Asymptotic monotonicity of the orthogonal speed and rate of convergence for semigroups of holomorphic self-maps of the unit disc
aa r X i v : . [ m a t h . C V ] A p r ASYMPTOTIC MONOTONICITY OF THE ORTHOGONAL SPEED AND RATE OFCONVERGENCE FOR SEMIGROUPS OF HOLOMORPHIC SELF-MAPS OF THEUNIT DISC
FILIPPO BRACCI † , DAVIDE CORDELLA † , AND MARIA KOUROUA BSTRACT . We show that the orthogonal speed of semigroups of holomorphic self-maps of theunit disc is asymptotically monotone in most cases. Such a theorem allows to generalize previousresults of D. Betsakos and D. Betsakos, M. D. Contreras and S. D´ıaz-Madrigal and to obtain newestimates for the rate of convergence of orbits of semigroups. C ONTENTS
1. Introduction 12. Semigroups in the unit disc 43. Speeds of semigroups 54. Orthogonal Speed and Harmonic Measure 85. Estimates of harmonic measures 96. Asymptotic monotonicity of orthogonal speed 107. Some applications 138. Final remarks and open questions 19References 201. I
NTRODUCTION
The theory of continuous semigroups of holomorphic self-maps of the unit disc D : = { z ∈ C : | z | < } —or just, for short, semigroups in D —is a flourishing subject of study since theearly nineteen century, both as a subject by itself and for many different applications, see, e.g.,[1, 2, 3, 10, 14, 18, 19] and bibliography therein.In this paper we are interested in considering the so-called “rate of convergence” of the orbitsof a non-elliptic semigroup in D to its Denoy-Wolff point. Estimates for the rate of convergenceof an orbit of a non-elliptic semigroup in D have been obtained in [5, 6, 7, 12, 13, 15, 16, 17]. Mathematics Subject Classification.
Primary 37C10, 30C35; Secondary 30D05, 30C80, 37F99, 37C25.
Key words and phrases.
Semigroups of holomorphic functions; hyperbolic geometry; dynamical systems. † Partially supported by PRIN Real and Complex Manifolds: Topology, Geometry and holomorphic dynam-ics n.2017JZ2SW5, by INdAM and by the MIUR Excellence Department Project awarded to the Department ofMathematics, University of Rome Tor Vergata, CUP E83C18000100006.
In particular, in [5], D. Betsakos proved that if ( φ t ) is a semigroup in D with Denjoy-Wolffpoint τ ∈ ∂ D , then there exists a constant K > | φ t ( ) − τ | ≤ Kt − / , t ≥ . The point 0 can be easily replaced with any z ∈ D . However, the exponent − / t is sharp, andcan be replaced with − ( φ t ) is either hyperbolic or parabolic with positive hyperbolicstep.In [7, Thm. 5.3] (see also [10, Thm. 16.3.1]), D. Betsakos, M. D. Contreras and S. D´ıaz-Madrigal, got an estimate of the previous type with the exponent − / − πα + β incase the image of the Koenigs function of the semigroup is contained in a sector of the form W α , β : = p + i { re i θ : r > , − α < θ < β } with α , β ∈ [ , π ] and α + β > p ∈ C .In [8] (see also [10, Ch. 16]), the first named author introduced three quantities, called speeds,which are defined in intrinsic terms using the hyperbolic distance and showed that the previousestimates can be translated in terms of one of such speeds. To be more concrete, let τ ∈ ∂ D be the Denjoy-Wolff point of ( φ t ) . Let π ( φ t ( )) ∈ ( − , ) τ be the closest point to φ t ( ) in thesense of hyperbolic distance k D in D . For t ≥
0, we let v o ( t ) : = k D ( , π ( φ t )) , and call it the orthogonal speed of ( φ t ) . It can be shown that v o ( t ) ∼ − log | τ − φ t ( ) | , andtherefore (1.1) can be translated in(1.2) lim inf t → + ∞ [ v o ( t ) −
14 log t ] > − ∞ , and, similarly, the estimate in [7, Thm. 5.3] can be obtained by replacing by π ( α + β ) .Now, in [8, Prop. 6.5] (see also [10, Cor. 16.2.6]) it is proved that the orthogonal speed of asemigroup whose image under the Koenigs function is a sector W α , β , goes like − π ( α + β ) log t as t → + ∞ . Therefore, (1.2) and [7, Thm. 5.3] can be rephrased as lim inf t → + ∞ [ v o ( t ) − w o ( t )] > − ∞ , where w ( t ) is the orthogonal speed of the semigroup whose image under the Koenigsfunction is a sector W α , β . Hence, the following natural question was raised in [8] (see Question4in [8, Sec. 8]):Question: Let ( φ t ) and ( ˜ φ t ) be non-elliptic semigroups in D with Koenigs functions h and ˜ h ,respectively, and denote by v o ( t ) (resp. ˜ v o ( t ) ) the orthogonal speed of ( φ t ) (resp. ( ˜ φ t ) ). Assume h ( D ) ⊂ ˜ h ( D ) . Is it true that lim inf t → + ∞ [ v o ( t ) − ˜ v o ( t )] > − ∞ ?In other words, is the orthogonal speed asymptotically monotone?In this paper we give a (partial) affirmative answer to the previous question. In particular, weprove that if one replaces the lim inf with lim sup, the answer is always yes. SYMPTOTIC MONOTONICITY 3
Theorem 1.1.
Let ( φ t ) , ( ˜ φ t ) be semigroups in D . Let h ( respectively , ˜ h) be the Koenigs functionof ( φ t ) ( resp. of ( ˜ φ t ) ). Suppose that h ( D ) ⊂ ˜ h ( D ) . Then lim sup t → + ∞ [ v o ( t ) − ˜ v o ( t )] > − ∞ , or, equivalently, lim inf t → + ∞ | φ t ( ) − τ || ˜ φ t ( ) − ˜ τ | < + ∞ , where τ ∈ ∂ D is the Denjoy-Wolff point of ( φ t ) and ˜ τ ∈ ∂ D is the Denjoy-Wolff point of ( ˜ φ t ) . Also, we are able to provide a (complete) affirmative answer to the question in many cases:
Theorem 1.2.
