑧∶Im𝑧>𝑎}o Π={𝑧∶Im𝑧<𝑎}.
No e ha (𝑆,𝐻,𝑧↦𝑧+1)is he canonical model o 𝜓1. Recall also ha 𝜑∈(𝜓1). The e o e,
acco ding o [5, Theo em 3.4], 𝜑=𝐻
−1◦g◦𝐻 o some g∈(𝑆) sa is ying
g(𝑧 + 1) = g(𝑧)+1 o all 𝑧∈𝑆.(7.1)
I 𝜑(Δ) = Δ, hen g(Π) = Π and hence, aking in o accoun (7.1), we conclude ha g(𝑧) = 𝑧 + 𝑐 o
all 𝑧∈Πand some cons an 𝑐∈ℝ. Clea ly, his equali y hold o all 𝑧∈𝑆. No e ha g(𝐻(𝔻))=
𝐻(𝜑(𝔻))⊂𝐻(𝔻).Since𝜓1is no an au omo phism o 𝔻,by[5, Theo em 3.1 (B)], we ha e 𝑐⩾0,
and o cou se, 𝑐≠0because 𝜑≠𝗂𝖽𝔻. The e o e, 𝜑=𝜓
𝑡0wi h 𝑡0∶= 𝑐 > 0.□
P oo o Theo em 1.9. Le (𝑆,𝐻,(𝑧↦𝑧+𝑡)
𝑡⩾0)s and o he canonical holomo phic model o (𝜓𝑡).
As in he p oo o Theo em 1.8, we see ha 𝜑is no a hype bolic au omo phism.
P oo o (I). The semig oup (𝜓𝑡)canno be ellip ic, because 𝜓1≠𝗂𝖽𝔻commu es wi h he non-
ellip ic sel -map 𝜑. Fu he mo e, (𝜓𝑡)canno be hype bolic because o he wise ha ing 𝜑∈(𝜓1)
and 𝜓1∉ 𝖠𝗎𝗍(𝔻)would imply 𝜑=𝜓
𝑡0 o some 𝑡0⩾0, see, o example, [5, Rema k 6.4]; bu his
di ec ly con adic s he hypo hesis. The e o e, (𝜓𝑡)is pa abolic, and by [4, Co olla y 4.1] so is 𝜑.
Mo eo e , by Lemma 5.6,(𝜓𝑡)can ha e a mos one pa abolic pe al.
P oo o (II). Le Δ∗be he unique pa abolic pe al o (𝜓𝑡).I isknown,see, o exam-
ple, [7, Theo em 13.5.7], ha Π∶=𝐻(Δ
∗)is a hal -plane o he o m Π={𝑧∶Im𝑧>𝑎} o
Π={𝑧∶Im𝑧<𝑎}. Below we gi e a p oo o he o me al e na i e. The o he case can be ea ed
in a simila way.
I (𝜓𝑡)is pa abolic o ze o hype bolic s ep, hen by [5, P oposi ion 4.3], we ha e 𝐻◦𝜑◦𝐻−1 =
(𝑤 ↦ 𝑤 + 𝑐), whe e 𝑐∉[0,+∞)because 𝜑isno anelemen o (𝜓𝑡).No ealso ha by[5, Theo-
em 3.1 (B)], 𝑐 ∉ (−∞, 0) because 𝜓1is no an au omo phism. Mo eo e , 𝜑(Δ∗)⊂Δ
∗by Lemma 5.4
and hence, Π+𝑐⊂Π. As a consequence, Im 𝑐 > 0. Since o each hype bolic pe al Δ,i simage
𝐻(Δ) is a ho izon al s ip, see, o example, [7, Theo em 13.5.5 (2)], i ollows ha he e exis s
𝑛∈ℕsuch ha 𝐻(Δ)+𝑛𝑐⊂Πand, as a esul , we ha e 𝜑◦𝑛(Δ)⊂Δ
∗. Choose he smalles o
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22 o 31 CONTRERAS e al.
such 𝑛s. To comple e he p oo o asse ion (II) o he case o ze o hype bolic s ep, i now su ices
o appeal o Lemmas 5.2 and 5.3.
