Angular extents and trajectory slopes in the theory of holomorphic semigroups in the unit disk

We study relationships between the asymptotic behaviour of a non-elliptic semigroup of holomorphic self-maps of the unit disk and the geometry of its planar domain (the image of the Koenigs function). We establish a sufficient condition for the trajectories of the semigroup to converge to its Denjoy-Wolff point with a definite slope. We obtain as a corollary two previously known sufficient conditions.


Introduction
A (one-parameter) semigroup (ϕ t ) t 0 of holomorphic self-maps of D -for short, a semigroup in D -is a continuous homomorphism t → ϕ t from the additive semigroup (R 0 , +) of non-negative real numbers to the semigroup Hol(D, D), • of holomorphic self-maps of D with respect to composition, endowed with the topology of uniform convergence on compacta. If ϕ t 0 is an automorphism of D for some t 0 > 0, then ϕ t is an automorphism for all t 0 and in such a case we will say that (ϕ t ) is a group, because indeed it can be extended to a group homomorphism R ∋ t → ϕ t ∈ Hol(D, D).
The theory of semigroups in D has a long history dating back to the early nineteen century. Moreover, nowadays, it is a flourishing branch of Analysis with strong connections with Dynamical Systems and with many applications in other areas (see [8] and the bibliography therein). Indeed, this paper is about a basic dynamical problem for semigroups in D. We refer the reader to [1], [8], or [17] for the results cited below without proof.
It is known that ϕ t 0 has a fixed point in D for some t 0 > 0 if and only if there exists τ ∈ D such that ϕ t (τ) = τ for all t 0. In such a case, the semigroup is called elliptic and there exists λ ∈ C with Re λ 0 such that ϕ ′ t (τ) = e −λ t for all t 0. The elliptic semigroup (ϕ t ) is a group if and only if Re λ = 0. Moreover, the above point τ is unique unless ϕ t = id D for all t 0, and it is called the Denjoy -Wolff point (DW-point in what follows) of the semigroup.
If the semigroup (ϕ t ) is not elliptic, then there exists τ ∈ ∂ D which is the Denjoy -Wolff point of ϕ t for all t > 0, i.e. ϕ t (τ) = τ and ϕ ′ t (τ) 1 in the sense of angular limits. As before τ is also called the Denjoy -Wolff point (DW-point in what follows) of the semigroup. In this case, there exists λ 0 such that ϕ ′ t (τ) = e −λ t for all t 0, where ϕ ′ t (τ) stands for the angular derivative of ϕ t at τ. A non-elliptic semigroup is said to be hyperbolic or parabolic depending on whether λ > 0 or λ = 0, respectively. Parabolic semigroups can be divided in two sub-types: a parabolic semigroup is of positive hyperbolic step if lim t→+∞ k D (ϕ t+1 (0), ϕ t (0)) > 0, where k D (·, ·) denotes the hyperbolic distance in D. Otherwise, (ϕ t ) is said to be of zero hyperbolic step.
A fundamental result for semigroups in D is the so called Continuous Denjoy -Wolff theorem which says that if (ϕ t ) is non-elliptic or elliptic but different from a group, then for any z ∈ D, ϕ t (z) → τ as t → +∞, where τ is the Denjoy -Wolff of the semigroup. Those functions t → ϕ t (z) can be properly named orbits (or trajectories) in the usual dynamical sense thanks to Berkson and Porta's celebrated theorem [4, Theorem (1.1)] which asserts that t → ϕ t (z) is real-analytic and there exists a unique holomorphic vector field G : D → C such that ∂ ϕ t (z) ∂t = G(ϕ t (z)), for all z ∈ D and all t 0.
For hyperbolic semigroups and parabolic semigroups of positive hyperbolic step, the arrival slope set is always a singleton (see [8,Sect. 17.4 and 17.5] for further information).
In contrast, for parabolic semigroups of zero hyperbolic step, the arrival slope set does not have to reduce to a unique point (see [10], [3], [14]). However, according to the following result by the first two authors, it does not depend on the initial point.
An (important) and open problem in the theory of semigroups in D has been (indeed, still is) how to detect whether the arrival slope set of a parabolic semigroup of zero hyperbolic step is a singleton or a specific kind of closed subinterval of − π 2 , π 2 . Here the word "detect" almost always means finding sufficient and/or necessary conditions of geometric nature. This is directly related to the second key notion (with the first one being the vector field) associated with each semigroup, namely, to its holomorphic model and its Koenigs function (see [4], [11], [18], [2], [8,Sect. 9]).
is a group of holomorphic automorphisms of U, and h : D → h(D) ⊂ U is a univalent holomorphic map (called a Koenigs function of the semigroup) satisfying the functional equation and the following absorbing property The set h(D) is called an associated planar domain of the semigroup.
Every semigroup in D admits a holomorphic model unique up to "holomorphic equivalence" (i.e., isomorphism of models). In particular, see e.g. [8,Theorem 9.3.5 on p. 245], a semigroup in D is non-elliptic if and only if one of its (mutually equivalent) holomorphic models is of the form (U, h, z → z + it). For such a holomorphic model, the functional equation (1.1) becomes Abel's classical equation Following the convention generally accepted in the literature, we will assume that all the considered holomorphic models for the non-elliptic semigroups are of the above canonical form. Then the Koenigs function becomes essentially unique: if h 1 , h 2 are two Koenigs functions of the same non-elliptic semigroup, then there exists a constant c ∈ C such that h 1 = h 2 + c.
Thanks to (1.3), planar domains of non-elliptic semigroups are complex domains of a very particular type: the so-called starlike-at-infinity domains.
Definition 1.5 A domain Ω ⊂ C is said to be starlike at infinity (in the direction of the imaginary axis) if Ω + it ⊂ Ω for any t 0.
Remark 1.6 Any domain Ω = C starlike at infinity is conformally equivalent to D and if h is a conformal mapping of D onto such a domain Ω , then the formula ϕ t := h −1 • (h + it) for t 0 defines a non-elliptic semigroup in D, whose Koenigs function is h.
In this context, our problem mentioned above can be rewritten as follows: to find geometrical properties of the planar domain of a parabolic semigroup of zero hyperbolic step which imply (or characterize) whether the corresponding arrival slope set is a singleton.
As far as we know, apart from examples and some folklore results concerning strong symmetry of the planar domain, the unique three papers dealing with the above question are [3], [5] and [7]. In [3], it is shown that whenever the boundary of the planar domain is included in a vertical half-strip, the arrival slope set is equal to {0}. Likewise, in [5], it is shown that if the boundary of the planar domain is included in a horizontal strip, the arrival slope set is also equal to {0}. In [7], the authors introduces some "boundary distance" functions, which measure the distance of a vertical straight line to the boundary of the planar domain, and use them to characterize geometrically when the arrival slope set coincides with the singleton {π/2} or {−π/2}. Moreover, they also show how these functions detect whether the convergence of the trajectories is non-tangential, i.e. whether the arrival slope set is a compact subset of − π 2 , π 2 . We would like to mention that there are also results treating the above problem in a non-geometrical way, i.e. without using planar domains. For instance, in [12] (see also [8,Proposition 7.5.5]), it is proved that the arrival slope sets of a parabolic semigroup of zero hyperbolic step is a singleton whenever its vector field has enough analytic regularity (in the angular sense) at its Denjoy -Wolff point.
In this paper, we introduce some new "angular extent" functions of a strongly geometrical meaning, which measure the angular displacement of the boundary of the planar domain with respect to a fixed vertical straight line (see Definition 3.5). Using these functions, we establish sufficient conditions for the arrival slope set of a nonelliptic semigroup to be a singleton (see Theorem 5.4 and Proposition 5.3). We also analyze the relationship between these functions and the non-tangential convergence of the orbits of the semigroup (see Proposition 5.1). Moreover, as a corollary, we recover results from [3] and [5].
The plan of the paper is as follows. In Section 2, we develop some new results about Carathéodory kernel convergence which can be of interest on their own and will be fundamental for the results of Section 5. In Section 3, we introduce and study those angular extent functions mentioned above. Section 4 is a brief review of the boundary distance functions introduced in [7]. We also study here their relationships with the angular extent functions from Section 3. In Section 5, we present our main results. Finally, in Section 6, we show a few examples dealing with some particularities of the angular extent functions, which, in particular, underline important differences between them and the (apparently quite similar) boundary distance functions.