Let ( φ t ) , ( ˜ φ t ) be semigroups in D . Let h ( respectively , ˜ h) be the Koenigs functionof ( φ t ) ( resp. of ( ˜ φ t ) ). Suppose that h ( D ) ⊂ ˜ h ( D ) and that (1) either h ( D ) is quasi-symmetric with respect to vertical axes, (2) or, ˜ h ( D ) is quasi-symmetric with respect to vertical axes, (3) or, ˜ h ( D ) is starlike with respect to some w ∈ ˜ h ( D ) .Then lim inf t → + ∞ [ v o ( t ) − ˜ v o ( t )] > − ∞ , or, equivalently, there exists K > such that for all t ≥ , | φ t ( ) − τ | ≤ K | ˜ φ t ( ) − ˜ τ | , where τ ∈ ∂ D is the Denjoy-Wolff point of ( φ t ) and ˜ τ ∈ ∂ D is the Denjoy-Wolff point of ( ˜ φ t ) . Here, we say that a starlike at infinity domain Ω is quasi-symmetric with respect to verticalaxes if there exists K > K − δ − ( t ) ≤ δ + ( t ) ≤ K δ − ( t ) , for all t ≥
0, where for some z ∈ Ω , we denote by˜ δ + ( t ) : = inf {| w − ( z + it ) | : Re w ≥ Re z , w ∈ C \ Ω } , ˜ δ − ( t ) : = inf {| w − ( z + it ) | : Re w ≤ Re z , w ∈ C \ Ω } , and δ ± ( t ) : = min { t , ˜ δ ± ( t ) } .It was proved in [9] that h ( D ) is quasi-symmetric with respect to vertical axes if and only if ( φ t ( )) converges non-tangentially to the Denjoy-Wolff point.Condition (3) in Theorem 1.2 is clearly satisfied by the sectors of type W α , β , with α , β ∈ [ , π ] and α + β >
0, hence, our theorem generalizes the results in [5] and [7, Thm 5.3].In [7, Thm. 5.6], the authors get some estimates in the case where h ( D ) = W α , β , with α , β ∈ [ , π ] and α + β >
0. Indeed, they prove that,(1) if α , β >
0, then there exists K > | ˜ φ t ( ) − τ | ≥ Kt − / ( α + β ) for all t ≥ α = β =
0, then there exists K > | ˜ φ t ( ) − τ | ≥ Kt − − / ( α + β ) ,for all t ≥ F. BRACCI, D. CORDELLA, AND M. KOUROU
Now, if α , β >
0, then h ( D ) = W α , β is quasi-symmetric with respect to vertical axes and then theresult can be obtained also from Theorem 1.2 (and the explicit computation of the orthogonalspeed of ( φ t ) ). While, if either α = β =
0, the picture does not enter into the hypotheses ofTheorem 1.2 because h ( D ) is not quasi-symmetric with respect to the vertical axes and we haveno information on ˜ h ( D ) . However, the estimate (2) in Theorem 1.2 is not a relation between theorthogonal speeds of ( φ t ) and ( ˜ φ t ) (but between the orthogonal speeds of ( ˜ φ t ) and the totalspeedof ( φ t ) ), and can be also obtained by the methods illustrated in this paper (see Remark 3.2).The proof of Theorem 1.2 is based on harmonic measure theory. Suppose Ω ⊂ C is a simplyconnected domain. The harmonic measure at a point w ∈ Ω with respect to D ⊂ ∂Ω is denotedby ω ( w , D , Ω ) . In Proposition 4.2, we prove that there exists a constant K > t ≥ | v ( t ) +
12 log ω ( , A t , D ) | ≤ K , where A t is defined as follows. For t ≥
1, let a t ∈ ∂ D ∩ { Im z > } be the intersection of ∂ D with the circle containing τφ t ( ) , orthogonal to ( − , ) and orthogonal to ∂ D at a t . Then let˜ A t ⊂ ∂ D be the closed arc containing 1 with end points a t and a t . Define A t : = τ ˜ A t .Then, in Section 5, we give some estimates of harmonic measures, based on Gaier’s Theoremand the Strong Markov Property. With these tools at hand, in the fundamental Lemma 6.1, weshow the (almost) monotonicity of the orthogonal speed, in the case where a certain harmonicmeasure along the orbit of the semigroup is bounded from below by zero. This lemma allowsus to prove Theorem 1.1 and Theorem 6.2, which is a more general version of Theorem 1.2(and, from which, Theorem 1.2 follows). In Section 7, we give some applications of our results.In particular, we discuss the rate of convergence in case the image of the Koenigs functioncontains/is contained in domains of type Π α : = { z ∈ C : Im z > | Re z | α } for α > Ξ ( α , θ ) : = ( − H ∩ Π α ) ∪ W ( θ ) , where W ( θ ) : = (cid:8) z ∈ C | arg ( z ) ∈ (cid:0) π − θ , π (cid:1)(cid:9) .We end the paper with Section 8 containing some open questions originating from this work.2. S EMIGROUPS IN THE UNIT DISC
In this section we briefly recall the basics of the theory of semigroups of holomorphic self-maps of the unit disc, as needed for our aims. We refer the reader to the books [1, 10, 14, 18]for details.
Definition 2.1.
A continuous semigroup ( φ t ) of holomorphic self-maps of D , or just a semi-group in D for short, is a semigroup homeomorphism between the semigroup of real non-negative numbers (with respect to sum) and the semigroup of holomorphic self-maps of D (withrespect to composition). Here, as usual, the chosen topology for R + is the Euclidean topologyand the space of holomorphic self-maps of D is endowed with the topology of uniform conver-gence on compacta.A semigroup ( φ t ) without fixed points in D is called non-elliptic. If ( φ t ) is a non-ellipticsemigroup, φ t has the same Denjoy-Wolffpoint τ ∈ ∂ D , for all t >
0. Moreover, lim t → + ∞ φ t ( z ) = τ ∈ ∂ D , for all z ∈ D . SYMPTOTIC MONOTONICITY 5
Let ( φ t ) be a non-elliptic semigroup in D . Up to conjugate with a rotation, we can assumethat the Denjoy-Wolff point of ( φ t ) is 1. The Denjoy-Wolff Theorem (see, e.g. [10, Thm. 1.8.4])implies that(2.1) φ t ( E ( , R )) ⊆ E ( , R ) , for all t ≥ R >
0, where E ( , R ) : = { z ∈ D : | − z | < R ( − | z | ) } .Let us denote the right half-plane by H : = { w ∈ C : Re w > } and let C : D → H be the Cayley transform defined by C ( z ) = ( + z ) / ( − z ) . Then (2.1) impliesthat for all s ≥ t ≥ Re ( C ( φ s ( ))) ≥ Re ( C ( φ t ( ))) . This is, in fact, the Denjoy-Wolff Theorem version in H (see also [10, Thm. 1.7.8]).If ( φ t ) is a non-elliptic semigroup in D , then there exists a (essentially unique) univalentfunction h : D → C such that(1) h ( φ t ( z )) = h ( z ) + it for all z ∈ D , t ≥ S t ≥ ( h ( D ) − it ) = Ω , where Ω is either a vertical strip, or a vertical half-plane or C .The function h is called the Koenigsfunction of ( φ t ) .3. S PEEDS OF SEMIGROUPS
Speeds of non-elliptic semigroups in D have been introduced in [8] (see also [10, Ch. 16]).We recall here the basic facts needed.Let τ ∈ ∂ D and let Γ : = ( − , ) τ . Then Γ is a geodesic for the hyperbolic distance k D in D .For every z ∈ D , there exists a unique point, π ( z ) ∈ Γ such that k D ( z , π ( z )) = min { k D ( z , w ) : w ∈ Γ } . Definition 3.1.