Suppose (𝜓𝑡)is o posi i e hype bolic s ep. We may suppose 𝑆=ℍ. The a gumen o he case
𝑆=−
ℍis simila . By [5, Theo em 3.4], he e exis s g∈(ℍ)sa is ying 𝐻◦𝜑=g◦𝐻and such ha
g(𝑤 + 1) = g(𝑤)+1 o any 𝑤∈ℍ. As in he p e ious case, gcanno be o he o m 𝑤↦𝑤+𝑐
wi h 𝑐∈ℝ. The e o e, see, o example, [5, Lemma 9.3 (II)], exp (2𝜋𝑖g(𝜁))=𝑓(𝑒
2𝜋𝑖𝜁), whe e 𝑓∈
𝖧𝗈𝗅(𝔻),𝑓(0) = 0,|𝑓′(0)|<1. Since he i e a es o such a sel -map 𝑓∶𝔻→𝔻con e ge o 0, see,
o example, [1, Theo em 3.1.13], i ollows ha o any 𝑤∈ℍ,Im g◦𝑛(𝑤) → +∞ as 𝑛→+∞.
The e o e, any poin 𝑧∈𝔻,
𝜑◦𝑛(𝑧) = 𝐻−1(g◦𝑛(𝐻(𝑧)))∈𝐻
−1(Π) = Δ∗
o 𝑛∈ℕla ge enough. As a consequence, o any hype bolic pe al Δ he e exis s 𝑛∈ℕsuch ha
𝜑◦𝑛(Δ) ∩ Δ∗≠∅. Relying on he ac ha pe als a e pai wise disjoin and applying Lemma 5.2
wi h 𝜑◦𝑛in place o 𝜑, we he e o e conclude ha 𝜑◦𝑛(Δ) ⊂ Δ∗. Now we can comple e he
p oo o (II) in he same manne as in he p e ious case: Pass o he smalles 𝑛∈ℕsa is ying
𝜑◦𝑛(Δ) ⊂ Δ∗and appeal o Lemmas 5.2 and 5.3.
P oo o (III). Suppose ha (𝜓𝑡)has no pa abolic pe al and le Δbe some hype bolic pe al
o (𝜓𝑡). Applying induc i ely Lemmas 5.2 and 5.3, we conclude ha he e exis s a sequence o
hype bolic pe als (Δ𝑛)wi h 𝛼-poin s 𝜎𝑛such ha Δ1=Δ,𝜑(Δ𝑛)⊂Δ
𝑛+1,and𝜑(𝜎𝑛)=𝜎
𝑛+1 o all
𝑛∈ℕ. In o de o see ha Δ𝑛sa e pai wise disjoin ,†we suppose on he con a y ha Δ𝑚+𝑘 =Δ
𝑚
o some 𝑚,𝑘 ∈ ℕ. Then 𝜑◦𝑘(Δ𝑚)⊂Δ
𝑚, and hence by Theo em 1.8(A) applied o Δand 𝜑 eplaced
by Δ𝑚and 𝜑◦𝑘we would ha e ha 𝜑◦𝑘=𝜓
𝑡0 o some 𝑡0>0. By Lemma 5.7, his would u he
imply ha 𝜑i sel is con ained in (𝜓𝑡), which con adic s he hypo hesis o he heo em. To com-
ple e he p oo , i emains o no ice ha acco ding o [6, Theo em 1.4], 𝜓′
1(𝜎𝑛)⩽𝜓′
1(𝜎1) o all
𝑛∈ℕand, as a consequence, he Cowen–Pomme enke inequali y [24, Theo em 4.1 (iii)]
+∞
∑
𝑛=1 |𝜏−𝜎
𝑛|2
𝜓′
1(𝜎𝑛)−1 ⩽2Re(1
𝜓1(0) −1
)<+∞
implies |𝜏−𝜎
𝑛|→0as 𝑛→+∞.□
8PETALS AND ISOGONALITY
Gi en 𝜑∈(𝔻)non-ellip ic and (𝜓𝑡) ⊄ 𝖠𝗎𝗍(𝔻)a con inuous one-pa ame e semig oup in he
disc such ha 𝜓1∈(𝜑) ⧵ {𝗂𝖽𝔻}, in Theo em 1.8, i was shown ha 𝜑maps pe als o (𝜓𝑡)in o
pe als o (𝜓𝑡). In ac , i such a pe al Δis pa abolic, we always ha e 𝜑(Δ) ⊂ Δ. Howe e , his is no
longe he case when Δis hype bolic. Apa om he same possibili y, ha is, apa om he case
𝜑(Δ) ⊂ Δ, and in gene al, we can ha e he o he wo ollowing si ua ions:
∙Si ua ion A: 𝜑(Δ) ⊂ Δ′and Δ′is a hype bolic pe al.