Kernel convergence
Recall the classical notion of kernel convergence of domains; for more details see e.g. [13,§II.5] or [15, §1.4]. Let (Ω n ) be a sequence of domains in C. Fix a point ω ∈ C. Suppose that ω ∈ Ω n for all n ∈ N large enough. Denote by G the (possibly empty) set of all points z ∈ C possessing the following property: there exists an open connected set ∆ ⊂ C containing the points z and ω and contained in Ω n for all sufficiently large n ∈ N.
The kernel K (Ω n ), ω of the sequence (Ω n ) with respect to the point ω is the union G ∪ {ω}. The following dichotomy holds: either G = / 0 and hence, trivially, K (Ω n ), ω = {ω}, or K (Ω n ), ω = G = / 0. In the latter case, K (Ω n ), ω coincides with the connected component of n∈N int m n Ω m that contains ω. Here int(·) stands for the topological interior of a set.
As a matter of convenience, we also define the kernel of (Ω n ) w.r.t. points ω ∈ C that fail to belong to all but a finite number of Ω n 's. In such a case, we define K (Ω n ), ω := {ω} if there exists a sequence (ω n ) converging to ω with ω n ∈ Ω n for all n ∈ N; otherwise, we put K (Ω n ), ω := / 0. The kernel of (Ω n ) w.r.t. ω is said to be non-trivial if it is different from / 0 and {ω}. In such a case, K (Ω n ), ω is a domain in C containing ω. Otherwise, i.e. if K (Ω n ), ω ∈ / 0, {ω} , we say that the kernel of (Ω n ) w.r.t. ω is trivial. Note that for any subsequence (Ω n k ), K (Ω n k ), ω ⊃ K (Ω n ), ω and, in general, the inclusion can be strict. A sequence (Ω n ) is said to converge to its kernel Ω * := K (Ω n ), ω w.r.t. a point ω ∈ C, if Ω * = / 0 and K (Ω n k ), ω = Ω * for every subsequence (Ω n k ).
The above "sequential" concepts can be extended to continuous indexes in a natural way. Consider a family (Ω r ) r>0 of domains in C and let ω ∈ C. If for some r 0 > 0, a fixed neighbourhood of ω is contained in Ω r whenever r r 0 , then K (Ω r ), ω , the kernel of the family (Ω r ) w.r.t. ω, is defined as the connected component of that contains ω. Otherwise, we put K (Ω r ), ω := {ω} or K (Ω r ), ω := / 0 depending on whether there exists a map (0, +∞) ∋ r → ω r ∈ C such that ω r ∈ Ω r for all r > 0 and ω r → ω as r → +∞.
Remark 2.1 It follows easily from the definition, that if K ⊂ K (Ω r ), ω is a compact set, then K ⊂ Ω r for all r > 0 large enough. Conversely, if a domain U is contained in Ω r for all r > 0 large enough, then U ⊂ K (Ω r ), ω for any ω ∈ U. Analogous statements hold for kernels of sequences of domains.
In the proof of our main result, Theorem 5.4, we make use of the following statement, which is an easily corollary of Carathéodory's classical Kernel Convergence Theorem; see e.g. [13, Theorem 1 in §II.5].
Proposition 2.2 Let (g n ) be a sequence of conformal mappings of D into C. If (g n ) converges locally uniformly in D to some function g, then (g n (D)) converges to its kernel w.r.t. ω := g(0). Moreover, g(D) = K (g n (D)), ω .
If the kernel K (g n (D)), ω is non-trivial, then g is conformal and on every compact set K ⊂ g(D), the sequence (g −1 n ) converges uniformly to g −1 .
As a consequence of Remark 1.6, in this paper, we will be especially interested in domains starlike at infinity. Simple "model examples" of such domains, relevant to the slope problem, are represented by angular sectors of the form where p ∈ C and 0 β 1 , β 2 π with β 1 + β 2 > 0.

