SECOND ANGULAR DERIVATIVES AND PARABOLIC ITERATION IN THE UNIT DISK 1 ...

32 Pages·2009·0.35 MB·English

TRANSACTIONSOFTHE
AMERICANMATHEMATICALSOCIETY
Volume362,Number1,January2010,Pages357–388
S0002-9947(09)04873-9
ArticleelectronicallypublishedonJuly24,2009
SECOND ANGULAR DERIVATIVES
AND PARABOLIC ITERATION IN THE UNIT DISK
MANUELD.CONTRERAS,SANTIAGOD´IAZ-MADRIGAL,
ANDCHRISTIANPOMMERENKE
Abstract. Inthispaperwedealwithsecondangularderivatives atDenjoy-
Wolff points for parabolic functions in the unit disc. Namely, we study and
analyze the existence and the dynamical meaning of this second angular de-
rivative. Forinstance,weprovideseveralcharacterizationsofthatexistencein
termsoftheso-calledKoenigsfunction. Itisworthpointingoutthatthereare
twoquitedifferentclassesofparaboliciteration: thosewithpositivehyperbolic
stepandthosewithzerohyperbolic step. Inthefirst case,theKoenigsfunc-
tionisintheCarath´eodoryclassbut,inthesecondcase,itisevenunknownif
itisnormal. Therefore,theideasandtechniquestoapproachthesetwocases
arereallydifferent. Intheend,wealsopresentseveralrigidityresultsrelated
tothesecondangularderivativesatDenjoy-Wolffpoints.
1. Introduction
The dynamical properties of a holomorphic self-map of the unit disk ϕ ∈
Hol(D;D) have attracted considerable interest since the beginning of the last cen-
tury. Now it is known that a major role is played by the collection of all inner and
boundaryfixedpointsofϕ.Werecallthatapointb∈Dissaidtobeaninnerfixed
point of ϕ if ϕ(b)=b; likewise, a point b∈∂D is said to be a boundary fixed point
of ϕ if limr→1ϕ(rb)=b.
In a natural way, the usual classification of the members of Hol(D;D) takes
into account the above set of fixed points. For instance, ϕ ∈ Hol(D;D) is said
to be elliptic if ϕ has at least one inner fixed point. It is a deep result, the so-
called Denjoy-Wolff Theorem, that any non-elliptic map ϕ∈Hol(D;D) always has
a boundary fixed point τ with a very strong property: all of their forward orbits
tend to τ. This point is clearly unique and it is called the Denjoy-Wolff point of
ϕ. Moreover, ϕ is somehow differentiable at τ. Since τ belongs to the boundary of
the unit disk, this comment requires a clarification and this leads us to recall the
concept of angular limit.
ReceivedbytheeditorsDecember9,2005and,inrevisedform,February20,2008.
2000 MathematicsSubjectClassification. Primary30D05,32H40;Secondary32H50.
Key words and phrases. Second angular derivative, parabolic functions, Denjoy-Wolff point,
Koenigsfunction,rigidity.
ThisresearchhasbeenpartiallysupportedbytheMinisteriodeCienciayTecnolog´ıa andthe
EuropeanUnion(FEDER)projectMTM2006-14449-C02-01andbyLa Consejer´ıade Educacio´n
y Ciencia de la Junta de Andaluc´ıa.
(cid:1)c2009 American Mathematical Society
357
358 M.D.CONTRERAS,S.D´IAZ-MADRIGAL,ANDC.POMMERENKE
Let f :D→C be an analytic function and b∈∂D. It is said that L∈C∞ is the
angular limit of f in b when z tends to b if, for every α∈(0,π/2),
lim f(z)=L,
z∈S(b,α),z→b
whereS(b,α)denotestheStolzanglecenteredinbandwithopeningα. Thenumber
L ismorecommonlydenotedby∠limz→bf(z). IfLisfiniteandtheangularlimit
f(z)−L
M :=∠zli→mb z−b ∈C∞
exists, then f is said to be angular differentiable at b and we write f(cid:4)(b)=M.
