Examples in Discrete Iteration of Arbitrary Intervals of Slopes

The Jou nal o Geome ic Analysis (2025) 35:99
h ps://doi.o g/10.1007/s12220-025-01934-4
Examples in Disc e e I e a ion o A bi a y In e als o
Slopes
Manuel D. Con e as1·F ancisco J. C uz-Zamo ano1·
Luis Rod íguez-Piazza2
Recei ed: 18 No embe 2024 / Accep ed: 2 Feb ua y 2025
© The Au ho (s) 2025
Abs ac
Gi en a compac in e al [a,b]⊂[0,π], we cons uc a pa abolic sel -map o he
uppe hal -plane whose se o slopes is [a,b]. The na u e o his cons uc ion is com-
ple ely disc e e and explici : we explici ly cons uc a sel -map and we explici ly show
in which way i s o bi s wande owa ds he Denjoy–Wol poin . We also analyze some
p ope ies o he He glo z measu e co esponding o such example, which yield he
egula i y o such sel -map in i s Denjoy–Wol poin .
Keywo ds Complex dynamics ·Disc e e i e a ion ·The slope p oblem
Ma hema ics Subjec Classi ica ion P ima y 30D05 ·37FXX
1 In oduc ion
Disc e e I e a ion in he uni disk Dis a b anch o he as ield known as Complex
Dynamics. Gi en a holomo phic sel -map g:D→D, he main aim is o analyze
asymp o ic p ope ies o he sequence o i e a ed sel -composi ions o g, gi en by
This esea ch was suppo ed in pa by Minis e io de Inno ación y Ciencia, Spain, p ojec
PID2022-136320NB-I00. The second au ho was 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.
BF ancisco J. C uz-Zamo ano
[email p o ec ed]
Manuel D. Con e as
[email p o ec ed]
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 ille, Spain
2Depa 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 ille, Spain
0123456789().: V,- ol 123
99 Page 2 o 15 M. D. Con e as e al.
g0=IdD, and gn+1=gn◦g,n∈N∪{0}. In his a icle we ocus on non-ellip ic sel -
maps, which a e he ones ha possess no ixed poin on D. The ollowing well-known
esul conce ns he dynamics o such sel -maps.
Theo em 1.1 [2, Theo em 3.2.1] (Denjoy–Wol Theo em) Le g :D→Dbe a non-
ellip ic sel -map. Then, he e exis s τ∈∂Dsuch ha gn→τlocally uni o mly, as
n→∞.
The poin τis usually known as he Denjoy–Wol poin o g. In pa icula , i
w0∈D, he Denjoy–Wol Theo em p o es ha he o bi wn=gn(w0)con e ges o
τ,asn→∞.
A classical p oblem in Disc e e I e a ion is o cha ac e ize he di ec ions h ough
which he o bi s o a non-ellip ic sel -map o Dcon e ge owa ds he Denjoy–Wol
poin . Namely, gi en w0∈D, calcula ing he numbe s s∈[−π/2,π/2]such ha
he e exis s a subsequence o he o bi sa is ying a g(1−τwnk)→s,ask→∞.This
is he Slope P oblem, which i s ly appea ed a ound 1930 in [18,19].
Dealing wi h his p oblem is usually easie in he uppe hal -plane se ing. To do
so, le H:= {z∈C:Im(z)>0}, and conside he con o mal mapping S:D→H
gi en by
S(w) =iτ+w
τ−w,w∈D,
whe e τ∈∂Dis he Denjoy–Wol poin o g. Then, cons uc he sel -map :H→H
gi en by =S◦g◦S−1. Simila ly as in he case o he uni disk, we can conside
he i e a ed sel -composi ions 0=IdH, and n+1= n◦ ,n∈N∪{0}.I is
easy o check ha n=S◦gn◦S−1. Some hing simila occu s in he case o o bi s.
Gi en z0∈H, i s o bi by is he sequence gi en by zn= n(z0). By cons uc ion
zn→∞,asn→∞.I w0∈Dand we choose z0=S(w0), hen zn=S(wn). Tha
is, n→∞,asn→∞, locally uni o mly, so ha he Denjoy–Wol poin o is
in ini y.
Conce ning he Slope P oblem, ixing a subsequence o he o bi s, we ha e ha