Remark 2.3
Clearly, when the above notions are applied to describing the limit behaviour of domains, much depends on the choice of the point ω involved in the definition of the kernel. Given a family (Ω r ) of domains and a sequence (r n ) ⊂ (0, +∞) tending to +∞, the limit behaviour of the sequence (Ω r n ) w.r.t. to some points ω ∈ C can be similar to that of the whole family (Ω r ), while for other choices of ω, (Ω r n ) and (Ω r ) can behave differently. Consider the following example. Let β ∈ (0, π], and define Ω r := 1 r Ω for all r > 0. It can be checked that if ω ∈ S 0 (0, β ), then K (Ω r ), ω = S 0 (0, β ), and for all ω ∈ C\S 0 (0, β ) the kernel K (Ω r ), ω is trivial. In particular, S 0 (0, β ) is the unique non-trivial kernel of the family (Ω r ). Moreover, (Ω r ) converges to its kernel S 0 (0, β ) w.r.t. any ω ∈ S 0 (0, β ).
It follows that the sequence (Ω 2 n ) converges to its kernel S 0 (0, β ) w.r.t. any ω ∈ S 0 (0, β ). However, (Ω 2 n ) has infinitely many other non-trivial kernels with respect to points in the left half-plane; namely, In fact, for any k ∈ Z, the sequence (Ω 2 n ) converges to its kernel D k w.r.t. ω k . Note also that in this example, Ω and hence all Ω r 's are starlike at infinity.
For families (Ω r ) generated, as in the above remark, by scaling a given domain Ω , the fact that the parameter r takes all positive real values imposes strong restrictions on possible non-trivial kernels.
Then the following assertions hold.
Proof of (C). Fix now ω ′ ∈ C and a sequence (r n ) ⊂ (0, +∞) converging to +∞ with the property that the kernel K (Ω r n ), ω ′ is non-trivial. Clearly As above, the domains D and D ′ := K (Ω r n ), ω ′ either coincide or do not intersect; moreover, D ′ + it ⊂ D ′ for all t 0. Since by hypothesis β 1 > 0 and β 2 > 0, it follows that D ′ = D. The proof is now complete.

