Examples in Discrete Iteration of Arbitrary Intervals of Slopes

The Journal of Geometric Analysis (2025) 35:99 https://doi.org/10.1007/s12220-025-01934-4 Examples in Discrete Iteration of Arbitrary Intervals of Slopes Manuel D. Contreras1·Francisco J. Cruz-Zamorano1· Luis Rodríguez-Piazza2 Received: 18 November 2024 / Accepted: 2 February 2025 © The Author(s) 2025 Abstract Given a compact interval [a,b]⊂[0,π], we construct a parabolic self-map of the upper half-plane whose set of slopes is [a,b]. The nature of this construction is completely discrete and explicit: we explicitly construct a self-map and we explicitly show in which way its orbits wander towards the Denjoy–Wolff point. We also analyze some properties of the Herglotz measure corresponding to such example, which yield the regularity of such self-map in its Denjoy–Wolff point. Keywords Complex dynamics ·Discrete iteration ·The slope problem Mathematics Subject Classification Primary 30D05 ·37FXX 1 Introduction Discrete Iteration in the unit disk Dis a branch of the vast field known as Complex Dynamics. Given a holomorphic self-map g:D→D, the main aim is to analyze asymptotic properties of the sequence of iterated self-compositions of g, given by This research was supported in part by Ministerio de Innovación y Ciencia, Spain, project PID2022-136320NB-I00. The second author was supported by Ministerio de Universidades, Spain, through the action Ayuda del Programa de Formación de Profesorado Universitario, reference FPU21/00258. BFrancisco J. Cruz-Zamorano [email protected] Manuel D. Contreras [email protected] Luis Rodríguez-Piazza [email protected] 1Departamento de Matemática Aplicada II and IMUS, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos, s/n, 41092 Seville, Spain 2Departmento de Análisis Matemático and IMUS, Facultad de Matemáticas, Universidad de Sevilla, Calle Tarfia, s/n, 41012 Seville, Spain 0123456789().: V,-vol 123 99 Page 2 of 15 M. D. Contreras et al. g0=IdD, and gn+1=gn◦g,n∈N∪{0}. In this article we focus on non-elliptic selfmaps, which are the ones that possess no fixed point on D. The following well-known result concerns the dynamics of such self-maps. Theorem 1.1 [2, Theorem 3.2.1] (Denjoy–Wolff Theorem) Let g :D→Dbe a nonelliptic self-map. Then, there exists τ∈∂Dsuch that gn→τlocally uniformly, as n→∞. The point τis usually known as the Denjoy–Wolff point of g. In particular, if w0∈D, the Denjoy–Wolff Theorem proves that the orbit wn=gn(w0)converges to τ,asn→∞. A classical problem in Discrete Iteration is to characterize the directions through which the orbits of a non-elliptic self-map of Dconverge towards the Denjoy–Wolff point. Namely, given w0∈D, calculating the numbers s∈[−π/2,π/2]such that there exists a subsequence of the orbit satisfying arg(1−τwnk)→s,ask→∞.This is the Slope Problem, which firstly appeared around 1930 in [18,19]. Dealing with this problem is usually easier in the upper half-plane setting. To do so, let H:= {z∈C:Im(z)>0}, and consider the conformal mapping S:D→H given by S(w) =iτ+w τ−w,w∈D, where τ∈∂Dis the Denjoy–Wolff point of g. Then, construct the self-map f:H→H given by f=S◦g◦S−1. Similarly as in the case of the unit disk, we can consider the iterated self-compositions f0=IdH, and fn+1=fn◦f,n∈N∪{0}.Itis easy to check that fn=S◦gn◦S−1. Something similar occurs in the case of orbits. Given z0∈H, its orbit by fis the sequence given by zn=fn(z0). By construction zn→∞,asn→∞.Ifw0∈Dand we choose z0=S(w0), then zn=S(wn). That is, fn→∞,asn→∞, locally uniformly, so that the Denjoy–Wolff point of fis infinity. Concerning the Slope Problem, fixing a subsequence of the orbits, we have that arg(1−τwnk)→sif and only if arg(zn)→π/2−s. Inspired by this relationship, we give the following definition. Definition 1.2 Let f:H→Hbe a holomorphic function whose Denjoy–Wolff point is infinity. Then, given z0∈H, we define the set of slopes of fas Slope[f,z0]={s∈[0,π]:There exists a subsequence {znk}with arg(znk)→s}, where arg denotes the principal branch of the argument function. By [2, Proposition 2.5.8], the Denjoy–Wolff point of f:H→His infinity if and only if α:= ∠lim z→∞ f(z) z∈[1,+∞). 