Turning to iteration, if τ is the Denjoy-Wolff point of some non-elliptic ϕ ∈
Hol(D;D), then ϕ is angular differentiable at τ and ϕ(cid:4)(τ)∈(0,1]. We recall that ϕ
is said to be parabolic whenever ϕ(cid:4)(τ)=1 and, otherwise, hyperbolic.
Hyperbolic iteration in the disk is quite well understood now (see for instance
the papers [10] and [3] and the references therein). However, in the parabolic case,
many interesting dynamical questions remain to be answered. Bearing in mind the
importanceoffirstorderangulardifferentiabilityinwhatisknowninbothcases,we
haveproposedheretostudythedynamicalbehaviorofparaboliciterationassuming
somekindofhighangulardifferentiabilityatthecorrespondingDenjoy-Wolffpoint.
In a certain sense and for different purposes (composition operators [2], rigidity
results [13], [14], ...), this approach has already been considered by several authors
and, indeed, some of our results are non-trivial extensions of some of their results.
Concerning the remarkable paper [2], we want to underline that their results and
techniques have been an important source of inspiration for this work. Anyway,
most of our results are, as far as we know, really new and we think this is the first
paper dealing explicitly with general questions about the existence or dynamical
meaning of the second angular derivative.
Since theconceptsofangular differentiability ofhighorder areintuitive but cer-
tainlytechnical,wehavedecidedtogroupthemalltogetherinthenextsection. As
usual, we make good use in our proofs of the pseudo-hyperbolic distance ρ(cid:1)D(z1,z2)
in the unit disk as well as the hyperbolic distance ρD(z1,z2). Sometimes, this will
bedoneindirectly. Wemeanbythisthatwewillprefertoworkwithρ(cid:1)H andρH,re-
spectively, the pseudo-hyperbolic distance and the hyperbolic distance in the right
half-plane H. In this respect, we recall that a holomorphic function f :H→C has
angular limit L∈C∞ at the point ∞, whenever, for every c>0,
lim f(w)=L,
w∈S(∞,c),w→∞
where S(∞,c)={u+iv ∈H:|v|≤cu}. As before, the number L will be denoted
by ∠limw→∞f(w).
The paper is divided into six sections, apart from this introduction. In the
next section, we present several not so widely known definitions and several minor
results (most of them only partially known) concerning angular differentiability of
arbitrary order at a boundary fixed point. In section three, we recall, also giving
new comments, the main types of classification of parabolic maps and give a short
review of the intertwining maps used in parabolic iteration. In addition, we prove
severaltechnicallemmasthatwewillneedrepeatedlyintheremainingsections. In
a certain sense, sections two and three can be thought of as preparatory sections.
Theotherfoursectionsarereallythecoreofthepaper. Namely,insectionfour,we
SECOND ANGULAR DERIVATIVES AND PARABOLIC ITERATION 359
treattheexistenceanddynamicalmeaningofsecondorderangulardifferentiability
at Denjoy-Wolff points for parabolic maps with positive hyperbolic step. Sections
five and six are devoted to parabolic maps with zero hyperbolic step. Basically,
we present some general results for those functions in section five and treat the
correspondingproblemsrelatedtosecondorderangulardifferentiabilityinthesixth
section. Finally, in the last section, we present several rigidity results when second
order angular derivatives vanish at Denjoy-Wolff points of non-elliptic maps.
2. High order angular differentiability at boundary fixed points
Consider amapϕ∈Hol(D;C)withaboundaryfixedpointb∈∂D. Wesaythat
ϕ is of angular-class of order p∈N at b, and we denote it by ϕ∈Cp(b), if
A
(cid:2)p
a
ϕ(z)=b+ j(z−b)j +γ(z), z ∈D,
j!
j=1
where a ,...,a ∈C and γ ∈Hol(D;C) with
1 p
γ(z)
∠lim =0.
z→b(z−b)p
It is clear that the numbers a ,...,a appearing in the above expression are neces-
1 p
sarily unique.