Remark 2.5
The condition β 1 β 2 = 0 in part (C) of the above proposition is essential. Just consider the example in Remark 2.3.

Remark 2.6
Fix w 0 ∈ C and let Ω ⊂ C be a domain. If the family of domains Ω r := 1 r (Ω − w 0 ) has a non-trivial kernel with respect to a point ω ∈ C, then also the family Ω r := 1 r Ω has the same kernel with respect to ω. A similar assertion holds for any sequence (Ω r n ) with r n → +∞ as n → +∞. Indeed, every point is contained in Ω r ′ along with some fixed neighbourhood for all r ′ > 0 large enough. Hence {ξ : |ξ − w| < ε} ⊂ Ω r ′ for some ε > 0 and all r ′ > 0 large enough. Taking into account that the same holds also with Ω r ′ and Ω r ′ interchanged, we conclude that

Angular extent functions for domains starlike at infinity
Recall that a domain starlike at infinity is a domain Ω of the complex plane such that Ω + it ⊂ Ω for all t 0.
Definition 3.1 Let Ω be a domain starlike at infinity. A point p ∈ C is said to be a natural point associated with Ω if there exists t 0 0 such that p + it ∈ Ω , for all t > t 0 . The set of all natural points associated with Ω will be denoted by NP(Ω ).  Remark 3.4 A useful subset of NP(Ω ), which will be denoted by NP 0 (Ω ), is formed by those points p ∈ ∂ Ω such that p + it ∈ Ω , for all t > 0.
Note that NP 0 (Ω ) can be empty, but this happens only for a narrow class of domains of the form Moreover, NP 0 (Ω ) can be reduced to a unique point: consider, e.g.
Definition 3.5 Let Ω be a domain starlike at infinity. Fix p ∈ NP(Ω ). For any t > 0 such that p + it ∈ Ω , we define the (normalized) left angular extent of Ω w.r.t. p by Likewise, the (normalized) right angular extent of Ω w.r.t. p is defined as The natural domain of definition for both functions α − Ω ,p and α + Ω ,p is the interval Remark 3.7 It is easy to see that for any domain Ω starlike at infinity and any p ∈ NP(Ω ), both functions α − Ω ,p and α + Ω ,p are continuous from the right on their natural domain of definition. However, in general, neither α − Ω ,p nor α + Ω ,p are continuous from the left. For instance, for Ω : which is discontinuous from the left at t = √ 2.
Remark 3.8 In general, for different points p, q ∈ NP(Ω ), we have different functions α + Ω ,p and α + Ω ,q . The same holds for the left angular extends. However, quite often the angular extends w.r.t. different points behave the same way in the limit as t → +∞.
Let Ω be a domain starlike at infinity and p ∈ NP(Ω ). Let t 0 (p) be defined by (3.1). Then the three conditions below are equivalent to each other.
(i) The following limits exist, and at least one of them is different from zero.
and at least one of them is different from zero.
(iii) There exist ω ∈ C with respect to which the family Ω r := 1 r Ω converges to a non-trivial kernel.
Moreover, if one and hence all of the above conditions are satisfied, then: For the proof of this theorem we need one technical lemma.
Proof Note that the hypothesis implies that t belongs to the natural domain of definition of α − Ω ,p and α + Ω ,p . For simplicity, we will assume, without loss of generality, that p = 0.
Proof of (ii) =⇒ (iii). Denote We are going to show that under condition (ii) the following two claims hold.
Claim 2: for any sequence {r n } ⊂ (0, +∞) tending to +∞ and any w ∈ ∂ S 0 (α − , α + ) there is a sequence {w n } converging to w and satisfying w n ∈ C \ Ω r n for all n ∈ N.
Suppose now that α + = π. The above argument works also in this case if 0 ∈ Ω , but if 0 ∈ Ω , then it might happen that α + Ω ,0 (t) = π and ite −iα + Ω,0 (t) = −it ∈ Ω for all t > 0. Hence we have to modify the above argument. To this end, fix some w * ∈ C \ Ω . Then Γ := {w * − it : t 0} ⊂ C \ Ω . Removing a finite number of terms in (r n ), we may suppose that ρr n > |w * | for all n ∈ N. Define w n to be the unique point of intersection Γ {z : |z| = ρr n } lying in the lower half-plane. Then clearly ζ n := w n /r n → −iρ = w as n → +∞. By construction, ζ n ∈ C \ Ω r n for all n ∈ N. Now Claim 2 and hence the implication (ii) =⇒ (iii) is proved.

