Characterization of finite shift via Herglotz’s representation

J. Math. Anal. Appl. 542 (2025) 128883 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa Regular Articles Characterization of finite shift via Herglotz’s representation ✩ Francisco J. Cruz-Zamorano Departamento de Matemática Aplicada II and IMUS, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos, s/n 41092, Sevilla, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 15 July 2024 Available online 14 September 2024 Submitted by C. Câmara Keywords: Parabolic functions Complex iteration Finite shift Rate of convergence A complete characterization of parabolic self-maps of finite shift is given in terms of their Herglotz’s representation. This improves a previous result due to Contreras, Díaz-Madrigal, and Pommerenke. We also derive some consequences for the rate of convergence of these functions to their Denjoy-Wolff point, improving a related result of Kourou, Theodosiadis, and Zarvalis for the continuous setting. © 2024 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/). 1. Introduction In the field of Discrete Complex Dynamics of the upper half-plane H={z=x +iy ∈C:y>0}, the main object of study is the behavior of the sequence of iterates of holomorphic self-maps f:H→H, that is, the sequence of functions given by fn+1 =f◦fn, n ∈N, where we define f0=Id H. In this paper we focus on non-elliptic self-maps f, that is, those with no fixed points on H. The dynamical behavior of non-elliptic functions is described in the following seminal result: Theorem 1.1 (Denjoy-Wolff). [2, Theorem 3.2.1] Let f:H→Hbe a non-elliptic holomorphic map. Then, there exists τ∈R ∪{∞}such that fn→τuniformly on compact sets of H. The point τis commonly known as the Denjoy-Wolff point of f, and it acts as a global attractor of its dynamics. Up to a conjugation with an isomorphism of H, one can always suppose that the Denjoy-Wolff point of a non-elliptic self-map is at infinity. Indeed, this might be useful to find different non-elliptic functions whose behavior varies. To this extent, we note that under this assumption one has [2, Corollary 2.5.5] ✩This research was supported by Ministerio de Innovación y Ciencia, Spain, project PID2022-136320NB-I00 and by Ministerio de Universidades, Spain, through the action Ayuda del Programa de Formación de Profesorado Universitario, reference FPU21/00258. E-mail address: [email protected]. https://doi.org/10.1016/j.jmaa.2024.128883 0022-247X/© 2024 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/). 2F.J. Cruz-Zamorano / J. Math. Anal. Appl. 542 (2025) 128883 ∠lim z→∞ f(z)=∞,∠lim z→∞ f(z) z=λ∈[1,+∞). If λ >1, fis known as hyperbolic. This work deals with parabolic functions, that is, those for which λ =1. The study of parabolic functions will be done through a representation result which is due to Herglotz [1, Theorem 6.2.1]: every holomorphic function f:H→Hcan uniquely be written as f(z)=αz +β+ R 1+tz t−zdμ(t),z∈H, where α≥0, β∈Rand μis a positive finite measure on R. Using [11, Chapter 5, Lemma 2], one can deduce the following: Theorem 1.2. A holomorphic self-map f:H→His parabolic with Denjoy-Wolff point at infinity if and only if it can be written as f(z)=z+β+ R 1+tz t−zdμ(t),z∈H,(1) where β∈Rand μis a positive finite measure on R, both of them not simultaneously null. Following the latter result, every property of fshould be mirrored on the corresponding representing parameters βand μ: this is the idea behind this paper. In particular, we are interested in characterizing the following property: Definition 1.3. Let f:H→Hbe