According to the Julia-Carath´eodory Theorem, we easily see that ϕ ∈ C1(b) if
A
and only if ϕ has a finite angular derivative at b. Clearly, when this happens, the
corresponding number a is exactly ϕ(cid:4)(b). In particular, parabolic and hyperbolic
1
membersofHol(D;D)areofangular-classoffirstorderattheirDenjoy-Wolffpoints.
ItisalsostraightforwardtocheckthatϕbelongstoC2(b)ifandonlyifϕ∈C1(b)
A A
and the following angular limit exists finitely:
ϕ(z)−b−ϕ(cid:4)(b)(z−b)
L:=∠lim .
z→b (z−b)2
Nowthecorrespondingnumbersa anda arejusta =ϕ(cid:4)(b)anda =2L.Inwhat
1 2 1 2
follows, a will be written as ϕ(cid:4)(cid:4)(b). Note that if ϕ is parabolic with Denjoy-Wolff
2
point τ ∈ ∂D, then ϕ ∈ C2(τ) if and only if the following angular limit exists
A
finitely:
ϕ(z)−z
∠ lim .
z→τ (z−τ)2
It is important to underline that there are examples showing that being of
angular-class of order two at a certain boundary fixed point is much stronger than
being of angular-class of order one. For instance, let us consider
(cid:3) (cid:4)
2
2z+(1−z)Log
1−z
ϕ(z):= (cid:3) (cid:4) , z ∈D,
2
2+(1−z)Log
1−z
whereLogistheprincipalbranchofthelogarithm. Itispossibletocheckthatϕisa
parabolic holomorphic self-mapoftheunitdiskhavingthepoint1asDenjoy-Wolff
point. Trivially, ϕ∈C1(1). However, ϕ∈/ C2(1) since
A A
ϕ(z)−z ϕ(z)−z
∠lim = lim =∞.
z→1 (z−1)2 z→1 (z−1)2
360 M.D.CONTRERAS,S.D´IAZ-MADRIGAL,ANDC.POMMERENKE
In the proofs of our results, we often pass to the right half-plane H. As usual,
given a non-elliptic map ϕ ∈ Hol(D;D) with Denjoy-Wolff point τ ∈ ∂D, we call
φ:=σ ◦ϕ◦(σ )−1 theiterationmapassociatedwithϕinH,whereσ istheusual
τ τ τ
Cayley map related to τ, that is,
τ +z
σ (z):= , z ∈D.
τ τ −z
It is well known (see [11] for more details) that φ ∈ Hol(H;H) and it has ∞ as
Denjoy-Wolff point. Moreover, the corresponding angular derivative at ∞ satisfies
that φ(cid:4)(∞)=ϕ(cid:4)(τ)−1 ∈[1,+∞) and, indeed,
φ(w)−ϕ(cid:4)(τ)−1w
∠ lim =0.
w→∞ w
Quite useful for the aim of this paper is the following result which links second
order differentiability of ϕ at its Denjoy-Wolff point and certain properties of the
associated iteration map in H. The proof basically involves quite standard compu-
tations with Cayley maps. In fact, with the same notation as in the proposition,
the key is the following identity:
(cid:5) (cid:6)
ϕ(z)−τ −α(z−τ) τα w+1 1 1
= φ(w)− w+1− ,
(z−τ)2 2 φ(w)+1 α α
where w =σ (z).
τ
Proposition 2.1. Let ϕ ∈ Hol(D;D) be non-elliptic with Denjoy-Wolff point τ ∈
∂D and let φ be the associated iteration map in H. Write α:=ϕ(cid:4)(τ)∈(0,1]. Then,
the map ϕ belongs to C2(τ) if and only if the following angular limit exists finitely:
A
1
a:=∠ lim (φ(w)− w);
w→∞ α
that is, for every w ∈H,
1
φ(w)= w+a+γ(w),
α
where γ ∈Hol(H;C) with ∠limw→∞γ(w)=0.
Moreover, when the above statements hold, we have that Re(a)≥0 and
ϕ(cid:4)(cid:4)(τ)τ =α2a+α(α−1).