Proof of (a) and (b).
We have already seen that if conditions (i) -(iii) hold, then the equality K (Ω r ), ω = S 0 α − (p), α + (p) takes place for at least one point ω ∈ C. By Proposition 2.4, this implies assertion (a). Now (b) follows from (a) and the fact that condition (iii) does not depend on the choice of the point p and take into account assertion (a). ⊓ ⊔

Remark 3.11
If Ω is a domain starlike at infinity with NP(Ω ) = C, then one of the following three mutually exclusive possibilities holds.

Remark 4.3
In contrast to the angular extents α ± Ω ,p , for any domain Ω starlike at infinity and any p ∈ C, the functions δ ± Ω ,p are continuous and non-decreasing on the whole interval (0, +∞).

Theorem 4.4 [7, Lemma 3.6]
Let Ω be a domain starlike at infinity. Then for any p, q ∈ C, there exist constants c 2 > c 1 > 0 such that for all t > 0, The following result obtained in [7] establishes a strong relationship between the slopes of the trajectories of a one-parameter semigroup at its DW-point and the limit behaviour of the boundary distance functions of the corresponding planar domain.
(B) lim n→+∞ Arg(1 − τϕ t n (z)) = π/2 (in particular, (ϕ t n (z)) converges tangentially to τ as n → +∞) for some (and hence all) z ∈ D if and only if for some (and hence all) p ∈ Ω , (C) lim n→+∞ Arg(1 − τϕ t n (z)) = −π/2 (in particular, (ϕ t n (z)) converges tangentially to τ as n → +∞) for some (and hence all) z ∈ D if and only if for some (and hence all) p ∈ Ω , As one might expect, for domains starlike at infinity, angular extent functions and boundary distance functions are closely related, see Proposition 4.6 below. At the same time, it is worth to mention that these two characteristics are not asymptotically equivalent, as demonstrated by Example 6.4 in the last section. Therefore, the information on the geometry of the planar domain near ∞ provided by the angular extents is not identical to that contained in the boundary distance functions.

Main results
In this section we prove our main results, which establish relationships between the trajectory slopes at the DW-point and the asymptotic behaviour of the angular extents in the planar domain of the semigroup for t → +∞.
As we mentioned in the introduction, essentially the slope problem has been solved for hyperbolic semigroups and for parabolic semigroups of positive hyperbolic step. Therefore, we might strict our attention to parabolic semigroups of zero hyperbolic step. At the same time, our methods do not require this assumption. That is why we will keep supposing only that the semigroup is non-elliptic.
Suppose now that condition (i) is satisfied. Then applying again Theorem 4.5 (A), we see that for there exists T t 0 and constants c 2 > c 1 > 0 such that Combining these inequalites with (5.1) and Proposition 4.6, we find that for all t > T . It follows that (5.2) holds with C 1 := c 1 π −1 ε and C 2 := ε −1 πc 2 .
⊓ ⊔ Remark 5.2 Example 6.5 in the next section shows that condition (5.1) in Proposition 5.1 is essential.