Let ( φ t ) be a non-elliptic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D . The(total)speed v ( t ) of ( φ t ) is v ( t ) : = k D ( , φ t ( )) , t ≥ . The orthogonalspeed v o ( t ) of ( φ t ) is v o ( t ) : = k D ( , π ( φ t ( ))) , t ≥ . The tangentialspeed v T ( t ) of ( φ t ) is v T ( t ) : = k D ( φ t ( ) , π ( φ t ( ))) , t ≥ . As a consequence of the “Pythagoras’ Theorem in hyperbolic geometry”, we have the fol-lowing relation for all t ≥ v o ( t ) + v T ( t ) −
12 log 2 ≤ v ( t ) ≤ v o ( t ) + v T ( t ) . F. BRACCI, D. CORDELLA, AND M. KOUROU
Also, as a consequence of the Julia’s Lemma and (3.1) (see, [10, eq. (16.1.3)] or [8, eq. (5.3)])(3.2) v T ( t ) ≤ v o ( t ) + . Moreover, the speeds of a semigroup are related to certain quantities, whose asymptotic esti-mates go under the name “rate of convergence” of a semigroup. For all t ≥
0, we have (cid:12)(cid:12)(cid:12)(cid:12) v ( t ) −
12 log 11 − | φ t ( ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤
12 log 2 , (cid:12)(cid:12)(cid:12)(cid:12) v o ( t ) −
12 log 1 | τ − φ t ( ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤
12 log 2 , (cid:12)(cid:12)(cid:12)(cid:12) v T ( t ) −
12 log | τ − φ t ( ) | − | φ t ( ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤
32 log 2 . (3.3)Since the definition of the speeds is given in hyperbolic terms, the speeds are invariant underconformal changes of coordinates. In particular, one can check that if ( φ t ) is a non-ellipticsemigroup in D with Denjoy-Wolff point 1 and C : D → H is the Cayley transform C ( z ) =( + z ) / ( − z ) , then (see [8, eq. (5.1)] or [10, Sec. 6.5]) the orthogonal speed of ( φ t ) is(3.4) v o ( t ) = k H ( , ρ t ) =
12 log ρ t , where we let C ( φ t ( )) = ρ t e i θ t for some ρ t > θ t ∈ ( − π , π ) , t ≥ v ( t ) ≤ v o ( t ) + . Since v ( t ) → + ∞ , as t → + ∞ , (because φ t ( ) → τ ∈ ∂ D ), it follows that lim t → + ∞ v o ( t ) = + ∞ and, in particular, lim t → + ∞ ρ t = + ∞ . Remark . Let ( φ t ) , ( ˜ φ t ) be semigroups in D . Let h (respectively, ˜ h ) be the Koenigs functionof ( φ t ) (resp. of ( ˜ φ t ) ). Let v ( t ) , v o ( t ) and ˜ v ( t ) , ˜ v o ( t ) denote the total and the orthogonal speedsof ( φ t ) and ( ˜ φ t ) , respectively. Suppose that h ( D ) ⊂ ˜ h ( D ) . Then clearly v ( t ) ≤ ˜ v ( t ) . Moreover,by (3.1), ˜ v o ( t ) ≤ ˜ v ( t ) +
12 log 2 ≤ v ( t ) + . Hence, lim inf t → + ∞ [ v ( t ) − ˜ v o ( t )] > − ∞ . For instance, if h ( D ) = W α , β , with α , β ∈ [ , π ] and α + β >
0, but either α = β =
0, then v ( t ) ∼ π + α + β ( α + β ) log t (see [8, Prop. 6.5] or [10, Cor. 16.2.6]). From this, condition (2) of [7, Thm.5.6] follows.It is presently unknown if we can replace π + α + β ( α + β ) log t with (the more natural) estimate v o ( t ) ∼ π ( α + β ) log t . SYMPTOTIC MONOTONICITY 7
In the final part of this section, we give some geometric conditions on the image of theKoenigs function of a semigroup, which assures that v o ( t ) is a non-decreasing function of t . Lemma 3.3.
Let ( φ t ) be a non-elliptic semigroup in D . Suppose that v ( t ) ≥ v ( t ) , for somet ≥ t ≥ . Then v o ( t ) ≥ v o ( t ) .Proof. Suppose t > t and v ( t ) ≥ v ( t ) . We can assume that the Denjoy-Wolff point of ( φ t ) is1. Let C ( z ) : = + z − z be the Cayley transform from D to H . Let ρ t e i θ t : = C ( φ t ( )) , with ρ t > θ t ∈ ( − π / , π / ) , t ≥
0. Then, v ( t ) = k H ( , ρ t e i θ t ) . By (2.2),(3.5) ρ t cos θ t ≥ ρ t cos θ t ≥ . This implies that ρ t e i θ t belongs to the set { w ∈ C : Re w ≥ ρ t cos θ t } .Let D ( , v ( t )) : = { w ∈ H : k H ( , w ) < v ( t ) } , which is a Euclidean disc of center a realnumber r ∈ ( , + ∞ ) , containing 1 in its interior and ρ t e i θ t on its boundary (in fact, the centeris cosh ( v ( t )) and the radius sinh ( v ( t )) = | cosh ( v ( t )) − ρ t e i θ t | , but we do not need thisexplicit computation). In particular, ∂ D ( , v ( t )) contains both ρ t e i θ t and ρ t e − i θ t . From this,a simple geometric consideration shows that { w ∈ C : Re w ≥ ρ t cos θ t , | w | ≤ ρ t } ⊂ D ( , v ( t )) . From the hypothesis, since ρ t e i θ t D ( , v ( t )) , the previous equation together with (3.5) im-plies immediately that ρ t ≥ ρ t , and, hence, v o ( t ) ≥ v o ( t ) . (cid:3) Proposition 3.4.
Let ( φ t ) be a non-elliptic semigroup with Koenigs function h. If h ( D ) is convex,then [ , + ∞ ) ∋ t v ( t ) is non-decreasing.Proof. Let 0 ≤ t ≤ t and assume by contradiction that v ( t ) < v ( t ) . Note that v ( t ) = k D ( , φ t ( )) = k h ( D ) ( h ( ) , h ( ) + it ) . Hence, if v ( t ) < v ( t ) , it follows that h ( ) + it ∈ D ( h ( ) , v ( t )) : = { w ∈ C : k h ( D ) ( h ( ) , w ) < v ( t ) } . Since the hyperbolic distance in a con-vex domain is a convex function, it follows that the hyperbolic discs are convex. Therefore,if h ( ) + it ∈ D ( h ( ) , v ( t )) , since h ( ) ∈ D ( h ( ) , v ( t )) as well, it follows that h ( ) + is ∈ D ( h ( ) , v ( t )) for all s ∈ [ , t ] . However, h ( ) + it ∈ D ( h ( ) , v ( t )) and equivalently, v ( t ) = k h ( D ) ( h ( ) , h ( ) + it ) < v ( t ) . We are led to a contradiction. (cid:3) More generally, we have the following result.
Proposition 3.5.
Let ( φ t ) be a non-elliptic semigroup with Koenigs function h. If h ( D ) is starlikewith respect to h ( ) then [ , + ∞ ) ∋ t v ( t ) is non-decreasing.Proof. Since the hyperbolic discs centered at h ( ) are also starlike with respect to h ( ) (see,e.g., [11, Thm. 2.10]), the proof is similar to the proof of Proposition 3.4 and we omit it. (cid:3) F. BRACCI, D. CORDELLA, AND M. KOUROU
4. O
RTHOGONAL S PEED AND H ARMONIC M EASURE
Lemma 4.1.