In the final part of the paper, we will also consider third order angular dif-
ferentiability at Denjoy-Wolff points and this will require a variant of the above
proposition for that third order. Once again, the proof of the corresponding result
involves computations with Cayley maps and, now, the corresponding key identity
is
ϕ(z)−τ −α(z−τ)− ϕ(cid:1)(cid:1)(τ)(z−τ)2 τ w+1
2 = (A−B+C),
(z−τ)3 4φ(w)+1
where w =σ (z) and
τ
1
A = (ϕ(cid:4)(cid:4)(τ)−τα)(φ(w)− w),
α
1
B = τα(wφ(w)− w2−aw),
α
C = ϕ(cid:4)(cid:4)(τ)−τ(α−1).
SECOND ANGULAR DERIVATIVES AND PARABOLIC ITERATION 361
Proposition 2.2. Let ϕ ∈ Hol(D;D) be non-elliptic with Denjoy-Wolff point τ ∈
∂D and let φ be the associated iteration map in H. Denote α:=ϕ(cid:4)(τ)∈(0,1] and
assume that ϕ belongs to C2(τ). Then, the map ϕ belongs to C3(τ) if and only if
A A
there exist two complex numbers a,b such that, for every w ∈H,
1 b
φ(w)= w+a+ +γ(w),
α w
where γ ∈Hol(H;C) with ∠limw→∞wγ(w)=0.
Moreover, when the above statements hold, we have that
3
ϕ(cid:4)(cid:4)(cid:4)(τ)τ2 = α[(aα+α−1)2−bα].
2
If, in addition, Rea=0, then Reb≥0.
As is indicated in [13] and in [2] for the parabolic case, the expression of the
above number b is quite related to what can be considered the angular Schwarzian
derivative S (τ) of ϕ at the corresponding Denjoy-Wolff point. Indeed, this fact
ϕ
is true in the general non-elliptic case. Namely, using the above proposition and
bearing in mind the algebraic rules for computing angular derivatives, it can be
checked that
(cid:7)(cid:3) (cid:4) (cid:3) (cid:4) (cid:8)
2τ2 ϕ(cid:4)(cid:4)(τ) (cid:4) 1 ϕ(cid:4)(cid:4)(τ) 2 2τ2
b=− − =− S (τ).
3 α ϕ(cid:4)(τ) 2 ϕ(cid:4)(τ) 3 α ϕ
3. Classification of parabolic iteration and intertwining maps
Given a parabolic map ϕ ∈ Hol(D;D), we say that ϕ is of zero hyperbolic step
(in short, zero h-step) if, for some z0 ∈D, ρD(zn,zn+1) →n 0, where zn =ϕn(z0). It
is well known that the word “some” can be replaced here by “all”. In other words,
the definition does not depend on the chosen initial point of the orbit. Using the
Scharwz-Pick Lemma, the parabolic maps which are not of zero h-step are those
ϕ∈Hol(D;D) such that
limρD(zn,zn+1)>0,
n
for some (resp. all) forward orbit (z ) of ϕ. That is the reason they are said to be
n
of positive hyperbolic step (in short, positive h-step).
Itisworthmentioningthatthiswayofclassifyingparabolicmapslosesitsmean-
ingwhenweconsidertheabovelimitsonlyintheangularsense. Thisisthecontent
of the next lemma, which will be used in section five. The explanation of this ap-
parently paradoxical fact is that, in the positive h-step case, all the forward orbits
tend tangentially to the Denjoy-Wolff point [9, Remark 1].
Lemma3.1. Letϕ∈Hol(D;D)beparabolicandτ ∈∂DthecorrespondingDenjoy-
Wolff point. Then ∠ lim ρD(ϕ(z),z)=0.
z→τ
Proof. As usual, we prove the above statement for the right half-plane H and we
denotebyφtheiterationmapin Hassociatedwithϕ.Werecallthatφ∈Hol(H;H)
and it is a parabolic map with ∞ as Denjoy-Wolff point. Write w = x +iy =
n n n
φ (1). We note that
n (cid:9) (cid:9)
(cid:9) (cid:9) (cid:9)(cid:9)φ(w)−w(cid:9)(cid:9)
ρ(cid:1)H(w,φ(w))=(cid:9)(cid:9)(cid:9)φφ((ww))+−ww(cid:9)(cid:9)(cid:9)= (cid:9)(cid:9)(cid:9)φ(cid:9)(w)R−eww (cid:9) (cid:9)(cid:9)(cid:9).
