Speeds of convergence of orbits of non-elliptic semigroups of holomorphic self-maps of the unit disc

We introduce three quantities related to orbits of non-elliptic continuous semigroups of holomorphic self-maps of the unit disc, the total speed, the orthogonal speed and the tangential speed and show how they are related and what can be inferred from those.


Introduction
Continuous semigroups of holomorphic self-maps of the unit disk D, or for short, semigroups in D, have been studied since the beginning of the previous century and are still a subject of interest, from the dynamical point of view, the analytic point of view, and the geometric point of view, and also, for different applications.
In this paper, we consider non-elliptic semigroups in D. For such a nonelliptic semigroup (φ t ) it is well known that there exists a unique point For semigroups, another interesting relation holds, namely, for all t ≥ 0, v T (t) ≤ v o (t) + 4 log 2.
The previous inequalities imply also that there exist universal constants The previous definitions of speeds have Euclidean counterparts and some previous results can be translated in terms of speeds using such a dictionary. It turns out that, for instance, a recent result of D. Betsakos [5] can be rephrased in terms of speeds, namely, for all non-elliptic semigroups, v o (t) ≥ 1 4 log t+C for all t ≥ 0 and a constant C ∈ R (while, for parabolic semigroups of positive hyperbolic step, 1/4 log t can be replaced by 1/2 log t).
Besides settling the notions of speeds and proving the aforementioned results, in this paper we provide a direct computation of total, orthogonal and tangential speeds in some cases (essentially when the image of the Koenigs function is a vertical angular sector).
The paper ends with a section of open questions which naturally arise from the developed theory.
If Ω C is a simply connected domain and z ∈ Ω, v ∈ C, given a Riemann map f : D → Ω, we let .
Similarly, we define the hyperbolic length Ω of a curve and the hyperbolic distance k Ω between points of Ω. By Schwarz's Lemma, all these hyperbolic quantities are invariant under biholomorphisms and are decreasing under the action of holomorphic functions. A geodesics for the hyperbolic distance is a smooth curve such that the hyperbolic length among any two points of the curve coincide with the hyperbolic distance between the two points. Using the conformal invariance of the hyperbolic distance, it follows studying the case of the unit disk that for every two points there exists a unique (up to parameterization) geodesic joining the two points.
Let H := {w ∈ C : Re w > 0} be the right half plane.
Since H is biholomorphic to D via a Cayley transform z → (1+z)/(1−z), one can easily prove that k H (w 1 , w 2 ) = 1 2 log Moreover, one can easily see that both lines parallel to the real axis, and arcs of circles orthogonal to the imaginary axis are geodesics in H.
Finally, using Carathéodory's prime-ends topology (see, e.g., [13]), one can see that for any z 0 ∈ Ω and any prime end x ∈ ∂ C Ω (here ∂ C Ω denotes the set of prime-ends of Ω endowed with the Carathéodory topology), there exists a unique geodesic γ : [0, +∞) → Ω, parametrized by hyperbolic arc length, so that γ(0) = z 0 and γ(t) converges to x in the Carathéodory topology. Indeed, this is true in D with the Euclidean topology, and since Riemann mappings are isometries for the hyperbolic distance and homeomorphisms for the Carathéodory topology and D is homeomorphic to D ∪ ∂ C D endowed with the Carathéodory topology, the result follows at once.

Hyperbolic projections, tangential and orthogonal speeds of curves in the disk
In what follows, for not burdening the notation, we will consider geodesics parameterized by (hyperbolic) arc length, but, as it will be clear, this is not relevant, and any parametrization of geodesics would work as well.
Let Ω C be a simply connected domain. Let γ : R → Ω be a geodesic parameterized by arc length. Let z ∈ Ω. The hyperbolic projection π γ (z) ∈ γ(R) of z onto γ is the closest point (in the hyperbolic distance) of γ to z, namely, Using conformal invariance, one can easily prove the following: Let Ω C be a simply connected domain. Let γ : R → Ω be a geodesic in Ω parameterized by arc length and let z ∈ Ω. Then π γ (z) is the point of intersection of γ with the geodesicγ containing z and intersecting γ orthogonally (in the Euclidean sense).