a g(1−τwnk)→si and only i a g(zn)→π/2−s. Inspi ed by his ela ionship,
we gi e he ollowing de ini ion.
De ini ion 1.2 Le :H→Hbe a holomo phic unc ion whose Denjoy–Wol poin
is in ini y. Then, gi en z0∈H, we de ine he se o slopes o as
Slope[ ,z0]={s∈[0,π]:The e exis s a subsequence {znk}wi h a g(znk)→s},
whe e a g deno es he p incipal b anch o he a gumen unc ion.
By [2, P oposi ion 2.5.8], he Denjoy–Wol poin o :H→His in ini y i and only
i
α:= ∠lim
z→∞
(z)
z∈[1,+∞).
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Examples in Disc e e I e a ion Page 3 o 15 99
In his si ua ion, is said o be pa abolic i α=1, and i is said o be hype bolic i
α>1.
Vali on [18] s udied he Slope P oblem o hype bolic sel -maps o Hin 1931. I
u ns ou ha i :H→His a hype bolic sel -map wi h Denjoy–Wol poin in ini y,
hen o e e y z∈H he e exis θ∈(0,π)such ha limn→∞ a g( n(z)) =θ;see[2,
Theo em 4.3.4]. His ideas ha e been ecen ly e isi ed by B acci and Poggi-Co adini.
Mo e p ecisely, hey p o ed ha he unc ion Slope[ ,·]: H→(0,π)is su jec i e
and ha monic [4, P ope y 2 (a) and P ope y 2 (b)].
The heo y is signi ican ly mo e di icul in he case o pa abolic sel -maps, which
a e ypically di ided in wo amilies. Fo a holomo phic sel -map :H→H, he
Schwa z-Pick Lemma assu es ha
kH( n+2(z), n+1(z)) ≤kH( n+1(z), n(z)), z∈H,n∈N,
whe e kHdeno es he hype bolic dis ance o he uppe hal -plane. In pa icula ,
limn→∞ kH( n+1(z), n(z)) exis s o all z∈H, and i is a non-nega i e eal numbe .
Fo non-ellip ic sel -maps, by [2, Co olla y 4.6.9.(i)], we know ha he la e limi is
ei he posi i e o e e y z∈Ho o no z∈H. Acco dingly, we say ha is o
posi i e o o ze o hype bolic s ep.
In 1979, Pomme enke con ibu ed o he Slope P oblem o pa abolic sel -maps o
H.In[17, Rema k 1], he p o ed ha limn→∞ a g( n(z)) exis s and i is ei he 0 o
πwhene e is a pa abolic sel -map o posi i e hype bolic s ep wi h Denjoy–Wol
poin in ini y. In [10, P oposi ion 2.6], using hype bolic geome y, we no iced ha he
la e limi does no depend on z. Namely, i holds ha ei he Slope[ ,z]={0}o
Slope[ ,z]={π} o all z∈H.
The main emphasis in his pape is pu on he case whe e is a pa abolic sel -
map o ze o hype bolic s ep. One o he i s su p ising con ibu ions o his case is in
he ema kable pape o Wol in 1929. In [19, Sec ion 6], he came up wi h he map
:H→Hgi en by
(z)=z+ieπ/2zi+ieπ/2,z∈H,(1)
whe e ziis de ined using he p incipal b anch o he loga i hm. I is easy o check ha
is pa abolic wi h Denjoy–Wol poin in ini y, since ∠limz→∞ (z)/z=1. Fo his
example, Wol managed o p o e ha Slope[ ,z0]con ains a leas wo poin s o all
ini ial poin s z0∈H.
The Slope P oblem has also been in oduced in he con inuous se ing o Complex
Dynamics, ha is, o semig oups {φ } ≥0o holomo phic sel -maps o he uppe
hal -plane. A mode n exposi ion o his opic can be ound in [5, Chap e 17]. In
[6], Con e as and Díaz-Mad igal ound ha he esul s o Vali on and Pomme enke
can be ansla ed o hype bolic and pa abolic semig oups o posi i e hype bolic s ep,
espec i ely. Fo a pa abolic semig oup o ze o hype bolic s ep, hey p o ed ha
he se o slopes is a compac in e al which does no depend on he ini ial poin .
Despi e Wol ’s example (1), hey conjec u ed ha , in he con inuous se ing, he se