(cid:9) +2(cid:9)
Rew
362 M.D.CONTRERAS,S.D´IAZ-MADRIGAL,ANDC.POMMERENKE
Moreover, by [9, Theorem 2], we have that
φ(w)
∠ lim =1.
w→∞ w
Now, fix 0 < c < +∞ and consider the Stolz angle with respect to ∞, S(∞,c) =
{u+iv :|v|<cu}. If w =u+iv ∈S(∞,c), then
(cid:9)(cid:9)(cid:9)(cid:9)φ(w)−w(cid:9)(cid:9)(cid:9)(cid:9)=(cid:9)(cid:9)(cid:9)(cid:9)φ(w)−w(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9) w (cid:9)(cid:9)(cid:9)≤(1+c)(cid:9)(cid:9)(cid:9)(cid:9)φ(w) −1(cid:9)(cid:9)(cid:9)(cid:9).
Rew w Rew w
φ(w)−w
Therefore, limw→∞,w∈S(∞,c) Rew = 0. Looking at the above expression for
ρ(cid:1)H(w,φ(w)), we conclude that ∠limw→∞ ρ(cid:1)H(w,φ(w))=0 as wanted. (cid:2)
It is also well known that if z ∈ D and ϕ ∈ Hol(D;D) is parabolic with
0
Denjoy-Wolff point τ ∈∂D, then the sequence of real numbers (Reφ (w )) is non-
n 0
decreasing, where φ is the iteration map associated with ϕ in H and w :=σ (z ).
0 τ 0
Therefore, we can consider L(z ) := lim Reφ (w ) ∈ (0,+∞]. Whenever L(z ) (cid:7)=
0 n n 0 0
+∞, the map ϕ is said to be of finite shift and, otherwise, of infinite shift with
respect to the point z . It is just a computation to check that the forward orbit
0
associated with z of ϕ is of finite shift type if and only if
0
1−|ϕ (z )|2
sup n 0 <+∞.
|τ −ϕ (z )|2
n n 0
Our first result shows that, indeed, this definition does not depend on the initial
pointoftheorbit,either. Weprovideaproofsincewearenotawareofanyreference
for it.
Proposition 3.2. Let ϕ∈ Hol(D;D) be parabolic with Denjoy-Wolff point τ ∈∂D.
If a forward orbit of ϕ is of finite shift type, then all of the forward orbits of ϕ
are also of finite shift type. In fact, for any two arbitrary points z ,z ∈ D, the
1 2
following limit always exists and is positive:
1−|ϕ (z )|2 |τ −ϕ (z )|2
L:=(L(z ,z )=)lim n 1 n 2 >0.
1 2 n |τ −ϕn(z1)|2 1−|ϕn(z2)|2
Moreover, if ϕ is of zero hyperbolic step, then L=1.
Proof. Asusual,weconsiderandprovetheabovestatementfortherighthalf-plane
H.Inthissense,wedenotebyφtheiterationmapin Hassociatedwithϕ.Wepoint
out that
1−|ϕ (z)|2
Reφ (w)= n ,
n |τ −ϕ (z)|2
n
where σ (z)=w. According to [9], we know that the limit
τ
φ (w)−iy
h(w):=lim n n, x =Reφ (1), y =Imφ (1),
n xn n n n n
exists,foreveryw ∈H.Moreover,ifφisofzeroh-step,thenhisconstantandequal
to one and if φ is of positive h-step, then h ∈ Hol(H;H). Now, take an arbitrary
point w ∈H. Therefore, if φ is of zero h-step,
1
Reφ (w )
lim n 1 =1,
n xn
SECOND ANGULAR DERIVATIVES AND PARABOLIC ITERATION 363
and, if φ is of positive h-step,
Reφ (w )
lim n 1 =Reh(w )>0.
n xn 1
From this, the result is clear. (cid:2)
Combining the concepts of zero/positive h-step and finite/infinite shift type, we
have a priori a classification of parabolic maps in four different classes. However,
we really only have three. This fact has been observed by P. Poggi-Corradini [8].