a holomorphic self-map which is parabolic and whose Denjoy-Wolff point is at infinity. The self-map fis said to be of finite (respectively, infinite) shift if there exists an initial point z0∈Hfor which the orbit zn=xn+iyn=fn(z0)is such that supnyn<+∞(respectively, supnyn=+∞). Remark 1.4. [9, Proposition 3.2] The latter definition does not depend on the initial point z0∈H. Functions of finite shift have attracted much attention in the area. For example, a characterization of this property in terms of angular regularity of the functions is given in [9, Theorem 4.1]. Poggi-Corradini also used a related property to work with backward orbits; see [14]. There are some references in the continuous version of Complex Dynamics (that is, the theory of holomorphic semigroups) as well; see [7] for an introduction to this topic. For example, in [3, Theorem 3], [12], and [10, Theorem 1.1], the authors characterize parabolic semigroups of finite shift in terms of different geometric quantities. Beyond functions of finite shift, in [8], Theorem 1.2 has also proved to be useful to characterize the hyperbolic step of parabolic functions (see Section 2for a definition). Indeed, many of the tools of this work have been previously derived there. The main result of this work is the following, which completely characterizes self-maps of finite shift: Theorem 1.5. Let f:H→Hbe a parabolic map with its Denjoy-Wolff point at infinity. The map fis of finite shift if and only if one of the following holds: (i) (−∞,0) |t|dμ(t) <+∞, (0,+∞)t2dμ(t) <+∞, and β>Rtdμ(t). (ii) (−∞,0) t2dμ(t) <+∞, (0,+∞)|t|dμ(t) <+∞, and β<Rtdμ(t). The proof of the first part of this result is given in Subsection 3.1. Then, in Subsection 3.2, the second part is deduced as a consequence of different results in [8,9]. Notice that this result improves the description of functions of finite shift given in [9, Theorem 4.1], as noted in Subsection 3.3. F.J. Cruz-Zamorano / J. Math. Anal. Appl. 542 (2025) 128883 3 As a consequence of Theorem 1.5, we obtain the following result: Corollary 1.6. Let f:H→Hbe a parabolic map of finite shift with its Denjoy-Wolff point at infinity. Then fn n→β− R tdμ(t),n→+∞, uniformly on compact sets of H. We end this paper with a discussion on the rate of convergence of a parabolic function of finite shift to its Denjoy-Wolff point. This issue has been extensively discussed in [4–6] under the general setting of continuous dynamics. In the specific setting of parabolic semigroups of finite shift, better estimates for the rate of convergence have been obtained in [13]. To introduce this topic in detail, we should move from the upper half-plane Hinto the unit disk D. Using a Möbius map S:D→H, this can be done by relating any non-elliptic self-map g:D→Dwith a self-map f=S◦g◦S−1:H→Hwhose Denjoy-Wolff point is infinity. Definition 1.3 can be translated into the setting of a self-map gof the unit disk, meaning that gis of finite shift if there exists some orbit of gand a horocycle (see [2, Definition 2.1.1]) of center τin which the orbit does not enter. If this is not the case, gis said o be of infinite shift. Moreover, Theorem 1.1 also remains valid. The point τ=S−1(∞) ∈∂Dis called the Denjoy-Wolff of gand it follows that gn→τas n →+∞ uniformly on compact sets of D. By discussing the rate of convergence to the Denjoy-Wolff point we mean to estimate how fast the latter convergence is. To this extent, we prove the next result in Section 4: Theorem 1.7. Let g:D→Dbe a parabolic map of finite shift whose Denjoy-Wolff point is τ∈∂D. Then, there exists C=C(g) >0such that n|gn−τ|→C, n →+∞, uniformly on compact sets of D. As we will see at the end, this result can be used to improve the rate of convergence given in [13, Theorem 1.1]. Acknowledgment. I would like to thank E. K. Theodosiadis and K. Zarvalis for their great hospitality and our fruitful conversations during my stay at the Aristotle University of Thessaloniki. 