∙Si ua ion B: 𝜑(Δ) ⊂ Δ′and Δ′is a pa abolic pe al.
In he ollowing wo esul s, we analyze when 𝜑maps Δon o Δ′in bo h si ua ions.
†An al e na i e way o p o e his ac is o combine [6, P oposi ion 5.2] wi h Lemma 5.3.
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EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 23 o 31
P oposi ion 8.1. Assume we a e in Si ua ion A and le 𝜎and 𝜎′be he 𝛼-poin s o Δand Δ′,
espec i ely. Then he ollowing condi ions a e equi alen :
(a) 𝜑(Δ) = Δ′;
(b) 𝜑is isogonal a 𝜎;
(c) he spec al alues o (𝜓𝑡)a 𝜎and 𝜎′coincide.
Mo eo e , i he abo e equi alen condi ions hold, hen 𝜑is a ine wi h espec o 𝜓1and hence,
(𝜓𝑡)⊂(𝜑).
P oposi ion 8.2. In Si ua ion B we always ha e 𝜑(Δ) ≠Δ′.
The p oo o hese wo p oposi ions equi e some backg ounds which we expose he ea e . We
begin by ecalling he de ini ion o isogonali y, which we ex end o he case o in ini e angula
limi .
De ini ion 8.3. A uni alen unc ion 𝑓∶𝔻→ℂis said o be isogonal o semi-con o mal a a
poin 𝜁∈𝜕
𝔻i he ollowing wo angula limi s exis :
𝑓(𝜁) ∶= ∠ lim
𝑧→𝜁 𝑓(𝑧) ∈ ℂ∪{∞} and ∠lim
𝑧→𝜁 a g 𝑓0(𝑧)
𝑧−𝜁,
whe e a g is o be unde s ood as a con inuous map om ℂ⧵{0} o ℝ∕(2𝜋ℤ)and
𝑓0(𝑧) ∶= {𝑓(𝑧)−𝑓(𝜁) i 𝑓(𝜁) ≠∞,
1∕𝑓(𝑧) i 𝑓(𝜁) = ∞.
Fu he , a uni alen unc ion Ψ∶ℍRe →ℂis said o be isogonal a 𝜔∈𝜕ℍRe i he composi ion
𝑓∶=Ψ◦𝐻𝜔, whe e 𝐻𝜔is he Cayley map o 𝔻on o ℍRe wi h 𝐻𝜔(0) = 0,𝐻𝜔(1) = 𝜔,isisogonal
a 𝜁=1.
Rema k 8.4. I 𝑓∈(𝔻)and 𝜁is a con ac poin o 𝑓, hen an elemen a y a gumen shows ha
he isogonali y condi ion equi es ha he image o he adial segmen [0, 𝜁] unde 𝑓app oaches
he poin 𝑓(𝜁) ∈ 𝜕𝔻o hogonally o 𝜕𝔻. The e o e, in his case, he isogonali y a 𝜁is equi alen
o
∠lim
𝑧→𝜁 A g 1−𝑓(𝜁)𝑓(𝑧)
1−𝜁𝑧
=0, (8.1)
whe e A g 𝑤 s ands o he unique alue o he a gumen o 𝑤≠0con ained in (−𝜋,𝜋]. Passing
his ema k o he igh hal -plane wi h he help o he Cayley map, we can say ha a uni alen
sel -map 𝑇∈(ℍRe)wi h a bounda y ixed poin a 0 is isogonal a 0 i and only i
∠lim
𝑤→0 A g 𝑇(𝑤)
𝑤=0. (8.2)
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24 o 31 CONTRERAS e al.