Let ( φ t ) be a non-elliptic semigroup in D with Denjoy-Wolff point . LetC ( z ) : = + z − z be the Cayley transform from D to H . Let ρ t e i θ t : = C ( φ t ( )) , with ρ t > and θ t ∈ ( − π / , π / ) , t ≥ . There exists K > such that for all t ≥ , (cid:12)(cid:12)(cid:12)(cid:12) v o ( t ) +
12 log ω ( , Θ t , H ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ K , where Θ t : = { iy : | y | ≥ ρ t } = { iy : | y | ≥ | C ( φ t ( )) |} .Proof. Let ω t : = ω ( , Θ t , H ) . By [10, Example 7.2.5], ω t = π Arg (cid:18) i ρ t − + i ρ t (cid:19) . Since lim t → + ∞ ρ t = + ∞ , there exists t > ρ t >
1, for all t ≥ t . Hence, from theprevious formula, for all t ≥ t , ω t = π arctan 2 ρ t ρ t − . Moreover, there exists t ≥ t such that ρ t ρ t − <
1, for all t ≥ t . For y ∈ [ , ] , we know that π y ≤ arctan y ≤ y . Hence, there exist constants 0 < c < c such that for all t ≥ t , c ρ t ≤
12 1 ρ t − ρ t ≤ ω t ≤ π ρ t − ρ t ≤ c ρ t . The above inequality and (3.4) lead to the result at once. (cid:3)
Let ( φ t ) be a non-elliptic semigroup in D with Denjoy-Wolff point 1. Note that, by theDenjoy-Wolff Theorem (see, e.g., [10, Thm. 1.8.4]), for every t > Re φ t ( ) >
0. Bearingthis in mind, we can state the following Proposition.
Proposition 4.2.
Let ( φ t ) be a non-elliptic semigroup in D with Denjoy-Wolff point . For t ≥ ,let a t ∈ ∂ D ∩ { Im z > } be the intersection of ∂ D with the circle containing φ t ( ) , orthogonalto ( − , ) and orthogonal to ∂ D at a t . Let A t ⊂ ∂ D be the closed arc containing and with endpoints a t and a t . Then there exists a constant K > such that (cid:12)(cid:12)(cid:12)(cid:12) v ( t ) +
12 log ω ( , A t , D ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ K , for all t ≥ . Proof.
It follows at once from Lemma 4.1 and the conformal invariance of the harmonic mea-sure under the Cayley transform. (cid:3)
SYMPTOTIC MONOTONICITY 9
5. E
STIMATES OF HARMONIC MEASURES
In all this section, ( φ t ) denotes a non-elliptic semigroup in D with Denjoy-Wolff point 1.Let C ( z ) : = + z − z be the Cayley transform from D to H . Let ρ t e i θ t : = C ( φ t ( )) , with ρ t > θ t ∈ ( − π / , π / ) , t ≥
0. For t ≥
1, let Γ t : = { ρ s e i θ s : s ≥ t } and Γ ∗ t : = { iy : | y | ≥ min s ≥ t ρ s } . In addition, set Θ t : = { iy : | y | ≥ ρ t } and note that Γ ∗ t = Θ t if and only if ρ s ≥ ρ t for all s ≥ t . Lemma 5.1.
There exists an increasing sequence { t n } , with t ≥ , converging to + ∞ such that Θ t n = Γ ∗ t n , for all n.Proof. Since [ , + ∞ ) ∋ t ρ t is continuous and lim t → + ∞ ρ t = + ∞ , there exists t ≥ ρ s ≥ ρ t for all s ≥ t . Then, by induction, we take t n ≥ t n − + [ t n − + , + ∞ ) ∋ t ρ t . (cid:3) Lemma 5.2.
Let t ≥ . For all s ≥ t, ω ( ρ s e i θ s , Γ ∗ t , H ) ≥ . Proof.
Let t ≥ t be such that ρ t : = min { ρ s : s ≥ t } . By definition, Γ ∗ t = { iy : | y | ≥ ρ t } . Considerthe automorphism T : H → H give by T ( w ) : = w ρ t . Let G : = { iy : | y | ≥ } . By the conformalinvariance of harmonic measure, we have ω ( ρ s e i θ s , Γ ∗ t , H ) = ω (cid:18) ρ s ρ t e i θ s , G , H (cid:19) . Since ρ s ρ t ≥
1, we have ω ( ρ s ρ t e i θ s , G , H ) ≥ / (cid:3) Lemma 5.3.
Fix θ ∈ ( , π / ) . Then there exists C = C ( θ ) > such that for all t ≥ , with | θ t | ≤ θ , we have ω ( ρ s e i θ s , Θ t , H ) ≥ C , for all s ≥ t.Proof. By (2.2), for every s ≥ t , ρ s cos θ s ≥ ρ t cos θ t . Therefore, ρ s ρ t e i θ s ∈ { w ∈ H : Re w > cos θ t } .Hence, repeating the argument in Lemma 5.2 with t = t , we obtain ω ( ρ s e i θ s , Θ t , H ) = ω (cid:18) ρ s ρ t e i θ s , G , H (cid:19) > C ( θ ) > , where C ( θ ) : = min { ω ( w , G , H ) : Re w > cos θ } . (cid:3) Lemma 5.4.
For all t ≥ , ω ( , Θ t , H ) < ω ( , Γ t , H \ Γ t ) . Proof.
This is essentially a consequence of the Hall’s (or Gaier’s) Theorem. To give some de-tails, let a t ∈ ∂ D and A t ⊂ ∂ D be as in Proposition 4.2. Let A ′ t ⊂ A t be the arc with end points 1and a t . Let W t : = { φ s ( ) : s ≥ t } , t ≥
1. Then, by Gaier’s Theorem (see, e.g., [10, Thm. 7.2.13]),for all t ≥ ω ( , W t , H \ W t ) > ω ( , A ′ t , H ) . Now, by definition of harmonic measure (or see, e.g. [10][Eq. (7.1.2)]), denoting by ℓ ( A ′ t ) theEuclidean length of A ′ t , we have ω ( , A ′ t , H ) = π ℓ ( A ′ t ) = π ℓ ( A t ) = ω ( , A t , H ) . Therefore, ω ( , W t , H \ W t ) > ω ( , A t , H ) . Using the conformal invariance of the harmonicmeasure and the Cayley transform, we have the result. (cid:3) Lemma 5.5.
Let t ≥ . Suppose that there exists c = c ( t ) > such that for all s ≥ t, (5.1) ω ( ρ s e i θ s , Θ t , H ) ≥ c . Then ω ( , Θ t , H ) ≥ c ω ( , Γ t , H \ Γ t ) . Proof.
By the Strong Markov Property for harmonic measure (see, [4, Lemma 3.7]), we have ω ( , Θ t , H ) = ω ( , Θ t , H \ Γ t ) + Z Γ t ω ( α , Θ t , H ) ω ( , d α , H \ Γ t ) , where, considering the measure λ : = ω ( , · , H \ Γ t ) on the boundary of H \ Γ t , we let ω ( , d α , H \ Γ t ) : = d λ (i.e., the integration with respect to the measure λ ).Therefore, by hypothesis (5.1), ω ( , Θ t , H ) ≥ Z Γ t ω ( α , Θ t , H ) ω ( , d α , H \ Γ t ) ≥ c Z Γ t ω ( , d α , H \ Γ t ) = ω ( , Γ t , H \ Γ t ) . (cid:3)
6. A
SYMPTOTIC MONOTONICITY OF ORTHOGONAL SPEED
In this section, ( φ t ) and ( ˜ φ t ) are non-elliptic semigroups in D with Koenigs functions h and˜ h , respectively. We assume that 1 is the Denjoy-Wolff point of both ( φ t ) and ( ˜ φ t ) .We use the notations introduced in the previous section, and we let Γ t , Γ ∗ t , Θ t be the setsassociated to φ t and ˜ Γ t , ˜ Γ ∗ t , ˜ Θ t the corresponding ones associated to ( ˜ φ t ) . SYMPTOTIC MONOTONICITY 11
Lemma 6.1.