Although orthogonal projections onto geodesics are not holomorphic maps, they do not increase the hyperbolic distance: Let Ω C be a simply connected domain, γ : R → Ω a geodesic parameterized by arc length. Then for every z, w ∈ Ω, we have k Ω (π γ (z), π γ (w)) ≤ k Ω (z, w).
Let P, Q ∈ R 2 two distinct points, and R any line through P -note that a line is a geodesic for the Euclidean metric. Let π R (Q) denote the (Euclidean) orthogonal projection of Q onto R. By Pythagoras' Theorem, |P −π R (Q)| 2 + |Q − π R (Q)| 2 = |P − Q| 2 . The next result tells that, in hyperbolic geometry, a Pythagoras' Theorem is true up to a universal constant without squaring the distances: Proposition 3.4 (Pytaghoras' Theorem in hyperbolic geometry). Let Ω C be a simply connected domain, γ : R → Ω a geodesic parameterized by arc length, x 0 ∈ γ and z ∈ Ω. Then where k Ω (z, γ) := inf t∈R k Ω (z, γ(t)) = k Ω (z, π γ (z)).
Proof. Since the statement is invariant under isometries for the hyperbolic distance, using a univalent map, we can transfer our considerations to H, and we can assume that γ(R) = (0, +∞) and x 0 = 1.
The previous proposition allows to make sense to the following definition and the subsequent remarks.
(1) The orthogonal speed and the tangential speed of a curve do not depend on the parameterization of the geodesic γ. Therefore, the definition of orthogonal speed and tangential speed depend only on Ω, z 0 and x.
(2) If Ω, Ω C are simply connected domains, z 0 ∈ Ω, z 0 ∈ Ω and f : Ω → Ω is a biholomorphism such that f This follows immediately since f is an isometry for the hyperbolic distances of Ω and Ω .
The reason for the name "tangential speed" follows from the following property: Then for all t ≥ t 0 , Proof. Since η(t) → σ as t → +∞, it follows that t 0 < +∞. The first equation follows immediately from the very definition of ω. Indeed, for every t ≥ 0, happened In order to prove the other two equations, up to change η with ση, we can assume without loss of generality that σ = 1. Let C : D → H be the Cayley transform given by C(z) = 1+z 1−z . For every t ≥ 0, let us write ρ t e iθt := C(η(t)), with ρ t > 0 and θ t ∈ (−π/2, π/2). This implies in particular, that where, the first equality follows from Remark 3.6(2), the second equality follows from the definition of orthogonal speed and since the orthogonal projection of ρ t e iθt onto the geodesic (0, +∞) is ρ t by Lemma 2.1(3), and the third equality follows from Lemma 2.1(1). Therefore, by (3.2), and taking into account that for t ≥ t 0 , we have As for the last inequality, from Proposition 3.4, we have and using the previous two inequalities for the estimates of ω(0, η(t)) and v o D,0 (η; t), we get the result. Remark 3.9. As a consequence of the previous proposition, we have that if η : [0, +∞) → D is a continuous curve such that lim t→+∞ η(t) = σ ∈ ∂D, then η converges to σ non-tangentially if and only if lim sup t→+∞ v T D,0 (η; t) < +∞.

Continuous non-elliptic semigroups of holomorphic self-maps of the unit disk
In this paper we consider only non-elliptic (continuous) semigroups of holomorphic self-maps of the unit disk. We refer the reader to, e.g., [1,2,7,16,22,24,19,20,4,8,9,10,18,21,23,25,26,27,28] for all unproved statements and more on the subject. A continuous non-elliptic semigroups of holomorphic self-maps of the unit disk, or just a non-elliptic semigroup for short, is a family (φ t ) such that for every t ≥ 0, φ t : D → D is holomorphic, with no fixed point in D for t > 0, φ t+s = φ t •φ s for all t, s ≥ 0, φ 0 (z) = z for all z ∈ D and [0, +∞) t → φ t is continuous with respect to the topology of uniform convergence on compacta of D.