De ini ion 8.5. Le 𝜎∈𝜕𝔻be a epelling ixed poin o a con inuous one-pa ame e semig oup
(𝜓𝑡)wi h associa ed in ini esimal gene a o 𝐺. The iple (ℍRe,Ψ,𝑄
𝑡)is called a p e-model o (𝜓𝑡)
a 𝜎i he ollowing condi ions a e me :
(i) o each 𝑡⩾0,𝑄𝑡is he au omo phism o ℍRe gi en by 𝑄𝑡(𝑧)∶=𝑒
𝜆𝑡𝑧, whe e 𝜆∶=𝐺
′(𝜎);
(ii) he map Ψ∶ℍRe →𝔻is holomo phic and injec i e, ∠lim
𝑤→0 Ψ(𝑤) = 𝜎,andΨis isogonal
a 0, ha is,
∠lim
𝑤→0 A g 1−𝜎Ψ(𝑤)
𝑤=0; (8.3)
(iii) Ψ◦𝑄𝑡=𝜓
𝑡◦Ψ o all 𝑡⩾0.
Rema k 8.6. I is known [25, Theo em 3.10] ha e e y con inuous one-pa ame e semig oup, a
each epelling ixed poin 𝜎, admi s a p e-model unique up o he ans o ma ion Ψ(𝑤) ↦ Ψ(𝑐𝑤),
whe e 𝑐is an a bi a y posi i e cons an . Mo eo e , Ψ(ℍRe)coincides wi h he hype bolic pe al
Δ(𝜎) associa ed wi h 𝜎.ThemapΨcan be exp essed ia he Koenigs unc ion ℎo (𝜓𝑡).Namely,i
he s ip ℎ(Δ(𝜎)) is 𝕊(𝑎,𝑏)={𝑤∶𝑎<Im𝑤<𝑏}, hen he in e wining map Ψin he p e-model
o (𝜓𝑡)a 𝜎is gi en by
Ψ(𝑤) ∶= ℎ−1(𝑏−𝑎
2𝜋 log 𝑤 + 𝑏+𝑎
2𝑖+𝑠
),𝑤∈ℍRe,
whe e 𝑠is an a bi a y eal cons an .
As he ollowing esul shows, i is also possible o alk abou p e-models associa ed wi h
pa abolic pe als.
Theo em 8.7 [7, P oposi ion 13.4.10 and i s p oo ]. Le (𝜓𝑡)be a pa abolic semig oup in he
uni disc wi h Denjoy–Wol poin 𝜏∈𝜕
𝔻and le Δbe a pa abolic pe al o (𝜓𝑡).Then, he eexis
Ψ∈(ℍRe,𝔻)wi h Ψ(ℍRe)=Δand ∠lim
𝑤→0 Ψ(𝑤) = 𝜏 and a pa abolic g oup (𝑄𝑡)in ℍRe wi h
Denjoy–Wol poin a 0 such ha , o all 𝑡⩾0,
Ψ◦𝑄𝑡=𝜓
𝑡◦Ψ.
De ini ion 8.8. In he no a ion o he abo e heo em, he iple (ℍRe,Ψ,𝑄
𝑡)is called a p e-model
o (𝜓𝑡)associa ed wi h he pa abolic pe al Δ.
Lemma 8.9. Le 𝑇be a uni alen sel -map o ℍRe.I 𝑇(0) = 0 in he sense o angula limi s and i
𝑇is isogonal a 0, hen o any 𝛽>0,
lim
𝑥→0+
𝑇(𝛽𝑥)
𝑇(𝑥) =𝛽. (8.4)
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EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 25 o 31
P oo . Since 𝑇is isogonal a 0 wi h 𝑇(0) = 0, he uni alen unc ion 𝑇(1−𝑧
1+𝑧 ),𝑧∈𝔻, sa is ies a
𝜁=1 he Visse –Os owski condi ion; see, o example, [26, P oposi ion 4.11 on p. 81]. I ollows†
ha
lim
𝑥→0+
𝑥𝑇′(𝑥)
𝑇(𝑥) =1.
As a consequence,
log 𝑇(𝛽𝑥)
𝑇(𝑥) =
𝛽
∫
1
𝑡𝑥 𝑇′(𝑡𝑥)
𝑇(𝑡𝑥)
d𝑡
𝑡⟶
𝛽
∫
1
d𝑡
𝑡= log 𝛽
as 𝑥→0
+, as desi ed. □
Recall ha a sequence (𝑧𝑛)⊂𝔻con e ging o some poin 𝜎∈𝜕𝔻is said o con e ge o 𝜎
non- angen ially i he limi se Slope[(𝑧𝑛), 𝑛 → +∞] o A g(1 − 𝜎𝑧𝑛)as 𝑛→+∞is compac ly
con ained in he open in e al (−𝜋∕2,𝜋∕2). Fu he , we say ha (𝑧𝑛)con e ges o 𝜎 angen ially,
i Slope[(𝑧𝑛), 𝑛 → +∞] ⊂ {−𝜋∕2,𝜋∕2}. Clea ly, (𝑧𝑛)con e ges o 𝜎non- angen ially i and only
i i has no subsequence con e ging angen ially.