o slopes should always be a single on. This conjec u e was disp o ed by Con e as,
Díaz-Mad igal, and Gumenyuk [8], and also by Be sakos [3], independen ly. Namely,
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99 Page 4 o 15 M. D. Con e as e al.
hey ound a semig oup whose se o slopes is he ull in e al [0,π]. Imp o ing he
echniques by Be sakos, o e e y p e ixed in e al [a,b]⊂[0,π], Kelgiannis [16]
cons uc ed a semig oup whose se o slopes is [a,b].
Qui e ecen ly, he la e ideas we e aken back o he disc e e se ing. Namely, e en
i he o bi s o a sel -map a e disc e e se s, we ha e shown ha he se o slopes o
pa abolic sel -maps o ze o hype bolic s ep is always a compac in e al which does
no depend on he ini ial poin .
Theo em 1.3 [10, Theo em 2.9] Le :H→Hbe a pa abolic unc ion o ze o
hype bolic s ep whose Denjoy–Wol poin is in ini y. Then, he e exis s 0≤a≤b≤π
such ha Slope[ ,z]=[a,b] o all z ∈H.
Examples wi h a bi a y se o slopes ha e also been cons uc ed h ough he use
o semig oups. Namely, i =φ1, whe e {φ }is he semig oup cons uc ed by Kel-
giannis, we ha e p o ed ha Slope[ ,z]=[a,b] o all z∈H. The e o e, we ha e
he ollowing esul .
Theo em 1.4 [10, Theo em 2.13] Gi en 0≤a≤b≤π, he e exis s a pa abolic
unc ion :H→Ho ze o hype bolic s ep whose Denjoy–Wol poin is in ini y such
ha Slope[ ,z]=[a,b] o all z ∈H.
This no e ocuses on cons uc ing examples o pa abolic sel -maps o ze o hype -
bolic s ep wi h a bi a y slope se s [a,b]⊂[0,π], wi h [a,b] =[0,π]. These
sel -maps a e o a disc e e na u e, and so we ex end a p e ious wo k ha p ima -
ily elied on semig oups. This yields a new p oo o Theo em 1.4 using ools om
Disc e e I e a ion, which is gi en in Sec .2. Indeed, ou cons uc ion is comple ely
explici : we explici ly cons uc he sel -map and we explici ly show in which way
he o bi s wande owa ds he Denjoy–Wol poin . This s ongly con as s wi h he
a gumen s in [16], whe e he semig oup is ob ained h ough i s Koenigs domain and
he p oo elies in sub le es ima es o some ha monic measu es. In Sec .3, we analyze
some p ope ies o he He glo z measu e co esponding o such examples. This leads
us o discuss he egula i y o such sel -maps a hei Denjoy–Wol poin s, and we
compa e his ac wi h p e ious e e ences.
In [10, Sec ion 4], we also ga e a disc e e and explici cons uc ion o he cases
[0,π]and [0,π/2]. Those sel -maps a e de ined h ough hei He glo z ep esen a ion
o mula, and we discuss hei egula i y a he Denjoy–Wol poin . Howe e , he new
cons uc ion is easie o ollow.
2 The Main Cons uc ion
The goal o his sec ion is, gi en 0 ≤a<b≤πwi h [a,b] =[0,π], o cons uc
a pa abolic sel -map :H→Ho ze o hype bolic s ep wi h Denjoy–Wol poin
in ini y o which
Slope[ ,z]=[a,b],z∈H.
No ice ha he limi case [a,b]=[0,π]has been co e ed in [10, Theo em 4.2] wi h
a di e en echnique.
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Examples in Disc e e I e a ion Page 5 o 15 99
We will s a wi h a gene al se ing, which we app op ia ely modi y la e o ge he
desi ed examples. To p esen i , le us ix some no a ion. Conside =C (−∞,0],
θ∈(0,π/2), and Aθ={z∈C:a g(z)∈(−θ,θ)}. We will cons uc a unc ion
F:→Csuch ha F(Aθ)⊂Aθ. The unc ion Fis gi en by
F(z)=z+p(z), z∈, (2)
whe e p:→Cis gi en by
p(z)=
∞