Anyway, for further reference and for the sake of completeness, we present here a
slight variation of his proof.
Proposition 3.3. Let ϕ∈ Hol(H;H) be parabolic of zero hyperbolic step. Then ϕ
is of infinite shift type.
Proof. Again, we consider and prove the above statement for the right half-plane
H. Let φ be the analytic map in H associated with ϕ. We know now (Proposition
3.2) thatbeingof finite/infinite shift typedoesnotdependontheinitial point. So,
suppose on the contrary, that
limx =L∈(0,+∞),
n
n
where x =Reφ (1) and y =Imφ (1). According again to [9], we have that
n n n n
φ (w)−iy
h (w)= n n −n→1, for every w ∈H.
n x
n
Therefore, (Reφ (w)) tends also to L, independently of the initial point w ∈ H.
n
Since(Reφ (w))isnon-decreasing,alsoforeveryw ∈H,wearriveatacontradiction
n
just by noting that
L+1≤Reφ (L+1)→n L.
n
(cid:2)
The other three classes are non-empty. In fact, φ(w)=w+1 is an example of a
parabolic map in H with zero h-step and infinite shift type and φ(w)=w+i is an
exampleofaparabolicmapinHwithpositiveh-stepandfiniteshifttype. Besides,
examples of parabolic maps in H with positive h-step and infinite shift type are
given by √
φ(w)=( w+a(1+i))2, with a>0 and w ∈H.
According to Theorem 4.1 below, any parabolic map of this type (positive h-
step and infinite shift type) provides an example of separation (with respect to
the Denjoy-Wolff point) between being of angular-class of order two and being of
angular-class of order one.
Tangential/non-tangential convergence, zero/positive h-step and finite/infinite
shift type are concepts which are known to be deeply connected for the important
and basic family of parabolic linear fractional self-maps in the unit disk. In the
next result and, for the sake of clarity, we have put together some of these charac-
terizations. In a certain sense, a large part of our work in sections four and five is
modelled upon this result.
Proposition 3.4. Let ϕ ∈ Hol(D;D) be a parabolic linear fractional map with
Denjoy-Wolff point τ ∈ ∂D. Then ϕ is analytic beyond the closed unit disk and
ϕ(cid:4)(cid:4)(z)(cid:7)=0, for every z ∈D. Moreover, the following statements are equivalent.
364 M.D.CONTRERAS,S.D´IAZ-MADRIGAL,ANDC.POMMERENKE
(1) ϕ∈Aut(D).
(2) The map ϕ is of positive hyperbolic step.
(3) The map ϕ is of finite shift.
(4) Re(ϕ(cid:4)(cid:4)(τ)τ)=0.
(5) Every (resp. some) forward orbit of ϕ tends tangentially to τ.
It is worth mentioning that, by Proposition 2.1, the opposite statement of the
above item 4 is Re(ϕ(cid:4)(cid:4)(τ)τ) > 0. We also want to point out that item 5 just says
that any forward orbit (z ) of ϕ cannot be asymptotically contained in any Stolz
n
angle of τ. However, it is also known that, for all the forward orbits (z ) of these
n
parabolic linear fractional self-maps of positive h-step, we have that the limits
lim Arg(1−τz )
n
n
π π
always exist, all of them have the same value and this value is or − .