If (φ t ) is a non-elliptic semigroup in D, there exists a point τ ∈ ∂D, the Denjoy-Wolff point of (φ t ) such that lim t→∞ φ t (z) = τ for all z ∈ D, and the convergence is uniform on compacta.
Moreover, the angular derivative φ t (τ ) of φ t at τ exists for all t ≥ 0 and there exists λ ≥ 0, the spectral value of (φ t ) such that for all z ∈ D and t ≥ 0. Moreover, Ω = t≥0 h(D) − it and we have the following cases: Ω is either a strip S r := {z ∈ C : 0 < Re z < r} (where r = π/λ with λ > 0 the spectral value of (φ t )), or the right half plane H, or the left half plane H − := {w ∈ C : Re w < 0} or C. The holomorphic model is universal in the sense that any other (semi)conjugation of (φ t ) factorizes through it (see [17,3]). The map h is called the Koenigs function of (φ t ).
The semigroup is hyperbolic if Ω is a strip, it is parabolic otherwise. Moreover, parabolic semigroups are of finite hyperbolic step if Ω is a half plane, or of zero hyperbolic step if Ω = C.
This definition is equivalent to the classical one, for which a semigroup (φ t ) is hyperbolic provided its spectral value is > 0, it is parabolic if its spectral value is 0, and the hyperbolic step is positive if for some -and hence anyz ∈ D. The last equivalence follows from the fact that k Ω (z, w) = lim t→∞ ω(φ t (z), φ t (w)) (see [3]).

Speeds of non-elliptic semigroups
Since the orbits of a non-elliptic semigroup converge to the Denjoy-Wolff point on ∂D, one might study the tangential and orthogonal speed of convergence. First of all, we show that the (asymptotic behavior of) orthogonal speed and the tangential speed of an orbit of a semigroup do not depend on the starting point: . Proof. Let γ : (−1, 1) → D be the geodesic of D defined by γ(r) = rτ . For z ∈ D let π γ (z) be the orthogonal projection of z onto γ. Then, by the very definition of orthogonal speed of curves and Proposition 3.3, we have A similar argument proves the second inequality. Namely, . Changing the role of z 1 and z 2 , we obtain the second inequality of the statement.
Lemmas 5.1 and 3.7 show that, in order to study the asymptotic behavior of the speed of convergence of semigroups' orbits to the Denjoy-Wolff point, it is enough to study the orbit starting at 0 and considering the speed with respect to 0. In other words, the following definition makes sense: and call v(t) the total speed of (φ t ).
It follows from Lemma 2.1 and the previous considerations that, if (φ t ) is a non-elliptic semigroup in D with the Denjoy-Wolff point τ ∈ ∂D, and C(z) = (τ + z)/(τ − z) (a biholomorphism from D to H), setting ρ t e iθt = C(φ t (C −1 (1))) with ρ t > 0 and θ t ∈ (−π/2, π/2), then By Proposition 3.4, if (φ t ) is a non-elliptic semigroup, we have A second less immediate relation between the orthogonal speed and the tangential speed is contained in the following proposition: is a non-elliptic semigroup in D, then, for every t ≥ 0, Proof. Let τ ∈ ∂D be the Denjoy-Wolff point of (φ t ) and let λ ≥ 0 be its spectral value. By Julia's Lemma, for every t ≥ 0 which is equivalent to Applying the function x → 1 2 log x to the previous inequality, we obtain for every t ≥ 0, Therefore, by Proposition 3.8, we have for all t ≥ 0, Hence, by (5.2), we have for all t ≥ 0, Finally, the previous equation implies that v T (t) ≤ v o (t) + 4 log 2 for all t ≥ 0, and we are done.
Note that Ω ± is a domain starlike at infinity. Moreover, for any open set D ⊂ C and p ∈ D, we let δ D (p) = inf{|z − p| : z ∈ C \ D}.