k=1
pk(z), pk(z)=akeiθk
(z+γk)k,z∈, (3)
whe e we a e using he main b anch o he a gumen o w o de ine wk. In he de ini ion
o p, he coe icien s ak,γk,kand θksa is y he ollowing assump ions:
ak>0,γ
k>0,
k>0,π
k≤θ, lim
k→∞ k=0,(4)
θ2k+π2k=θ, θ2k−1−π2k−1=−θ, (5)
∞

l=1
al
γl
l
≤γ1
2,(6)
γ1≥2,γ
k+1>γ2
k,(7)
k−1

l=1
al
γl
k
≤(θ)ak
2kγ2k
k
,
∞

l=k+1
al
γl
l
≤(θ)ak
2kγ2k
k
,(8)
whe e
(θ) =1
4+ an2(θ)
.
Example 2.1 The la e condi ions a e ul illed o he coe icien s
k=θ
π
1
2k,ak=Ck
1(k!)2,γ
k
k=(C2k!)3k,k∈N,
whe e C1,C2>1 a e la ge enough cons an s, and θkis au oma ically de ined by (5).
Lemma 2.2 Assume (4)–(7). Then, p is a well-de ined holomo phic unc ion on .
Mo eo e , he image o p is con ained in Aθ.
P oo No ice ha pis a sum o holomo phic maps which a e well-de ined on . Then,
o see ha pis well-de ined and holomo phic on , i is enough o ind ha he sum
de ining pis uni o mly con e gen on e e y compac subse o . Indeed, i K⊂
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99 Page 6 o 15 M. D. Con e as e al.
is a compac se , hen he e mus exis k0∈Nsuch ha K⊂{z∈C:|z|≤γk0/2}.
Then, he e exis s M>0 such ha
k0

l=1
|pl(z)|≤M,z∈K.
Fo he o he e ms, i l>k0,by(7)weha e
γl≥γ2
l−1≥2γl−1≥2γk0.
Then, i l>k0and z∈K, we ge
|z+γl|≥γl−γk0
2=γl1−γk0
2γl≥γl
2.
Then, using he la e inequali y and (4), o l>k0we ha e ha
|pl(z)|=al
|z+γl|l≤2lal
γl
l
≤2al
γl
l
.
All in all, using also (6) and he Weie s ass Theo em, we conclude ha he sum
de ining pis uni o mly con e gen on K.
Le us now p o e ha he image o pis con ained in Aθ. Since Aθis a con ex cone,
o all z,w ∈Aθwe ha e ha z+w∈Aθ. Mo eo e , i z∈Aθand w∈Aθ,i is
easy o check ha z+w∈Aθ. Summing up, i is enough o p o e ha he image o
each pkis con ained in Aθ. To do so, le us assume ha k∈Nis e en (i i is odd,
hen he a gumen is simila ). Conside he map
z→ 1
(z+γk)k,z∈.
By (4), i s image is he angula egion A={z∈C:a g z∈(−πk,π
k)}. Then, by
(4) and (5), he image o pkis he angula egion delimi ed by he a gumen s
θk+πk=θ, θk−πk=θ−2πk≥θ−2θ≥−θ.
The e o e, i is clea ha he image o pkis con ained in Aθ.
Rema k 2.3 Le us no ice ha he i s pa o he la e p oo can easily be adap ed o
see ha pis a well-de ined and holomo phic map on C (−∞,−γ1].
Rema k 2.4 Lemma 2.2 implies ha he map Fgi en in (2) sa is ies F(Aθ)⊂Aθ,
since Aθis a con ex cone. Mo eo e , he same a gumen yields ha F(H)⊂H o
e e y hal -plane H⊂such ha Aθ⊂H.
The nex esul is he main idea behind he examples ha we will cons uc la e .
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Examples in Disc e e I e a ion Page 7 o 15 99
Lemma 2.5 Assume (4)–(8). Le z0=γ1/2, and conside zn=xn+iyn=Fn(z0),
n∈N. The e exis wo subsequences znkand zmksuch ha a g(znk)→θand
a g(zmk)→−θas k →∞.
P oo Fi s o all, since z0∈Aθ, i ollows om Rema k 2.4 ha zn∈Aθ o all n∈N.
Mo eo e , since θ∈(0,π/2), we ha e ha Re(p(z)) > 0 o all z∈. This means
ha xnis an inc easing sequence and ha Fhas no ixed poin on Aθ. The e o e, using
Rema k 2.4, we conclude ha he Denjoy–Wol poin o F:Aθ→Aθis in ini y.
Then, zn→∞,asn→∞. Since |z|and Re(z)a e compa able quan i ies o all
z∈Aθ, we deduce ha xn→+∞,asn→∞.
Following his idea, we de ine
k:= {z∈Aθ:γk≤Re(z)≤γ2
k},k∈N.
By (7), k∩l=∅ o all k= l. We claim ha , o e e y k∈N he e exis s Nk∈N
such ha zNk∈kand zn/∈k o all n<Nk. To see his, use (6) and no ice ha
xn+1−xn=Re(p(zn)) ≤|p(zn)|≤
∞