Lemma 8.10. Suppose 𝑓∈(𝔻)is isogonal a some poin 𝜁∈𝜕𝔻and ha 𝑓(𝜁) ∈ 𝜕𝔻.I (𝑧𝑛)⊂𝔻
con e ges o 𝜁 angen ially, and i (𝑤𝑛)gi en by 𝑤𝑛∶= 𝑓(𝑧𝑛) o all 𝑛∈ℕcon e ges o 𝑓(𝜁), hen
he con e gence o (𝑤𝑛)is also angen ial.
P oo . Composing 𝑓wi h sui able con o mal mappings we may eplace 𝔻wi h ℍRe and suppose
𝑓(𝜁)=𝜁=0.Fixsome𝜂∈(0,𝜋∕2)and se 𝜃∶=1
2(𝜂 + 𝜋∕2).Since(𝑧𝑛)⊂ℍRe con e ges o 0
angen ially, he e exis s 𝑟>0such ha 𝐴𝜃,𝑟 ∶= {𝑧 ∶ |A g 𝑧|<𝜃, |𝑧|<𝑟}does no con ain any
poin o (𝑧𝑛). The isogonali y o 𝑓a 0 implies, see, o example, [26, P oposi ion 4.10 on p. 81],
ha he e exis s 𝜌>0such ha 𝐴𝜂,𝜌 ⊂𝑓
(𝐴𝜃,𝑟).Since𝑓is uni alen by hypo hesis, i ollows ha
𝐴𝜂,𝜌 does no con ain any poin o (𝑤𝑛). Taking in o accoun ha 𝜂∈(0,𝜋∕2)in his a gumen is
a bi a y, we conclude ha i (𝑤𝑛)con e ges o 0, hen he con e gence mus be angen ial. □
Lemma 8.11. Le 𝐷1,𝐷
2,𝐷
3∈{𝔻,ℍRe}and le 𝑓3∶= 𝑓2◦𝑓1,whe e𝑓1∶𝐷
1→𝐷
2and
𝑓2∶𝐷
2→𝐷
3a e wo (holomo phic) uni alen unc ions. The ollowing wo s a emen s hold.
(A) I 𝑓1is isogonal a some 𝜁1∈𝜕𝐷
1wi h 𝜁2∶= 𝑓1(𝜁1)∈𝜕𝐷
2and i 𝑓2is isogonal a 𝜁2wi h
𝜁3∶= 𝑓2(𝜁2)∈𝜕𝐷
3, hen𝑓3is isogonal a 𝜁1wi h 𝑓3(𝜁1)=𝜁
3.
(B) Con e sely, i 𝑓3is isogonal a some 𝜁1∈𝜕𝐷
1and i 𝜁3∶= 𝑓3(𝜁1)∈𝜕𝐷
3, hen: (i) 𝑓1is isogonal
a 𝜁1, (ii) 𝜁2∶= 𝑓1(𝜁1)belongs o 𝜕𝐷2, (iii) 𝑓2is isogonal a 𝜁2, and (i ) 𝑓2(𝜁2)=𝜁
3.
P oo . Fi s o all, wi hou loss o gene ali y, we may suppose 𝐷1=𝐷
2=𝐷
3=𝔻.
P oo o (A). Taking in o accoun Rema k 8.4, we ha e o show ha o any sequence (𝑧𝑛)⊂𝔻
con e ging o 𝜁1non- angen ially, 𝑓3(𝑧𝑛)→𝜁
3and A g ((1 − 𝜁3𝑓3(𝑧𝑛))∕(1 − 𝜁1𝑧𝑛))→0.This
†In [27] one can ind ano he p oo o his ac (no assuming global uni alence o 𝑇), see [27, Lemma 8.1] applied
o 𝑧 ↦ 1∕𝑇(1∕𝑧), as well as some closely ela ed esul s connec ing asymp o ic beha io o he hype bolic de i a i e o
isogonali y and o exis ence o ini e angula de i a i e.
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