2. Preliminaries: parabolic dynamics Parabolic self-maps of Hare further classified in terms of their hyperbolic step. To define it, let ρbe the pseudo-hyperbolic distance on H. For any initial point z0∈H, the Scharwz-Pick Lemma assures that the sequence ρ(fn+1(z0), fn(z0)) is non-increasing with respect to n ∈N. Therefore, it must converge. Indeed, if it converges to zero, then it does so for every initial point [2, Corollary 4.6.9.(i)]. Thus, the following definition is given: Definition 2.1. A parabolic self-map f:H→His said to be of zero hyperbolic step if ρ(fn+1(z0), fn(z0)) → 0as n →+∞for some (every) z0∈H. Otherwise, it is said to be of positive hyperbolic step. Determining the hyperbolic step of a given function can be a difficult task. However, some theoretical results are known: 4F.J. Cruz-Zamorano / J. Math. Anal. Appl. 542 (2025) 128883 Theorem 2.2. [15]Let f:H→Hbe a parabolic self-map whose Denjoy-Wolff point is at infinity. Let z0∈H be an initial point, and consider its orbit zn=xn+iyn=fn(z0), n ∈N. The following limit b= lim n→∞ xn+1 −xn yn ∈R exists. Furthermore, b =0if and only if fis of zero hyperbolic step. Also, zn+1/zn→1(see [15, Eq. (3.16)]), and yn+1/yn→1(see [15, Eq. (3.17)]). Theorem 2.3. [9, Proposition 3.3] Parabolic self-maps of finite shift are also of positive hyperbolic step. Remark 2.4. There are also self-maps of infinite shift and positive hyperbolic step. For examples of such maps with the aid of the representation given in Theorem 1.2, we refer to [8, Theorem 1.6]. 3. Proof of Theorem 1.5 and Corollary 1.6 In this section, from now on, the function f:H→Hwill always be a holomorphic map which is parabolic and whose Denjoy-Wolff point is at infinity. Indeed, with the aid of Theorem 1.2, we will identify fwith (β, μ). 3.1. Proof of the first part of Theorem 1.5 Assume that fis of finite shift and let us show that R|t|dμ(t) <+∞. Notice that, by Theorem 2.3, f is of positive hyperbolic step. In particular, Theorem 2.2 assures that for a given initial point z0∈H, the orbit zn=xn+iyn=fn(z0)is such that lim n→∞ xn+1 −xn yn =L∈R\{0}. But, as fis of finite shift, there must exists Y>0such that lim n→+∞yn=Y. (2) Thus, lim n→+∞xn+1 −xn=LY =: Δ =0.(3) In the sequel, we will suppose that Δ >0 (otherwise, the proof will apply with slight modifications). In that case, Stolz’s lemma assures that xn/(nΔ) →1as n →+∞. Under this setting, we can show the following: Lemma 3.1. (0,+∞)t2dμ(t) <+∞. Proof. First of all, notice that yn+1 =yn1+yn+1 −yn yn=y0 n  k=0 1+yk+1 −yk yk,n∈N. Since fis of finite shift, ynis bounded. Therefore, F.J. Cruz-Zamorano / J. Math. Anal. Appl. 542 (2025) 128883 5 ∞  n=0 1+yn+1 −yn yn<+∞, or equivalently, ∞  n=0 yn+1 −yn yn <+∞. But, from (1), ∞  n=0 yn+1 −yn yn = ∞  n=0 Im(f(zn)−zn) yn = R∞  n=0 1+t2 (t−xn)2+y2 ndμ(t). Using (3), let N∈Nbe such that Δ 2≤xn+1 −xn≤3Δ 2,n≥N. Thus, for every t >x None can find k=k(t) ≥Nsuch that |t−xk|≤Δ. In that case, using also (2), ∞  n=0 1+t2 (t−xn)2+y2 n ≥1+t2 Δ2+y2 k ≥1+t2 Δ2+Y2,t>x N. It follows that, +∞> R∞  n=0 1+t2 (t−xn)2+y2 ndμ(t)≥ (xN,+∞) 1+t2 Δ2+Y2dμ(t). Since μis a positive finite measure, the result follows.  