2 2
In our analysis of second angular differentiability, an important role is played
by the intertwining maps developed by the third author alone [9] and jointly with
Baker [1]. Namely, given a parabolic map φ ∈ Hol(H;H) with Denjoy-Wolff point
∞ and of positive h-step, it is proved in [9] that the limit
φ (w)−iy
h(w):=lim n n, w ∈H, x =Reφ (1), y =Imφ (1)
n xn n n n n
exists uniformly on compact subsets of H. Moreover, h∈ Hol(H;H) and it satisfies
h◦φ=h+iδ
y −y
for a certain non-zero real number δ = δ(φ,1) = lim n+1 n. In other words,
n x
n
h is an intertwining map between the iteration couples (φ,H) and (w+iδ,H). In
what follows, this function h will be called the Koenigs map of φ (normalized with
respect to 1). Clearly, we can normalize with respect to another point w ∈ H. In
0
this case,
φ (w)−iImφ (w )
h (w) = lim n n 0
w0 n (cid:3) Reφn(w0) (cid:4)
φ (w)−iy Imφ (w )−y x
= lim n n −i n 0 n n
n xn xn Reφn(w0)
h(w)−iImh(w ) h(w)−iImh(w )
= 0 = 0 .
Reh(w ) Reh(w )
0 0
δ
Therefore,thecorrespondingconjugationequationisnowh ◦φ=h +i .
w0 w0 Reh(w )
0
δ
We note that the signs of and δ are exactly the same. It is important to
Reh(w )
0
isolate this fact and we say that the above parabolic of positive h-step map φ is
of type I whenever δ > 0 and of type II if δ < 0. All of these definitions pass to
the unit disk context in the usual way by using Cayley maps. For instance, given
an arbitrary parabolic map ϕ ∈ Hol(D;D) with Denjoy-Wolff point τ ∈ ∂D and of
positive h-step, the corresponding Koenigs map of ϕ will be hD :=h◦στ, where h
is the Koenigs map of φ, that is, the analytic map in H associated with ϕ.
SECOND ANGULAR DERIVATIVES AND PARABOLIC ITERATION 365
Given a parabolic map φ ∈ Hol(H;H) with Denjoy-Wolff point ∞ and of zero
hyperbolic step, it is proved in [1] that the limit
φ (w)−w
h(w):=lim n n, w ∈H, w =φ (1)
n wn+1−wn n n
exists uniformly on compact subsets of H. Thus, h ∈ Hol(H;C) and, in general,
that is all we can say about the image of h. In what follows, this function h will
also be called the Koenigs map of φ (normalized with respect to 1) and it satisfies
h◦φ=h+1.
In other words, h is an intertwining map between the iteration couples (φ,H) and
(w+1,C). Again, these definitions pass to the unit disk context in the usual way.
So,givenanarbitraryparabolicmapϕ∈Hol(D;D)withDenjoy-Wolffpointτ ∈∂D
and of zero h-step, the corresponding Koenigs function of ϕ will be hD = h◦στ,
where his theKoenigsfunctionof φ, theanalyticmap in Hassociatedwithϕ. For
a unified approach to Koenigs map for parabolic iteration, we refer the reader to
[6].
We end this section by showing a technical lemma.
Lemma 3.5. Let φ ∈ Hol(H;H) be parabolic with ∞ as Denjoy-Wolff point and
denote p(w) := φ(w)−w, w ∈ H. For any two points w,w∗ ∈ H, the following
inequality holds:
|p(w)−p(w∗)|≤2 ρ(cid:1)H(w,w∗) min{Rep(w),Rep(w∗)}.
1−ρ(cid:1)H(w,w∗)
Moreover, given an arbitrary forward orbit (w ) of φ there exists a constant C >0
n
(not depending on n) such that, for every n∈N and every w ∈[w ,w ], we have
n n+1
|p(w)−p(w )|≤CRep(w ).
n n
Proof. Since φ ∈ Hol(H;H), we have that Rep(w) > 0, for all w ∈ H, or p is
constantly equal toiα, for some α∈R. Inthis secondcase, the lemmais trivial, so
we assume that p∈Hol(H;H). Then, by the Schwarz-Pick Lemma, we obtain that
for any pair of points w,w∗ ∈H,
|p(w)−p(w∗)|≤(cid:9)(cid:9)(cid:9)(cid:9)w−w∗(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)p(w)+p(w∗)(cid:9)(cid:9)(cid:9)
w+w∗
(cid:9) (cid:9)
≤(cid:9)(cid:9)(cid:9)w−w∗(cid:9)(cid:9)(cid:9)(2Rep(w∗)+|p(w)−p(w∗)|).
w+w∗
Hence,
|p(w)−p(w∗)|≤ 2ρ(cid:1)H(w,w∗) Rep(w∗).
