On the Hardy number of Koenigs domains

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 . Funding Funding o open access publishing: Uni e sidad de Se illa/CBUA Da a A ailabili y No da ase s we e gene a ed o analysed du ing he cu en s udy. Decla a ions Con lic o in e es The au ho s decla e no compe ing in e es s. Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which pe mi s use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence, and indica e i changes we e made. The images o o he hi d pa y ma e ial in his a icle a e included in he a icle’s C ea i e Commons licence, unless indica ed o he wise in a c edi line o he ma e ial. I ma e ial is no included in he a icle’s C ea i e Commons licence and you in ended use is no pe mi ed by s a u o y egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission di ec ly om he copy igh holde . To iew a copy o his licence, isi h p://c ea i ecommons.o g/licenses/by/4.0/. Re e ences 1. Aba e, M.: Holomo phic Dynamics on Hype bolic Riemann Su aces, olume 89. Wal e de G uy e GmbH & Co KG, (2022). 2. 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Camb idge Uni e si y P ess, Camb idge (1995) 19. Sa , E., To ik, V.: Loga i hmic Po en ials wi h Ex e nal Fields. Sp inge , Be lin (1997) 20. Vali on, G.: Su l’i é a ion des onc ions holomo phes dans un demi-plan. Bull. Sci. Ma h 55, 105–128 (1931) Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a ilia ions.