Suppose h ( D ) ⊂ ˜ h ( D ) . Let c > . Then there exists a constant H ∈ R such that,for every t ≥ and for every s ≥ t, (6.1) ω ( ˜ ρ s e i ˜ θ s , ˜ Θ t , H ) ≥ c , we have v o ( t ) − ˜ v o ( t ) ≥ H . Proof.
Let C : D → H be the Cayley transform given by C ( w ) = ( + w ) / ( − w ) . Hence, h ◦ C − : H → h ( D ) is a biholomorphism such that h ( ) = r + it , for some r , t ∈ R and h ( C − ( Γ t )) = r + i [ t + t , + ∞ ) . Similarly, ˜ h ◦ C − : H → ˜ h ( D ) is a biholomorphism mapping˜ Γ t onto ˜ r + i [ ˜ t + t , + ∞ ) , with ˜ h ( ) = ˜ r + i ˜ t , for some ˜ r , ˜ t ∈ R .Case 1.Assume r = ˜ r and t = ˜ t .Let T : = r + i [ t + t , + ∞ ) . By (in order of usage) Lemma 5.4, conformal invariance, domainmonotonicity and again conformal invariance, we obtain ω ( , Θ t , H ) (Lemma 5 . ) < ω ( , Γ t , H \ Γ t ) (conformal inv.) = ω ( r + it , T , h ( D ) \ T ) (domain monoton.) ≤ ω ( r + it , T , ˜ h ( D ) \ T ) (conformal inv.) = ω ( , ˜ Γ t , H \ ˜ Γ t ) (Lemma 5 . ) ≤ c ω ( , ˜ Θ t , H ) . Therefore, by Lemma 4.1 (denoting by ˜ K > ( ˜ φ t ) and by K > ( φ t ) ), we have v o ( t ) ≥ −
12 log ω ( , Θ t , H ) − K ≥ −
12 log ω ( , ˜ Θ t , H ) − K −
12 log 2 c ≥ ˜ v o ( t ) + ˜ K − K −
12 log 2 c . Setting H : = ˜ K − K − log c , we have the result in this case.Case 2.General case.Let w ∈ D be such that ˜ h ( w ) = r + it (this is possible because h ( D ) ⊂ ˜ h ( D ) ). Let A : D → D be an automorphism such that A ( ) = A ( w ) =
0. Let ˜ ϕ t : = A ◦ ˜ φ t ◦ A − . Hence, ( ˜ ϕ t ) isa non-elliptic semigroup in D with Denjoy-Wolff point 1, and it is easy to check that ˜ h ◦ A − isthe Koenigs function of ( ˜ ϕ t ) . Moreover, ˜ h ◦ A − ( ) = ˜ h ( w ) = r + it . Therefore, by Case 1, v o ( t ) − ˜ w ( t ) ≥ H , where ˜ w ( t ) denotes the orthogonal speed of ( ˜ ϕ t ) . By [10, Prop. 16.1.6], there exists H ′ > | ˜ v o ( s ) − ˜ w o ( s ) | ≤ H for all s ≥
0, hence v o ( t ) − ˜ v o ( t ) ≥ H − H ′ . (cid:3) Proof of Theorem 1.1.
By [10, Prop. 16.1.6], up to conjugation, we can assume without loss ofgenerality that 1 is the Denjoy-Wolff point of both ( φ t ) and ( ˜ φ t ) .By Lemma 5.1 and Lemma 5.2, there exists an increasing sequence { t n } , t ≥
1, convergingto + ∞ such that ω ( ˜ ρ s e i ˜ θ s , ˜ Θ t n , H ) ≥ /
2, for all s ≥ t n . Therefore, by Lemma 6.1, there exists H ∈ R such that v o ( t n ) − ˜ v o ( t n ) ≥ H for all n . The wanted statement follows at once from (3.3). (cid:3) Theorem 6.2.
Let ( φ t ) , ( ˜ φ t ) be semigroups in D . Let h ( respectively , ˜ h) be the Koenigs functionof ( φ t ) ( resp. of ( ˜ φ t ) ). Suppose that h ( D ) ⊂ ˜ h ( D ) and that (1) either { φ t ( ) } converges non-tangentially to the Denjoy-Wolff point, (2) or, { ˜ φ t ( ) } converges non-tangentially to the Denjoy-Wolff point, (3) or, [ , + ∞ ) ∋ t ˜ v o ( t ) is (eventually) non-decreasing, (4) or, [ , + ∞ ) ∋ t ˜ v ( t ) is (eventually) non-decreasing.Then lim inf t → + ∞ [ v o ( t ) − ˜ v o ( t )] > − ∞ . Proof.
By [10, Prop. 16.1.6], up to conjugation, we can assume without loss of generality that1 is the Denjoy-Wolff point of both ( φ t ) and ( ˜ φ t ) .(1) In this hypothesis, lim sup t → + ∞ v T ( t ) < + ∞ , hence, by (3.1), there exists c > | v ( t ) − v o ( t ) | ≤ c , for all t ≥ h ( D ) ⊂ ˜ h ( D ) , then v ( t ) ≥ ˜ v ( t ) + c , for some c ∈ R and for all t ≥
0. Taking intoaccount again (3.1), we have v o ( t ) ≥ v ( t ) + c ≥ ˜ v ( t ) + c + c ≥ ˜ v o ( t ) + c + c −
12 log 2 , for all t ≥
0, and we are done.(2) In this hypothesis, by Lemma 5.3, there exists C > t ≥
1, we have ω ( ˜ ρ s e i ˜ θ s , ˜ Θ t , H ) ≥ C , for all s ≥ t . Therefore, by Lemma 6.1, there exists H ∈ R such that v o ( t ) − ˜ v o ( t ) ≥ H , for all t ≥ t ˜ v o ( t ) is (eventually) non-decreasing if and only if t log ˜ ρ r is (even-tually) non-decreasing, if and only if t ˜ ρ r is (eventually) non-decreasing. By definition,the latter condition is eventually equivalent to ˜ Γ ∗ t = ˜ Θ t . If this is satisfied, by Lemma 5.2, ω ( ˜ ρ s e i ˜ θ s , ˜ Θ t , H ) ≥ /
2, for all s ≥ t and for all t large enough. Again, the result follows thenfrom Lemma 6.1.(4) It follows at once from Lemma 3.3 and (3). (cid:3) Proof of Theorem 1.2. (1) (respectively (2)) follows at once by Theorem 6.2 (1) (resp. (2)) and[9, Thm. 1.1] (or [10, Thm. 17.3.1]).(3) In case w = ˜ h ( ) , the result follows from Proposition 3.5 and Theorem 6.2.(4). SYMPTOTIC MONOTONICITY 13
In case w = ˜ h ( ) , let A : D → D be an automorphism of D such that A ( w ) =
0. Let ˜ ϕ t : = A ◦ ˜ φ t ◦ A − . Hence, ( ˜ ϕ t ) is a non-elliptic semigroup in D , and it is easy to check that h : = ˜ h ◦ A − is the Koenigs function of ( ˜ ϕ t ) . Since h ( D ) is starlike with respect to 0 by construction andhypothesis, it follows by Proposition 3.5 that the total speed w ( t ) of ( ˜ ϕ t ) is non-decreasing.Hence, by Theorem 6.2.(4), lim inf t → + ∞ [ v o ( t ) − w o ( t )] > − ∞ , where w o ( t ) denotes the orthogonal speed of ( ˜ ϕ t ) . By [10, Prop. 16.1.6], there exists a constant K > | ˜ v o ( t ) − w o ( t ) | ≤ K for all t ≥
0. The wanted statement follows at once from(3.3). (cid:3)
7. S
OME APPLICATIONS
As it is clear from Theorem 1.1 or Theorem 1.2 (and (3.3)), in order to obtain explicit esti-mates for the rate of convergence of orbits in terms of the geometry of the image of the Koenigsfunction of a semigroup, the main issue is to have estimates of the rate of convergence in specialdomains.In this section, we estimate the orthogonal speed of semigroups whose Koenigs function hasimage given by some special forms and apply our main results to get general applications. Fix α > Π α : = { z ∈ C | Im z > | Re z | α } The domain Π α is starlike at infinity. Therefore, if h α : D → Π α is a Riemann map, it turns outthat h α is the Koenigs function of the semigroup ( φ α t ) where φ α t ( z ) : = h − ( h ( z ) + it ) , z ∈ D , t ≥
0. Clearly, ( φ α t ) is a non-elliptic semigroup in D . Since S t ≥ ( Π α − it ) = C , the semigroup is iit q t − + i (cid:0) t − (cid:1) − q s − + i (cid:0) t − (cid:1) F IGURE