123 Examples in Discrete Iteration Page 3 of 15 99 In this situation, fis said to be parabolic if α=1, and it is said to be hyperbolic if α>1. Valiron [18] studied the Slope Problem for hyperbolic self-maps of Hin 1931. It turns out that if f:H→His a hyperbolic self-map with Denjoy–Wolff point infinity, then for every z∈Hthere exist θ∈(0,π)such that limn→∞ arg(fn(z)) =θ;see[2, Theorem 4.3.4]. His ideas have been recently revisited by Bracci and Poggi-Corradini. More precisely, they proved that the function Slope[f,·]: H→(0,π)is surjective and harmonic [4, Property 2 (a) and Property 2 (b)]. The theory is significantly more difficult in the case of parabolic self-maps, which are typically divided in two families. For a holomorphic self-map f:H→H,the Schwarz-Pick Lemma assures that kH(fn+2(z), fn+1(z)) ≤kH(fn+1(z), fn(z)), z∈H,n∈N, where kHdenotes the hyperbolic distance of the upper half-plane. In particular, limn→∞ kH(fn+1(z), fn(z)) exists for all z∈H, and it is a non-negative real number. For non-elliptic self-maps, by [2, Corollary 4.6.9.(i)], we know that the latter limit is either positive for every z∈Hor for no z∈H. Accordingly, we say that fis of positive or of zero hyperbolic step. In 1979, Pommerenke contributed to the Slope Problem for parabolic self-maps of H.In[17, Remark 1], he proved that limn→∞ arg(fn(z)) exists and it is either 0 or πwhenever fis a parabolic self-map of positive hyperbolic step with Denjoy–Wolff point infinity. In [10, Proposition 2.6], using hyperbolic geometry, we noticed that the latter limit does not depend on z. Namely, it holds that either Slope[f,z]={0}or Slope[f,z]={π}for all z∈H. The main emphasis in this paper is put on the case where fis a parabolic selfmap of zero hyperbolic step. One of the first surprising contributions to this case is in the remarkable paper of Wolff in 1929. In [19, Section 6], he came up with the map f:H→Hgiven by f(z)=z+ieπ/2zi+ieπ/2,z∈H,(1) where ziis defined using the principal branch of the logarithm. It is easy to check that fis parabolic with Denjoy–Wolff point infinity, since ∠limz→∞ f(z)/z=1. For this example, Wolff managed to prove that Slope[f,z0]contains at least two points for all initial points z0∈H. The Slope Problem has also been introduced in the continuous setting of Complex Dynamics, that is, for semigroups {φt}t≥0of holomorphic self-maps of the upper half-plane. A modern exposition of this topic can be found in [5, Chapter 17]. In [6], Contreras and Díaz-Madrigal found that the results of Valiron and Pommerenke can be translated to hyperbolic and parabolic semigroups of positive hyperbolic step, respectively. For a parabolic semigroup of zero hyperbolic step, they proved that the set of slopes is a compact interval which does not depend on the initial point. Despite Wolff’s example (1), they conjectured that, in the continuous setting, the set of slopes should always be a singleton. This conjecture was disproved by Contreras, Díaz-Madrigal, and Gumenyuk [8], and also by Betsakos [3], independently. Namely, 123 99 Page 4 of 15 M. D. Contreras et al. they found a semigroup whose set of slopes is the full interval [0,π]. Improving the techniques by Betsakos, for every prefixed interval [a,b]⊂[0,π], Kelgiannis [16] constructed a semigroup whose set of slopes is [a,b]. Quite recently, the latter ideas were taken back to the discrete setting. Namely, even if the orbits of a self-map are discrete sets, we have shown that the set of slopes of parabolic self-maps of zero hyperbolic step is always a compact interval which does not depend on the initial point. Theorem 1.3 [10, Theorem 2.9] Let f :H→Hbe a parabolic function of zero hyperbolic step whose Denjoy–Wolff point is infinity. Then, there exists 0≤a≤b≤π such that Slope[f,z]=[a,b]for all z ∈H. Examples with arbitrary set of slopes have also been constructed through the use of semigroups. Namely, if