Koenigs function h, and planar domain Ω := h(D). Fix some p ∈ NP(Ω ) and denote
Then the following assertions hold: (A) If α − p > 0 and α + p > 0, then the trajectories t → ϕ t (z) converge to τ non-tangentially for all z ∈ D.
Proof Assertion (A) is a corollary of Proposition 5.1. Indeed, using a simple observation that we see that under the hypothesis of (A), for all t > 0 large enough we have which implies (5.1), and (i) There exists the limit θ := lim n→+∞ Arg(1 − τϕ t n (z)) ∈ − π 2 , π 2 .
(ii) There exists the limit m := lim (iii) There exists the limit µ := lim n→+∞ Im(τϕ t n (z)) Moreover, if one and hence all of the above hold, then e iθ = m and µ = − tan θ .
For such a sequence (a n ) consider the following automorphisms of D, T n (z) := a n + z 1 + a n z , z ∈ D, and univalent functions Therefore, (F n ) is a normal family in D. Since by construction |F n (0)| = 1 for all n ∈ N, (F n ) is indeed relatively compact in Hol(D, C). Moreover, by Hurwitz's Theorem, any accumulation point of (F n ) is either a constant or a univalent function in D. Let g : D → C be one of those accumulation points, i.e. suppose that g is the limit of a some subsequence (F n k ). Denote g k := F n k , k ∈ N.
Since 0 ∈ ∂ g k (D) for all k ∈ N, inequality (5.6) with ρ := a n k and s = s(x) := T n k (x) leads to On the other hand, applying again (5.6) with ρ = ρ(x) := T n k (x) and s := a n k , we have for all x ∈ [0, 1). (5.8) Recall that |g k (0)| = 1 for all k ∈ N. Hence from (5.7) with x = 0, we obtain Therefore, see e.g. [8, Theorem 3.4.9], It follows that g cannot be constant and thus it is univalent in D. In particular, by Proposition 2.2, this means that the sequence of domains D k := g k (D) converges to a non-trivial kernel D * w.r.t. g(0) and that g(D) = D * . Denote Ω r := 1 r Ω , r > 0. On the one hand, by Theorem 3.9, there exists ω ∈ C w.r.t. which (Ω r ) converges to its kernel K (Ω r ), ω = S 0 (α − , α + ).
Therefore, according to the Riemann Mapping Theorem, for a suitable U ∈ Aut(D). We can determine U using (5.7) and (5.8). Indeed, passing in these inequalities to the limit as k → +∞ and taking into account that |g k (0)| = 1 for all k ∈ N and that lim n→+∞ a n = 1, we get It follows that g(x) → ∞ as x → −1 + and g(x) → 0 as x → 1 − . Taking into account that |g(0)| = 1 and using (5.9), we therefore conclude that U = id D .
We have proved that every converging subsequence of (F n ) has the same limit. Recalling that (F n ) is a normal family in D, we may conclude that (F n ) converges locally uniformly in D to (5.10) Note that i ∈ F n (D) for all n ∈ N and that i ∈ F(D). Hence by Proposition 2.2, Furthermore, by (5.5) with z := z n , for all n ∈ N, we have 1 − H −1 (ix n ) = 1 − T n (z n ) = (1 − a n )(1 − z n )/(1 + a n z n ). Therefore, Finally, according to (5.10), we have This completes the proof. ⊓ ⊔ Now, we are going to apply the above results to domains starlike at infinity whose boundary is contained in a "neighbourhood" of the boundary of a sector S p (β 1 , β 2 ).
Since Corollaries 5.10 and 5.11 follow directly from Corollary 5.8, we only need to prove the latter one. Two examples making use of Corollary 5.8 with a non-constant function ρ can be found at the beginning of Section 6.
Proof of Corollary 5.8 Clearly, without loss of generality we may assume that p = 0.
Case 3: 0 < β 1 < π, β 2 = π. As in the previous case, we see that A R 0 ⊂ Ω . Thanks to (5.13), we also have B R 1 ⊂ C \ Ω . The rest of the proof is the same as in Case 2.
In particular, δ + Ω ,0 (t) and t → tα + Ω ,0 (t) are not asymptotically equivalent at +∞. ⊓ ⊔ The next example shows that it is not possible to get a result similar to Theorem 4.5 using the functions α ± Ω ,p instead of δ ± Ω ,p .
Example 6.5 There exists a parabolic semigroup (ϕ t ) in D of zero hyperbolic step with the associated planar domain Ω and a sequence (t n ) ⊂ (0, +∞) tending to +∞ such that (ϕ t n (z)) converges to the DW-point of the semigroup non-tangentially, but α − Ω ,0 (t n ) and α + Ω ,0 (t n ) are not asymptotically equivalent.