1−ρ(cid:1)H(w,w∗)
Now, take an arbitrary forward orbit (w ) of φ. At this point, we recall the
n
following fact on pseudo-hyperbolic distances on the right half-plane: If a,b ∈ H,
then
max{ρ(cid:1)H(w,a),ρ(cid:1)H(w,b)}≤ρ(cid:1)H(a,b)
for every w belonging to the segment [a,b]. Now, applying this fact to the different
couples {w ,w }, we deduce that, for every n and for every w belonging to the
n n+1
segment [w ,w ],
n n+1
|p(w)−p(w )|≤ 2ρ(cid:1)H(wn,wn+1) Rep(w ).
n 1−ρ(cid:1)H(wn,wn+1) n
366 M.D.CONTRERAS,S.D´IAZ-MADRIGAL,ANDC.POMMERENKE
Moreover, we know that limnρ(cid:1)H(wn,wn+1) always exists and belongs to [0,1).
Therefore,wecanfindC >0whichdependsonthechosenorbitbutnotonn,such
that
sup 2ρ(cid:1)H(wn,wn+1) ≤C.
n 1−ρ(cid:1)H(wn,wn+1)
(cid:2)
4. Second order angular differentiability: The positive step case
Throughoutthissection,weonlydealwithparaboliciterationofpositiveh-step.
Inourfirstresult,westudytherelationshipamongstbeingofangular-classoforder
two, beingoffiniteshifttypeandtheangulardifferentiabilityofKoenigsfunctions.
Theorem 4.1. Let ϕ ∈ Hol(D;D) be parabolic of positive hyperbolic step and let
τ ∈∂D be the corresponding Denjoy-Wolff point. The following are equivalent.
(1) The map ϕ belongs to C2(τ).
A
(2) The map ϕ is of finite shift type.
(3) The map hD belongs to CA1(τ), where hD is the Koenigs function associated
with ϕ.
Proof. As usual, we will mainly work in the right half-plane H. Moreover, given a
forward orbit (z ) of ϕ, the sequence (w ) will be the corresponding forward orbit
n n
of φ and vice versa. That is, σ (z ) = w . Finally, h will be the corresponding
τ n n
Koenigs map of φ. We recall that h◦φ=h+iδ, for some real number δ (cid:7)=0.
(1)⇒(2) Suppose that (2) is false and take an arbitrary orbit (w ):=(φ (w ))
n n 0
of φ such that lim Rew = +∞. Since ϕ ∈ C2(τ) and, according to Proposition
n n A
2.1, we can assure that
w =w +a+γ(w ),
n+1 n n
where a = ϕ(cid:4)(cid:4)(τ)τ ∈ C and γ ∈ Hol(H;C). Note that we cannot give any infor-
mation about lim γ(w ), since we do not know if (w ) tends non-tangentially to
n n n
∞.
Now, let us set p := a+γ ∈ Hol(H;C). As usual, there are two possibilities:
Rep(w) > 0, for all w ∈ H, or p is constantly equal to iα, for some α ∈ R. In this
second case, the implication is clear, so we assume that p∈Hol(H;H).
According to Lemma 3.5, we can find C > 0 such that, for every n ∈ N and
every w ∈[w ,w ] we have that
n n+1
|p(w)−p(w )|≤CRep(w ).
n n
In addition, we also know that
Rep(w ) Rew
n = n+1 −1→n 0.
Rew Rew
n n
So we obtain that
|p(w)−p(w )|
lim sup n =0.
n→∞w∈[wn,wn+1] Rewn
Besides, we have (see the comments in section three about normalizations in the
positive h-step case)
Imp(w ) Imφ (φ(w ))−Imw δ
lim n =lim n 0 n = (cid:7)=0.
n Rewn n Rewn Reh(w0)