The following result is a consequence of [12] and Remark 5.3: Note that this implies that, in particular, for hyperbolic semigroup the orthogonal speed is essentially monotone, in the sense that, if (φ t ) is a hyperbolic semigroup with the Koenigs function h, total speed v(t) and orthogonal speed v o (t) and (φ t ) is a hyperbolic semigroup with the Koenigs functionh and h(D) ⊂h(D), total speedṽ(t) and orthogonal speedṽ o (t), then by (5.2), v o (t) ≥ṽ o (t) + C for all t ≥ 0 and some C > 0, since in the previous case, v(t) ≥ṽ(t) for all t ≥ 0 by the monotonicity of the hyperbolic distance.

Total speed of convergence
In this section we consider the total speed of convergence of orbits of hyperbolic and parabolic semigroups to the Denjoy-Wolff point. Proposition 6.1. Let (φ t ) be a non-elliptic semigroup in D, with the Denjoy-Wolff point τ ∈ ∂D and φ t (τ ) = e −λt for λ ≥ 0 and t ≥ 0 (in particular, (φ t ) is hyperbolic if λ > 0, parabolic otherwise). Then In case λ = 0, that is, (φ t ) is parabolic, it follows immediately from (5.2) that In case λ > 0, that is, (φ t ) is hyperbolic, we have already noticed that lim sup t→+∞ v T (t) < +∞. Thus from (5.2) we have the result.
According to the type of the semigroup, we have also a simple lower bound on the total speed: Proposition 6.2. Let (φ t ) be a non-elliptic semigroup in D, with the Denjoy-Wolff point τ ∈ ∂D.
• If (φ t ) is hyperbolic with spectral value λ > 0, then where the last equality follows from a direct computation and taking into account that k S π/λ (h(0) + it, π 2λ + it) = k S π/λ (h(0), π 2λ ) for all t ∈ R since z → z + it is an automorphism of S π λ . From this, the result for hyperbolic semigroups follows at once. Now, assume that (φ t ) is parabolic of positive hyperbolic step. We can assume that its canonical model is (H, h, z + it) (in case the canonical model is (H − , h, z +it) the argument is similar). Arguing as in the hyperbolic case, we see that v(t) ≥ k H (1, 1 + it) + C, for some constant C ∈ R and every t ≥ 0. Now, write 1 + it = ρ t e iθt for ρ t > 0 and θ t ∈ [0, π/2). A simple computation shows that ρ t = √ 1 + t 2 and cos θ t = 1 √ 1+t 2 . Therefore, by Lemma 2.1(1) and (2), we have k H (1, 1 + it) ≥ k H 1, 1 + t 2 + 1 2 log 1 + t 2 = log 1 + t 2 ≥ log t, and the result follows in this case as well. Finally, in case (φ t ) is parabolic of zero hyperbolic step, the canonical model is (C, h, z + it). Since h(D) is starlike at infinity and is different from C, there exists p ∈ C such that p − it ∈ h(D) for all t ≥ 0 and p + it ∈ h(D) for all t > 0. Hence, h(D) ⊆ K p , where K p is the Koebe domain C \ {ζ ∈ C : Re ζ = Re p, Im ζ ≤ Im p}. Therefore, arguing as in the previous cases, we find C ∈ R such that for every t ≥ 0, Taking into account that the map K 0 z → √ −iz ∈ H is a biholomorphism, where the branch of the square root is chosen so that √ 1 = 1, we have by Lemma 2.1(1) and we are done.
Remark 6.3. The bound given by Proposition 6.2 is sharp. Indeed, as it is clear from the proof, if (φ t ) is a hyperbolic group in D with spectral value λ > 0, then there exists C > 0 such that |v(t) − λ 2 t| < C for every t ≥ 0, while, if (φ t ) is a parabolic group, then there exists C > 0 such that |v(t) − log t| < C for every t ≥ 0 -so that, in this sense, non-elliptic groups in D have the lowest total speed. Moreover, the semigroup (φ t ) in D defined as φ t (z) := h −1 (h(z) + it), z ∈ D, where h : D → K 0 is a Riemann map for the Koebe domain K 0 , has the property that there exists C > 0 such that |v(t) − 1 4 log t| < C for all t ≥ 0. A direct consequence of Proposition 6.1 and Proposition 6.2 is the following: As it is clear from the proof of the previous proposition, one can get lower or upper estimates on the total speed of convergence according to the geometry of the image of the Koenigs function using the domain monotonicity of the hyperbolic distance. We provide here an example of such situation by studying a particular case.
goes like 1 2 log t as t → +∞ and the result follows.