1. The domain Π α with α = h α ( ) = i and 1 is the Denjoy-Wolff point of ( φ α t ) . The domain Π α is symmetric with respect to theimaginary axis, and therefore by [9, Thm. 1.1] (or [10, Thm. 17.3.3]), the orbits of ( φ t ) convergenon-tangentially to 1. Moreover, γ : [ , + ∞ ) −→ Π α with γ ( t ) : = i ( t + ) is a geodesic for the hyperbolic distance of h ( D ) (see, e.g., [10, Prop. 6.1.3]) and h ([ , )) = γ ([ , + ∞ )) (since h − ( γ ( t )) →
1, as t → + ∞ , hence [ , ) and h − ( γ ([ , + ∞ )) are geodesics in D , whose closure contain both 0 and 1, hence, they are equal). In particular, the tangential speedof ( φ α t ) is identically zero, the orthogonal speed v o α ( t ) coincides with the total speed v α ( t ) and,since γ is a geodesic, v α ( t ) = k Π α ( i , i ( + t )) = Z + t κ Π α ( is ; i ) d s . By the Distance Lemma for convex simply connected domains (see, e.g., [10, Thm. 5.2.2]),(7.1) 12 Z + t d s δ α ( is ) ≤ v α ( t ) ≤ Z + t d s δ α ( is ) , where δ α ( ir ) denotes the Euclidean distance of ir from the boundary of Π α . Lemma 7.1.
Let α > . For any c ∈ ( , ) , there exists s ≥ such that for all s ≥ s ,cs / α ≤ δ α ( is ) ≤ s / α . Proof.
Fix s ≥
1. Since s / α is the distance of is to the point s / α + is ∈ ∂Π α , it is clear that δ α ( is ) ≤ s / α . By the symmetry of Π α , there exists x ≥ δ α ( is ) = | ( x + ix α ) − is | = x + ( x α − s ) . In fact, the point x is the largest positive root of the equation(7.2) x α + α x − α − s = . Note that, if 1 < α ≤
2, this equation has a unique positive root for any s ≥
1, while, if α > s ≥
1, there are two positive roots.Now let x α ( s ) : = x be the point defined above. The function s x α ( s ) is strictly increasingand when s goes to infinity, x α ( s ) diverges to + ∞ , as well. By (7.2), s = x α ( s ) α (cid:18) + α x α ( s ) ( α − ) (cid:19) and one deduces that there exists a positive strictly increasing function g α ( s ) : [ , + ∞ ) → ( , ) such that lim s → + ∞ g α ( s ) = δ α ( is ) ≥ x α ( s ) = g α ( s ) · s / α . Thus the proof is completed. (cid:3)
Remark . If α =
2, we have δ ( is ) = s s − + (cid:18) − (cid:19) = r s − . SYMPTOTIC MONOTONICITY 15
Now we can apply Lemma 7.1 to (7.1). Since Z + t s − / α d s = (cid:18) αα − (cid:19) h − + ( + t ) − α i , for any ε > t (depending on ε and α ),(7.3) 12 (cid:18) αα − (cid:19) t − α . v α ( t ) = v o α ( t ) . ( + ε ) (cid:18) αα − (cid:19) t − α , where f ( t ) . f ( t ) means that there exists λ ∈ R , such that f ( t ) − f ( t ) ≥ λ for all t .As a direct application of Theorem 1.2, (7.3) and (3.3), we get the following result. Proposition 7.3.
Suppose ( φ t ) is a non-elliptic semigroup in D with Denjoy-Wolff point τ andKoenigs function h. Let v o ( t ) be the orthogonal speed of ( φ t ) . (1) Suppose that h ( D ) ⊆ p + Π α , for some α > and p ∈ C . Then lim inf t → + ∞ (cid:20) v o ( t ) − α ( α − ) t − α (cid:21) > − ∞ , or, equivalently, there exists K > such that for all t ≥ , | φ t ( ) − τ | ≤ K exp (cid:18) − αα − t − α (cid:19) . (2) Suppose that p + Π α ⊆ h ( D ) , for some α > and p ∈ C . Then for any ε > , lim sup t → + ∞ (cid:20) v o ( t ) − ( + ε ) α ( α − ) t − α (cid:21) < + ∞ or, equivalently, there exists K ( ε ) > such that for all t ≥ , | φ t ( ) − τ | ≥ K ( ε ) exp (cid:18) − ( + ε ) αα − t − α (cid:19) . Let α > θ ∈ ( , π ] . We let Ξ ( α , θ ) : = ( − H ∩ Π α ) ∪ W ( θ ) , where W ( θ ) : = (cid:8) z ∈ C | arg ( z ) ∈ (cid:0) π − θ , π (cid:1)(cid:9) .Once again, such a domain is starlike at infinity. It is convex when 0 < θ ≤ π , otherwise itis starlike with respect to any point z ∈ H with arg ( z ) >
0. If h α , θ : D → Ξ ( α , θ ) is a Riemannmap, then it is the Koenigs function of the semigroup φ α , θ t ( z ) : = h − α , θ ( h α , θ ( z ) + it )) definedfor any z ∈ D and t ≥
0. As S t ≥ ( Ξ ( α , θ ) − it ) is the whole complex plane, the semigroup ( φ α , θ t ) is parabolic with zero hyperbolic step. Again, we can assume h α , θ ( ) = i , without lossof generality. For any t ≥ δ + α , θ ( it ) : = min { inf {| z − it | | Re z ≥ , z ∈ C \ Ξ ( α , θ ) } , t } = ( ( sin θ ) t θ ∈ (cid:0) , π (cid:1) t θ ∈ (cid:2) π , π (cid:3) iit H t + i t √ − q t − + i (cid:0) t − (cid:1) F IGURE
2. The domain Ξ ( , π ) while δ − α , θ ( it ) : = min { inf {| z − it | | Re z ≤ , z ∈ C \ Ξ ( α , θ ) } , t } = δ α ( it ) , where δ α ( it ) is the distance from the boundary of Π α , considered in the first example. ByLemma 7.1, δ α ( it ) = O ( t / α ) , so it follows that the domain is not quasi-symmetric with respectto vertical axes. In particular, by [9, Thm. 1.1(2)], each orbit of the semigroup ( φ α , θ t ) convergestangentially to its Denjoy-Wolff point and we can assume that up to conjugation with a rotation,it is equal to 1.Let us recall the following result. Lemma 7.4. [10, Corollary 16.2.6]
Let be θ , η ∈ [ , π ] , not both equal to zero. Consider thedomain W ( θ , η ) = { z ∈ C | arg ( − iz ) ∈ ( − θ , η ) } . Let ( φ t ) be a semigroup of holomorphic self-maps in D with Koenigs map h and h ( D ) = p + W ( θ , η ) , for some p ∈ C . (1) If both θ and η are non-zero, the tangential speed v T ( t ) of ( φ t ) is bounded, while forthe total and orthogonal speeds one hasv ( t ) ∼ v o ( t ) ∼ (cid:18) πθ + η (cid:19) log t . (2) If otherwise θ ∈ ( , π ] and η = , the speeds of ( φ t ) have the following behaviorv T ( t ) ∼
12 log t , v o ( t ) ∼ π θ log t , v ( t ) ∼ π + θ θ log t . When θ = and η ∈ ( , π ] , the result is analogous, just replace θ with η . Returning to our domain Ξ ( α , θ ) , we have that W ( θ ) ⊂ Ξ ( α , θ ) . Moreover, for any η ∈ ( , π ] we can find a point p η ∈ C , for which Ξ ( α , θ ) ⊂ p η + W ( θ , η ) . So by Lemma 7.4 andTheorem 1.2, it follows that for the orthogonal speed v o α , θ of ( φ α , θ t ) one has(7.4) π θ ( − ε ) log t . v o α , θ ( t ) . π θ log t , SYMPTOTIC MONOTONICITY 17 where ε : = ηθ + η ∈ (cid:0) , πθ + π (cid:3) is arbitrarily small, for η sufficiently close to zero. More generally,by the same argument, we have an analogous outcome to Proposition 7.3. Proposition 7.5.