f=φ1, where {φt}is the semigroup constructed by Kelgiannis, we have proved that Slope[f,z]=[a,b]for all z∈H. Therefore, we have the following result. Theorem 1.4 [10, Theorem 2.13] Given 0≤a≤b≤π, there exists a parabolic function f :H→Hof zero hyperbolic step whose Denjoy–Wolff point is infinity such that Slope[f,z]=[a,b]for all z ∈H. This note focuses on constructing examples of parabolic self-maps of zero hyperbolic step with arbitrary slope sets [a,b]⊂[0,π], with [a,b] =[0,π]. These self-maps are of a discrete nature, and so we extend a previous work that primarily relied on semigroups. This yields a new proof of Theorem 1.4 using tools from Discrete Iteration, which is given in Sect.2. Indeed, our construction is completely explicit: we explicitly construct the self-map and we explicitly show in which way the orbits wander towards the Denjoy–Wolff point. This strongly contrasts with the arguments in [16], where the semigroup is obtained through its Koenigs domain and the proof relies in subtle estimates of some harmonic measures. In Sect.3, we analyze some properties of the Herglotz measure corresponding to such examples. This leads us to discuss the regularity of such self-maps at their Denjoy–Wolff points, and we compare this fact with previous references. In [10, Section 4], we also gave a discrete and explicit construction for the cases [0,π]and [0,π/2]. Those self-maps are defined through their Herglotz representation formula, and we discuss their regularity at the Denjoy–Wolff point. However, the new construction is easier to follow. 2 The Main Construction The goal of this section is, given 0 ≤a<b≤πwith [a,b] =[0,π], to construct a parabolic self-map f:H→Hof zero hyperbolic step with Denjoy–Wolff point infinity for which Slope[f,z]=[a,b],z∈H. Notice that the limit case [a,b]=[0,π]has been covered in [10, Theorem 4.2] with a different technique. 123 Examples in Discrete Iteration Page 5 of 15 99 We will start with a general setting, which we appropriately modify later to get the desired examples. To present it, let us fix some notation. Consider =C\(−∞,0], θ∈(0,π/2), and Aθ={z∈C:arg(z)∈(−θ,θ)}. We will construct a function F:→Csuch that F(Aθ)⊂Aθ. The function Fis given by F(z)=z+p(z), z∈, (2) where p:→Cis given by p(z)= ∞  k=1 pk(z), pk(z)=akeiθk (z+γk)k,z∈, (3) where we are using the main branch of the argument of wto define wk. In the definition of p, the coefficients ak,γk,kand θksatisfy the following assumptions: 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) where (θ) =1 4+tan2(θ) . Example 2.1 The latter conditions are fulfilled for the coefficients k=θ π 1 2k,ak=Ck 1(k!)2,γ k k=(C2k!)3k,k∈N, where C1,C2>1 are large enough constants, and θkis automatically defined by (5). Lemma 2.2 Assume (4)–(7). Then, p is a well-defined holomorphic function on . Moreover, the image of p is contained in Aθ. Proof Notice that pis a sum of holomorphic maps which are well-defined on . Then, to see that pis well-defined and holomorphic on , it is enough to find that the sum defining pis uniformly convergent on every compact subset of . Indeed, if K⊂ 123 99 Page 6 of 15 M. D. Contreras et al. is a compact set, then there must exist k0∈Nsuch that K⊂{z∈C:|z|≤γk0/2}. Then, there exists M>0 such that k0  l=1 |pl(z)|≤M,z∈K. For the other terms, if l>k0,by(7)wehave γl≥γ2 l−1≥2γl−1≥2γk0. Then, if l>k0and z∈K, we get |z+γl|≥γl−γk0 2=γl1−γk0 2γl≥γl 2. Then, using the latter inequality and (4), for l>k0we have that |pl(z)|=al |z+γl|l≤2lal γl l ≤2al γl l . All in all, using also (6) and the Weierstrass Theorem, we conclude that the sum defining pis uniformly convergent on K. Let us now prove that the image of pis contained in Aθ. Since Aθis a convex cone, for all z,w ∈Aθwe have that z+w∈Aθ. Moreover, if z∈Aθand w∈Aθ,itis easy to check that z+w∈Aθ. Summing up, it is enough to prove that the image of each pkis contained in Aθ. To do so, let us assume that k∈Nis even (if it is odd, then the argument is similar). Consider the map z→ 1 (z+γk)k,z∈. By (4), its image is