Let us now introduce two new positive measures related with μ. One of them is the measure ωgiven by ω(A)=μ(A∩(−∞,0]), where A ⊂Ris any measurable set. Notice that, since μis a finite measure, so is ω. The other one, ν, is absolutely continuous with respect to μ, and its associated density is given by dν dμ(t)=(1+t2)χ(0,+∞)(t),t∈R. Notice that Lemma 3.1 implies that νis also a finite measure. Using these measures, fcan be decomposed as f(z)=z+˜ β+p1(z)+p0(z),z∈H, where ˜ β=β− (0,+∞) tdμ(t)∈R,p 1(z)= R 1+tz t−zdω(t),p 0(z)= R dν(t) t−z,z∈H. 6F.J. Cruz-Zamorano / J. Math. Anal. Appl. 542 (2025) 128883 Note that, by Lemma 3.1, ˜ βis well-defined. It is also clear that p1is a well-defined holomorphic map on C\(−∞, 0]. With these ideas, we can show that the different terms in the decomposition of fbehave as follows: Lemma 3.2. (a) For any α>0, consider Hα:= {z∈H:Im(z) ≥α}. Then, supz∈Hα|Re(p0(z))|<+∞. (b) Re(p1(x +iy)) ≥Re(p1(x)) for all x, y>0. (c) lim infx→+∞Re(p1(x)) ≥ (−∞,0) (−t)dω(t). Proof. (a) Let z=x +iy ∈Hα. It suffices to notice that |Re(p0(z))|≤|p0(z)|≤ R dν(t) |t−z|≤ν(R) y≤ν(R) α. (b) Write Re(p1(x+iy)) =  R t−x+t2x−t(x2+y2) (t−x)2+y2dω(t), and notice that ∂ ∂y t−x+t2x−t(x2+y2) (t−x)2+y2=−2y(1 + t2)(t−x) ((t−x)2+y2)2≥0, whenever t ≤0 <x, y≥0. (c) Notice that Re(p(x)) =  (−∞,0] 1+tx t−xdω(t),x>0. In particular, the integrand is positive if and only if t <−1/x. By Fatou’s lemma, one has lim inf x→+∞ (−∞,−1/x) 1+tx t−xdω(t)≥ (−∞,0) (−t)dω(t). On the other hand,  [−1/x,0]  1+tx t−x dω(t)≤1 xω([−1/x, 0]) →0,x→+∞. Thus, the result holds.  To end with the proof of the first part of Theorem 1.5, notice that if we suppose that  R |t|dμ(t)=+∞,(4) F.J. Cruz-Zamorano / J. Math. Anal. Appl. 542 (2025) 128883 7 then, by Lemma 3.1, it has to be that  (−∞,0) |t|dμ(t)=+∞.(5) In that case, by (3), let N∈Nbe such that xn>0for all n ≥N. Using Lemma 3.2.(a,b), we have xn+1 −xn=˜ β+Re(p1(zn)) + Re(p0(zn)) ≥˜ β+Re(p1(xn)) −sup z∈Hy0 |Re(p0(z))|,n≥N. Therefore, if we assume (4), then (3), (5)and Lemma 3.2.(c) imply that xn+1 −xn→+∞as n →+∞, which contradicts (3). Thus, (4) cannot hold, and so the first part of Theorem 1.5 follows. 3.2. Proof of the second part of Theorem 1.5 In [8], the authors fully characterize the hyperbolic step of a family of parabolic self-maps of H. That is, the following holds: Theorem 3.3. [8, Theorem 1.4] Let f:H→Hbe a parabolic self-map with its Denjoy-Wolff point at infinity such that R|t|dμ(t) <+∞. Then, fis of positive hyperbolic step if and only if one of the following holds: (i) (0,+∞)t2dμ(t) <+∞, and β>Rtdμ(t). (ii) (−∞,0) t2dμ(t) <+∞, and β<Rtdμ(t). Moreover, fis of positive hyperbolic step if and only if it is of finite shift. In Subsection 3.1 we have proven that, if f:H→His a parabolic self-map of finite shift whose DenjoyWolff point is at infinity, then R|t|dμ(t) <+∞. Therefore, the characterization of self-maps of finite shift given in Theorem 1.5 can be directly deduced from the latter result. 3.3. Further remarks Let us recall that functions of finite shift are also in connection with self-maps enjoying some regularity in an angular sense. To introduce these ideas, the following definition is given in [9]: Definition 3.4. Consider a holomorphic map φ:D→Cwith a boundary fixed point τ∈∂D, that is, ∠lim z→τφ(z)=τ. Then, φis said to be of angular-class of order p ∈Nat τ, denoted as φ ∈Cp A(τ), if there exist c1, ..., cp∈C and a holomorphic function γ:D→Csuch that φ(z)=τ+ p  k=1 ck k!