Suppose ( φ t ) is a non-elliptic semigroup in D with Denjoy-Wolff point τ andKoenigs function h. Let v o ( t ) be the orthogonal speed of ( φ t ) . (1) Suppose that h ( D ) ⊆ p + Ξ ( α , θ ) , for some α > , θ ∈ ( , π ] and p ∈ C . Then for any ε ∈ (cid:0) , πθ + π (cid:3) lim inf t → + ∞ h v o ( t ) − π θ ( − ε ) log t i > − ∞ , or, equivalently, there exists K ( ε ) > such that for all t ≥ , | φ t ( ) − τ | ≤ K ( ε ) t ( − + ε ) π / θ . (2) Suppose that p + Ξ ( α , θ ) ⊆ h ( D ) , for some α > , θ ∈ ( , π ] and p ∈ C . Let’s assumethat h ( D ) is starlike with respect to an inner point. Then lim sup t → + ∞ h v o ( t ) − π θ log t i < + ∞ or, equivalently, there exists K > such that for all t ≥ , | φ t ( ) − τ | ≥ Kt − π / θ . Remark . The results above do not depend on α . This is not a deficiency of the methods weuse, but a natural fact, due to (7.4). In other words, in the previous setting, the “non-tangential”side controls the orthogonal speed. Indeed, Condition (2) of Proposition 7.5 is equivalent toassume the (weaker) hypothesis that p + W ( θ ) ⊆ h ( D ) and h ( D ) is starlike.On the other hand, it is interesting to note that the exponent α controls the tangential speedof the semigroup ( φ α , θ t ) , which is not influenced by the angle θ . Proposition 7.7.
For the tangential speed of the semigroup ( φ α , θ t ) , the following estimates (upto real constants) hold (cid:18) − α (cid:19) log t . v T α , θ ( t ) . (cid:18) − α (cid:19) log t . Hence for any ε ∈ (cid:0) , πθ + π (cid:3) , we have the following bounds for the total speed (cid:18) π θ ( − ε ) + − α (cid:19) log t . v α , θ ( t ) . (cid:18) π θ + − α (cid:19) log t . Proof.
We divide the proof into steps.
Step 1. Lower bound for tangential speed .Let H be the curve H : [ , ∞ ) −→ Ξ ( α , θ ) with H ( r ) = re i ( π − θ ) / . This curve is a quasi-geodesic, as its hyperbolic length is ℓ Ξ ( α , θ ) ( H , [ r , r ]) ≤ ℓ W ( θ ) ( H , [ r , r ]) ≤ Z r r d r δ W ( θ ) ( H ( r )) = θ log r r and by the Distance Lemma for simply connected domains (see, e.g. [9, Thm. 3.5]) k Ξ ( α , θ ) ( H ( r ) , H ( r )) ≥
14 log + r − r (cid:0) sin θ (cid:1) r ! ≥
14 log r r . By means of the Gromov shadowing Lemma (see, e.g., [10, Thm. 6.3.8]), it is enough to findbounds for inf r ≥ k Ξ ( α , θ ) ( it , H ( r )) , since the same bounds, up to constants not depending on t , will hold also for v T α , θ ( t ) .Now, once t is chosen big enough, δ Ξ ( α , θ ) ( it ) = δ α ( it ) = O ( t / α ) . If (cid:0) sin θ (cid:1) r ≤ δ α ( it ) , then k Ξ ( α , θ ) ( it , H ( r )) ≥
14 log + | it − H ( r ) | (cid:0) sin θ (cid:1) r ! =
14 log + q r + t − rt cos θ (cid:0) sin θ (cid:1) r ≥
14 log + t − r (cid:0) sin θ (cid:1) r ! , which is a decreasing function of r , soinf ≤ r ≤ ( sin θ ) − δ α ( it ) k Ξ ( α , θ ) ( it , H ( r )) ≥
14 log + t − (cid:0) sin θ (cid:1) − δ α ( it ) δ α ( it ) ! ∼ (cid:18) − α (cid:19) log t . On the other hand, if (cid:0) sin θ (cid:1) r > δ α ( it ) , then k Ξ ( α , θ ) ( it , H ( r )) ≥
14 log (cid:18) + | it − H ( r ) | δ α ( it ) (cid:19) ≥
14 log + (cid:0) sin θ (cid:1) t δ α ( it ) ! and so, one concludes that v T α , θ ( t ) & ( − / α ) log t . Step 2. Upper bound for tangential speed .For every t greater than some fixed t ≥ δ Ξ ( α , θ ) ( it ) = δ α ( it ) and the point q t : = it + δ α ( it ) belongs to W ( θ ) ⊂ Ξ ( α , θ ) . So for any t ≥ t and r ≥
1, we define the path σ t , r given by theconcatenation of the Euclidean segment from it to q t L t : [ , ] −→ Ξ ( α , θ ) with L t ( s ) = it + δ α ( it ) s , where γ t , r is the geodesic arc with respect to the hyperbolic metric of W ( θ ) joining q t with H ( r ) = re i ( π − θ ) / . By possibly increasing t , we may also assume that δ Ξ ( α , θ ) ( L t ( s )) ≥ δ α ( it ) , SYMPTOTIC MONOTONICITY 19 for any 0 ≤ s ≤
1. Therefore we have k Ξ ( α , θ ) ( it , H ( r )) ≤ ℓ Ξ ( α , θ ) ( σ t , r ) = ℓ Ξ ( α , θ ) ( L t ) + ℓ Ξ ( α , θ ) ( γ t , r ) ≤ ℓ Ξ ( α , θ ) ( L t ) + ℓ W ( θ ) ( γ t , r ) ≤ Z δ α ( it ) d s δ Ξ ( α , θ ) ( L t ( s )) + k W ( θ ) ( q t , H ( r )) ≤ Z δ α ( it ) d s δ α ( it ) + k W ( θ ) ( q t , H ( r ))= + k W ( θ ) ( q t , H ( r )) . Now let β t : = π − arg q t , so that t · tan β t = δ α ( it ) . Thus, considering the conformal map z z π / θ which sends e W ( θ ) : = e i ( θ − π ) / W ( θ ) onto H , and using known estimates for k H (see for instance[8, Lemma 2.1]) k W ( θ ) ( q t , H ( r )) = k W ( θ ) ( | q t | e i ( π − β t ) , re i ( π − θ ) / ) = k e W ( θ ) ( | q t | e i ( θ − β t ) , r )= k e W ( θ ) (cid:18) , | q t | r e i ( θ − β t ) (cid:19) = k H , | q t | π / θ r π / θ e i ( π − πθ β t ) ! ≤ π θ log | q t | r +