In Proposition 6.1, we showed that if (φ t ) is a parabolic semigroup in D, then v(t)/t → 0 as t → +∞. This is essentially the only possible upper bound, as the following proposition shows: = +∞.
Proof. Let {a j } be a strictly increasing sequence of positive real numbers, a 1 > 0, lim j→+∞ a j = +∞. Let {b j } be a strictly increasing sequence of positive real numbers to be chosen later on. Let Note that Ω is simply connected and starlike at infinity. Let h : D → Ω be a Riemann map such that h(0) = 0, and let φ t (z) := h −1 (h(z) + it) for z ∈ D and t ≥ 0. Then (φ t ) is a semigroup in D and, since t≥0 (Ω − it) = C, it follows that (φ t ) is parabolic of zero hyperbolic step. In order to estimate the total speed v(t) of (φ t ), note that Ω is symmetric with respect to the imaginary axis iR, hence the orbit [0, +∞) t → it is a geodesic in Ω, and so is [0, +∞) t → φ t (0) in D.
In particular, if we set γ(t) = it, we have v(t) = ω(0, φ t (0)) = k Ω (0, it) = t 0 κ Ω (γ(r); γ (r))dr where the last inequality follows from the classical estimates on the hyperbolic metric (see, e.g., [10]) Now, we claim that we can choose the b j 's in such a way that for every j ≥ 1 there exists x j ∈ (b j , b j+1 ) such that δ Ω (it) = a j+1 for every t ∈ [x j , b j+1 ] and such that Indeed, set b 1 = 1. Let x 1 > 1 be such that |ix 1 − (a 1 + ib 1 )| = a 2 . Simple geometric consideration shows that, if we take b 2 > x 1 then δ Ω (it) = a 2 for every t ∈ [x 1 , b 2 ]. Moreover, since g(t)/t → 0 as t → +∞, we can find b 2 > x 1 such that Therefore, there exist x 1 , b 2 such that (6.3) is satisfied for j = 1. Now, we can argue by induction is a similar way. Suppose we constructed b 1 , . . . , b j and x 1 , . . . , x j−1 for j > 1. Then we select x j in such a way that |ix j − (a j + ib j )| = a j+1 and, again since g(t)/t → 0 as t → +∞, we choose b j+1 > x j such that g(t) = +∞, and we are done.

Orthogonal speed of convergence of parabolic semigroups
In this section we give estimates on the orthogonal speed of convergence of semigroups. Since the orbits of hyperbolic semigroups converge nontangentially to the Denjoy-Wolff point, it follows from (5.2) that the total and the orthogonal speeds of hyperbolic semigroups have the same asymptotic behavior. Therefore, we concentrate on parabolic semigroups. In order to simplify the notation, for any α ∈ (0, π], we write V (α) := V (α, 0) = w = ρe iθ : ρ > 0, |θ| < α .
The first part of the following result follows immediately from the fact that h(D) is contained in the Koebe domain C \ {z ∈ C : Re z = Re p, Im z ≤ Im p}, where p ∈ C \ h(D) and Proposition 6.5. Whereas, the second part is a deep result in [6], where the analogue Euclidean expression is estimated using harmonic measure theory (and then the result in terms of speed follows from Proposition 3.8).  In general, we have the following bounds (which was proved in its Euclidean counterpart by D. Betsakos [5]): (1) Take a point p ∈ C \ h(D). Since h is starlike at infinity, h(D) ⊂ p + iV (π) and the result follows immediately from Theorem 7.1.

Open Questions
The previous results give rise to the following questions: Question 1: Suppose (φ t ) is a non-elliptic semigroup in D. Is it true that lim sup t→∞ [v T (t) − 1 2 log t] < +∞? Question 2: Suppose (φ t ) is a parabolic semigroup in D of positive hyperbolic step. Is it true that |v T (t)− 1 2 log t| < C for some constant C > 0? If so, does this condition characterize parabolic semigroups of positive hyperbolic step?