(z−τ)k+γ(z),z∈D,∠lim z→τ γ(z) (z−τ)p=0. In particular, as a consequence of Julia-Wolff-Carathéodory Theorem [2, Corollary 2.5.5] (also known as the Julia-Wolff Theorem), every non-elliptic function is of angular-class of first order at its Denjoy-Wolff point. These angular-classes are intimately linked to functions of finite shift, as the following result suggests: 8F.J. Cruz-Zamorano / J. Math. Anal. Appl. 542 (2025) 128883 Theorem 3.5. [9, Theorem 4.1] Let φ:D→Dbe a parabolic self-map of positive hyperbolic step with DenjoyWolff point τ∈∂D. Then, φis of finite shift if and only if φ ∈C2 A(τ). Theorem 1.5 can be seen as a refinement of the latter result. To see this, one should link [9, Propositions 2.1] and [8, Lemma 3.3] to state the following: Proposition 3.6. Let f:H→Hbe a parabolic self-map with its Denjoy-Wolff point at infinity such that R|t|dμ(t) <+∞. Consider the self-map φ:D→Dwhich is conjugated to fand has Denjoy-Wolff point τ∈∂D. Then, φ ∈C2 A(τ). However, the converse is not true. These remarks have already been noticed in [8, p. 15]. 3.4. Proof of Corollary 1.6 We will prove that there exists a non-empty open set U⊂Hsuch that fn(z) n→β− R tdμ(t),n→+∞, for all z∈U. Then, the result will follow from the identity principle and the fact that the sequence {fn/n} is a normal family. To do so, as fis of finite shift, let us suppose that fis as in Theorem 1.5.(i) (if it is as in Theorem 1.5.(ii), the proof will also work with slight modifications). Write fas f(z)=z+β− R tdμ(t)+p(z),z∈H. By [8, Lemma 3.9], there exists a, b >0such that p(fn(z)) →0as n →+∞if z∈Ω:={z=x+iy ∈C:x>a,y>b}. In particular, if z∈Ω, it follows that Re(fn+1(z)) −Re(fn(z)) →β− R tdμ(t),n→+∞. Applying Stolz’s lemma, we obtain Re(fn(z)) n→β− R tdμ(t),n→+∞. Moreover, since fis of finite shift, Im(fn(z)) is bounded, and so Im(fn)/n →0as n →+∞. Then, the result follows. 4. Application: rate of convergence on the unit disk This section aims to develop a proof of Theorem 1.7 and to discuss its implications on the theory of continuous dynamics. To do so, recall that any map g:D→Dwhose Denjoy-Wolff point is τ∈∂Dcan be F.J. Cruz-Zamorano / J. Math. Anal. Appl. 542 (2025) 128883 9 conjugated to a map f=S◦g◦S−1:H→Hwhose Denjoy-Wolff point is infinity, where S:D→His the Möbius map given by S(z)=iτ+z τ−z,z∈D,S −1(w)=τw−i w+i,w∈H. From this, a computation shows: Lemma 4.1. With the above notations, gn(z)−τ=−2iτ fn(S(z)) + i,z∈D. 4.1. Proof of Theorem 1.7 It follows from Lemma 4.1 and Corollary 1.6, having in mind that fn(S(z)) →∞as n →+∞for all z∈D. 4.2. Continuous dynamics The uniform convergence on compact sets given in Corollary 1.6 lets us relate the “discrete” rate of convergence in Theorem 1.7 with its analogous “continuous” one. That is, the next result (which improves [13, Theorem 1.1] and complements [5, Theorem 1.(b)]) can also be derived: Theorem 4.2. Let {φt:D→D}t>0be a parabolic semigroup of self-maps of finite shift whose Denjoy-Wolff point is τ∈∂D. There exists C=C(φ1) >0such that t|φt−τ|→C, t →+∞, uniformly on compact sets of D. Proof. Set g=φ1. Notice that gis a parabolic self-map of Dwhich is of finite shift and has Denjoy-Wolff point τ∈∂D. Moreover, for every t >0, consider the decomposition t =n +s, where n ∈Nand s ∈[0, 1). Notice that, given a compact set K⊂Dand z∈K, we have min w∈K|gn(w)−τ|≤|φt(z)−τ|≤max w∈K|gn(w)−τ|, where K={φs(z) :s ∈[0, 1], z∈K} ⊂Dis a compact set. Since n =n(t)and t/n →1as t →+∞, the result follows from Theorem 1.7. Remark 4.3. The constant in Theorem 1.7 (respectively, in Theorem 4.2) is C=2 β−Rtdμ(t) , where βand μare the representing parameters in terms of Theorem 1.2 of the map f:H→Hwhich is conjugated to g(respectively, to φ1) through the Möbius map Sused in Lemma 4.1.