12 log 1sin (cid:0) πθ β t (cid:1) +
12 log 2 . By choosing r = | q t | = t p + tan β t and by observing that, since lim t → + ∞ β t = (cid:0) πθ β t (cid:1) ∼ log θπβ t ∼ log 1tan β t = log t δ α ( it ) ∼ (cid:18) − α (cid:19) log t , we conclude that v T α , θ ( t ) ≤ k Ξ ( α , θ ) ( it , H ( | q t | )) . (cid:18) − α (cid:19) log t . Step 3. Total speed .The statement for the total speed v α , θ follows directly from (3.1) and (7.4). (cid:3)
8. F
INAL REMARKS AND OPEN QUESTIONS
Theorem 1.1 and Theorem 6.2 move towards the direction of giving an affirmative answer toQuestion 4 in [8]. However, the complete answer is still unknown, and, as it follows from theresults in Section 5, if counterexamples exist, they are rather peculiar.Note that (using the same notation as in Section 5), given a semigroup ( φ t ) of D , since ρ s → + ∞ , as s → + ∞ , by the same argument of Lemma 5.2, for all t ≥
1, there exists s t ≥ t such thatinf s ≥ t ω ( ρ s e i θ s , Θ t , H ) = ω ( ρ s t e i θ st , Θ t , H ) and lim inf s → + ∞ ω ( ρ s e i θ s , Θ t , H ) > . Question(i): Does there exist a semigroup ( φ t ) of D so thatlim inf t → + ∞ ω ( ρ s t e i θ st , Θ t , H ) = ( φ t ) of D so that the orthogonal speed is not(eventually) non-decreasing? Note that this is equivalent to ask if t ρ t is not (eventually)non-decreasing, for a semigroup ( φ t ) of D .R EFERENCES [1] M. Abate, Iterationtheoryofholomorphicmapsontautmanifolds, Mediterranean Press, 1989.[2] D. Aharonov, M. Elin, S. Reich and D. Shoikhet, Parametricrepresentationofsemi-completevectorfieldsontheunitballsof C n andinHilbertspace, Atti Accad. Naz. Lincei 10 (1999), 229–253.[3] E. Berkson, H. Porta, Semigroupsof holomorphicfunctionsand composition operators. Michigan Math. J., (1978), 101–115.[4] D. Betsakos, Harmonicmeasureonsimplyconnecteddomainsoffixedinradius. Ark. Mat., 36, (1998), 275–306.[5] D. Betsakos, On the rate of convergence of parabolic semigroups of holomorphic functions. Anal. Math.Phys., 5 (2015), 207–216.[6] D. Betsakos, On the rate of convergence of hyperbolic semigroups of holomorphic functions. Bull. Lond.Math. Soc. 47 (2015), 493–500.[7] D. Betsakos, M. D. Contreras, S. D´ıaz-Madrigal, On the rateof convergenceof semigroupsof holomorphicfunctionsattheDenjoy-Wolffpoint. Rev. Mat. Iberoamericana, (to appear).[8] F. Bracci, Speeds of convergenceof orbits of non-elliptic semigroups of holomorphicself-maps of the unitdisc, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 73, 2, 21–43 (2019)[9] F. Bracci, M. D. Contreras, S. Diaz-Madrigal, H. Gaussier, A. Zimmer, Asymptotic behavior of orbits ofholomorphicsemigroups, J. Math. Pures Appl. (9) 133, 263–286 (2020)[10] F. Bracci, M. D. Contreras, S. Diaz-Madrigal, ContinuousSemigroupsofHolomorphicSelf-MapsoftheUnitDisc. Springer Monographs in Mathematics, Springer Nature Switzerland AG 2020.[11] P. L. Duren, UnivalentFunctions. Springer-Verlag 1983.[12] M. Elin, F. Jacobzon, Parabolic type semigroups: asymptotics and order of contact. Anal. Math. Phys 4(2014), 157–185.[13] M. Elin, D. Khavinson, S. Reich, D. Shoikhet, Linearization models for parabolic dynamical systems viaAbel’sfunctionalequation. Ann. Acad. Sci. Fenn. Math. 35 (2010), 439–472.[14] M. Elin and D. Shoikhet, LinearizationModelsforComplexDynamicalSystems. Birkh¨auser, 2010.[15] M. Elin, S. Reich, D. Shoikhet and F. Yacobzon, Rates of convergence of one-parameter semigroups withboundaryDenjoy-Wolfffixedpoints, Fixed Points Theory and its Applications, Yokohama Publishers, 2008,43–58.[16] M. Elin, D. Shoikhet, Dynamic extension of the Julia-Wolff-Carath´eodorytheorem Dynam. Systems Appl.10 (2001), 421–437.[17] F. Jacobzon, M. Levenshtein, S. Reich, Convergencecharacteristicsofone-parametercontinuoussemigroups,Analysis and Mathematical Physics, (2011), 311–335. SYMPTOTIC MONOTONICITY 21 [18] D. Shoikhet, SemigroupsinGeometricalFunctionTheory. Kluwer Academic Publishers, 2001.[19] A. G. Siskakis, Semigroupsofcompositionoperatorsonspacesofanalyticfunctions,areview, Amer. Math.Soc., 229–252, 1998.F. B
RACCI : D
IPARTIMENTO DI M ATEMATICA , U
NIVERSIT ` A DI R OMA “T OR V ERGATA ”, V
IA DELLA R ICERCA S CIENTIFICA
1, 00133, R
OMA , I
TALIA . E-mail address : [email protected] D. C
ORDELLA : D
IPARTIMENTO DI M ATEMATICA , U
NIVERSIT ` A DI R OMA “T OR V ERGATA ”, V
IA DELLA R ICERCA S CIENTIFICA
1, 00133, R
OMA , I
TALIA . E-mail address : [email protected] M. K
OUROU : D
EPARTMENT OF M ATHEMATICS , U
NIVERSITY OF
W ¨
URZBURG , E
MIL F ISCHER S TRASSE
40, 97074, W ¨
URZBURG , G
ERMANY . E-mail address ::