k=1
ak
|zn+γk|k≤
∞

k=1
ak
γk
k
≤γ1
2,(9)
whe e we ha e used ha Re(zn)≥0. Since
γ2
k−γk≥γk≥γ1>γ1
2,
whe eweha ealsoused(7), he claim ollows.
Recall ha xn→+∞,asn→∞. Since {zn}⊂Aθ, we can ind 
Nk∈Nsuch
ha zn∈k o all Nk≤n<
Nkand z
Nk/∈k.
The cons uc ions o he subsequences znkand zmka e e y simila , so we will only
cons uc znk. To do so, assume ha k∈Nis e en. No ice ha , by (5),
a g(pk(zn)) ∈(θ −2πk,θ), Nk≤n<
Nk.
Mo eo e , i l<kand Nk≤n<
Nk, hen
|pl(zn)|≤al
γl
k
.
Simila ly, i l>kand Nk≤n<
Nk, hen
|pl(zn)|≤al
γl
l
.
Howe e , i Nk≤n<
Nk, we use ha kis a apezium whose u hes poin s om
−γka e γ2
k±iγ2
k an(θ) o conclude ha
|pk(zn)|≥ak
γ2
k+γk+iγ2
k an(θ)k≥ak
2γ2
k+iγ2
k an(θ)k
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99 Page 8 o 15 M. D. Con e as e al.
=ak
γ2k
k|2+i an(θ)|k
=(θ)kak
γ2k
k
≥(θ)ak
γ2k
k
,(10)
whe e we ha e used ha (θ) ∈(0,1)and ha k≤θ/π < 1, by (4).
All in all, le us de ine
Rk(zn):= l∈N,l=kpl(zn)
pk(zn),Nk≤n<
Nk.
Using (8) and (10), i is clea ha
|Rk(zn)|≤l∈N,l=k|pl(zn)|
|pk(zn)|≤1
k,Nk≤n<
Nk.
In pa icula ,
a g(p(zn)) =a g(pk(zn)) +a g (1+Rk(zn)),
whe e
|a g (1+Rk(zn))|≤a csin 1
k≤2
k.
We conclude ha
a g(p(zn)) ≥θ−2πk−2/k,Nk≤n<
Nk.
The e o e,
yn+1−yn
xn+1−xn
≥ an (θ−2πk−2/k),Nk≤n<
Nk.(11)
By (9), i is clea ha
xNk≤γk+γ1
2.
Since zNk∈Aθ, his means ha
yNk≥−γk+γ1
2 an(θ).
All in all, using (11), o k∈Nla ge enough so ha an (θ−2πk−2/k)≥0, we
ha e ha
y
Nk=yNk+(y
Nk−yNk)≥yNk+(x
Nk−xNk) an(θ −2πk−2/k)
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Examples in Disc e e I e a ion Page 9 o 15 99
≥−γk+γ1
2 an(θ) +γ2
k−γk−γ1
2 an θ−2πk−2
k.
Then,
y
Nk
x
Nk
≥
−γk+γ1
2 an(θ) +γ2
k−γk−γ1
2 an θ−2πk−2
k
γ2
k+γ1
2
.
In pa icula , since we chose k∈N o be an e en numbe , we use (4) and (7) o
conclude ha
θ≥lim in
k→∞ a g(z
N2k)≥θ.
Namely,
lim
k→∞ a g(z
N2k)=θ,
and so i is clea ha i is enough o choose nk=
N2k.
Le us now add ess he main cons uc ion.
Theo em 2.6 Le 0≤a<b≤π,[a,b] =[0,π]. The e exis s a pa abolic sel -map
:H→Ho ze o hype bolic s ep wi h Denjoy–Wol poin in ini y o which
Slope[ ,z]=[a,b],z∈H.
P oo Conside he angle
θ=b−a
2∈(0,π/2),
and he map Fas in (2), whe e we a e assuming (4)–(8). Le us also conside he hal -
plane H={z∈C:a g(z)∈(−θ−a,π−θ−a)}. No ice ha Aθ⊂H. Then, as seen
in Rema k 2.4,F(H)⊂H. In ha case, conside he es ic ion g=F|H:H→H.
Le us now ema k ha he map p:H→Aθ, as de ined in (3), is bounded. To see
his, no ice ha
min
z∈∂H|z+γk|=γksin(a+θ).
Then, by (6), gi en ha z∈H,weha e
|p(z)|≤
∞

k=1
|pk(z)|≤1
sin(a+θ)
∞

k=1
ak
γk
k
<+∞,(12)
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