the angular region A={z∈C:arg z∈(−πk,π k)}. Then, by (4) and (5), the image of pkis the angular region delimited by the arguments θk+πk=θ, θk−πk=θ−2πk≥θ−2θ≥−θ. Therefore, it is clear that the image of pkis contained in Aθ. Remark 2.3 Let us notice that the first part of the latter proof can easily be adapted to see that pis a well-defined and holomorphic map on C\(−∞,−γ1]. Remark 2.4 Lemma 2.2 implies that the map Fgiven in (2) satisfies F(Aθ)⊂Aθ, since Aθis a convex cone. Moreover, the same argument yields that F(H)⊂Hfor every half-plane H⊂such that Aθ⊂H. The next result is the main idea behind the examples that we will construct later. 123 Examples in Discrete Iteration Page 7 of 15 99 Lemma 2.5 Assume (4)–(8). Let z0=γ1/2, and consider zn=xn+iyn=Fn(z0), n∈N. There exist two subsequences znkand zmksuch that arg(znk)→θand arg(zmk)→−θas k →∞. Proof First of all, since z0∈Aθ, it follows from Remark 2.4 that zn∈Aθfor all n∈N. Moreover, since θ∈(0,π/2), we have that Re(p(z)) > 0 for all z∈. This means that xnis an increasing sequence and that Fhas no fixed point on Aθ. Therefore, using Remark 2.4, we conclude that the Denjoy–Wolff point of F:Aθ→Aθis infinity. Then, zn→∞,asn→∞. Since |z|and Re(z)are comparable quantities for all z∈Aθ, we deduce that xn→+∞,asn→∞. Following this idea, we define k:= {z∈Aθ:γk≤Re(z)≤γ2 k},k∈N. By (7), k∩l=∅for all k= l. We claim that, for every k∈Nthere exists Nk∈N such that zNk∈kand zn/∈kfor all n<Nk. To see this, use (6) and notice that xn+1−xn=Re(p(zn)) ≤|p(zn)|≤ ∞  k=1 ak |zn+γk|k≤ ∞  k=1 ak γk k ≤γ1 2,(9) where we have used that Re(zn)≥0. Since γ2 k−γk≥γk≥γ1>γ1 2, wherewehavealsoused(7), the claim follows. Recall that xn→+∞,asn→∞. Since {zn}⊂Aθ, we can find  Nk∈Nsuch that zn∈kfor all Nk≤n< Nkand z Nk/∈k. The constructions of the subsequences znkand zmkare very similar, so we will only construct znk. To do so, assume that k∈Nis even. Notice that, by (5), arg(pk(zn)) ∈(θ −2πk,θ), Nk≤n< Nk. Moreover, if l<kand Nk≤n< Nk, then |pl(zn)|≤al γl k . Similarly, if l>kand Nk≤n< Nk, then |pl(zn)|≤al γl l . However, if Nk≤n< Nk, we use that kis a trapezium whose furthest points from −γkare γ2 k±iγ2 ktan(θ) to conclude that |pk(zn)|≥ak γ2 k+γk+iγ2 ktan(θ)k≥ak 2γ2 k+iγ2 ktan(θ)k 123 99 Page 8 of 15 M. D. Contreras et al. =ak γ2k k|2+itan(θ)|k =(θ)kak γ2k k ≥(θ)ak γ2k k ,(10) where we have used that (θ) ∈(0,1)and that k≤θ/π < 1, by (4). All in all, let us define Rk(zn):= l∈N,l=kpl(zn) pk(zn),Nk≤n< Nk. Using (8) and (10), it is clear that |Rk(zn)|≤l∈N,l=k|pl(zn)| |pk(zn)|≤1 k,Nk≤n< Nk. In particular, arg(p(zn)) =arg(pk(zn)) +arg (1+Rk(zn)), where |arg (1+Rk(zn))|≤arcsin 1 k≤2 k. We conclude that arg(p(zn)) ≥θ−2πk−2/k,Nk≤n< Nk. Therefore, yn+1−yn xn+1−xn ≥tan (θ−2πk−2/k),Nk≤n< Nk.(11) By (9), it is clear that xNk≤γk+γ1 2. Since zNk∈Aθ, this means that yNk≥−γk+γ1 2tan(θ). All in all, using (11), for k∈Nlarge enough so that tan (θ−2πk−2/k)≥0, we have that y Nk=yNk+(y Nk−yNk)≥yNk+(x Nk−xNk)tan(θ −2πk−2/k) 123 Examples in Discrete Iteration Page 9 of 15 99 ≥−γk+γ1 2tan(θ) +γ2 k−γk−γ1 2tan θ−2πk−2 k. Then, y Nk x Nk ≥ −γk+γ1 2tan(θ) +γ2 k−γk−γ1 2tan θ−2πk−2 k γ2 k+γ1 2 . In particular, since we chose k∈Nto be an even number, we use (4) and (7)to conclude that θ≥lim inf k→∞ arg(z N2k)≥θ. Namely, lim k→∞ arg(z N2k)=θ, and so it is clear that it is enough to choose nk= N2k. Let us now address the main construction. Theorem 2.6 Let 0≤a<b≤π,[a,b] =[0,π]. There exists a parabolic self-map f:H→Hof zero hyperbolic step with Denjoy–Wolff point infinity for which Slope[f,z]=[a,b],z∈H. Proof Consider the angle θ=b−a 2∈(0,π/2), and the map Fas in (2), where we are assuming (4)–(8). Let us also consider the halfplane H={z∈C:arg(z)∈(−θ−a,π−θ−a)}. Notice that Aθ⊂H. Then, as seen in Remark 2.4,F(H)⊂H. In that case, consider the restriction g=F|H:H→H. Let us now remark that the map p:H→Aθ, as defined in (3), is bounded. To see this, notice that min z∈∂H|z+γk|=γksin(a+θ). Then, by (6), given that z∈H,wehave |p(z)|≤ ∞  k=1 |pk(z)|≤1 sin(a+θ) ∞  k=1 ak γk k <+∞,(12) 123