Analytic flows on the unit disk: angular derivatives and boundary fixed points

35 Pages·2006·0.3 MB·English

Pacific
Journal of
Mathematics
ANALYTIC FLOWS ON THE UNIT DISK: ANGULAR
DERIVATIVES AND BOUNDARY FIXED POINTS
MANUEL D. CONTRERAS AND SANTIAGO D´IAZ-MADRIGAL
Volume222 No.2 December2005
PACIFICJOURNALOFMATHEMATICS
Vol.222,No.2,2005
ANALYTIC FLOWS ON THE UNIT DISK: ANGULAR
DERIVATIVES AND BOUNDARY FIXED POINTS
MANUEL D. CONTRERAS AND SANTIAGO DÍAZ-MADRIGAL
We use the concept of angular derivative and the hyperbolic metric in the
unitdiskD,tostudythedynamicalaspectsoftheequilibriumpointsbelong-
ing to ∂D of some complex-analytic dynamical systems on D. Our results
showadeepconnectionbetweenthedynamicalpropertiesofthoseequilib-
rium points and the geometry of certain simply connected domains of C.
As a consequence, and in the context of semigroups of analytic functions,
we give some geometric insight to a well-known inequality of Cowen and
Pommerenkeabouttheangularderivativeofananalyticfunction.
1. Introduction
A (one-parameter) semigroup of analytic functions is any continuous homomor-
phism8:t7→8(t)=ϕ fromtheadditivesemigroupofnonnegativerealnumbers
t
intothecompositionsemigroupofallanalyticfunctionswhichmapDintoD. That
is,8satisfiesthefollowingthreeconditions:
(a) ϕ istheidentityinD,
0
(b) ϕt+s =ϕt ◦ϕs,forallt,s ≥0,
(c) ϕ (z)tendsto z ast tendsto0,uniformlyoncompactsubsetsofD.
t
Itiswell-knownthatcondition(c)canbereplacedby
(c’) Forevery z∈D,limt→0ϕt(z)=z.
Semigroups of this type appear in many areas of analysis, such as the theory
ofcompositionoperators,thetheoryofMarkovstochasticprocesses,optimization
theoryandthetheoryofplanarvectorfields. Inthispaper,weareinterestedinthis
lastaspect,whichwediscussindetailforcompleteness.
MSC2000: primary30C20,30D05,37F99;secondary30F45,37E35.
Keywords: angularderivative,fixedpoints,planarvectorfields,semigroupsofanalyticfunctions.
This research has been partially supported by the Ministerio de Ciencia y Tecnología (projects
BFM2000-1062 and BFM2003-07294-C02-02) and by La Consejería de Educación y Ciencia de
laJuntadeAndalucía.
253
254 MANUELD.CONTRERASANDSANTIAGODÍAZ-MADRIGAL
Givenasemigroup8=(ϕ ),itcanbeproved(see[Shoikhet2001;Berksonand
t
Porta 1978]) that there exists a unique analytic function G :D→C such that, for
each z∈D,thetrajectory
γ :[0,+∞)→R2, t 7→γ (t):=ϕ (z)
z z t
isthesolutionoftheCauchyproblem
(cid:26) x˙ = f(x,y), y˙ =g(x,y),
(x(0),y(0))=z,
where f =ReG and g =ImG. Usually, this planar dynamical system is written
in the form w˙ =G(w), that is, as a complex-analytic dynamical system in D. We
call G the vector field, and w˙ = G(w) the dynamical system, associated with 8.
In operator theory, G is also known as the infinitesimal generator of 8. There
is a very nice representation, due to Berkson and Porta [1978], of those analytic
functionsonthediskwhicharegeneratedinthisway:
An analytic function G : D → C is the vector field of a semigroup of analytic
functions8ifandonlyifthereisapointb∈Dandananalyticfunction p:D→C
withRe p≥0suchthat
G(z)=(b−z)(1−b¯z)p(z), z∈D.
Moreover,sucharepresentationisunique. Thepointb iscalledtheDenjoy–Wolff
point(DW-point)of8and p iscalledtheCarathe´odoryfunctionassociated to8.
Looking at this representation, we see that, whenever b ∈ D, the associated
dynamical system has a critical point. However, G never vanishes if b ∈ ∂D.
Moreover, it is well known that complex-analytic dynamical systems in simply
connected domains cannot have limit cycles [Needham and King 1994], so it is
natural to wonder about the “elements” which govern (and how they do it) the
dynamics of this system without critical points and without limit cycles. In the
rest of this section, we will see that the key for understanding this dynamics is
the study of some “critical points” belonging to ∂D. Since, in our setting, G and
thefunctions(ϕ )areonlydefinedinD,thatcommentrequiresaclarificationand
t
thisleadsustorecallsomeconceptsfromcomplexfunctiontheory[Pommerenke
1975;1992;Shapiro1993].
Letϕ:D→Cbeananalyticfunctionandtakea∈∂D. Wesaythat L ∈C∞ is
theangularlimit ofϕ ina when z tendstoa if,foreveryα∈(0,π/2),
lim ϕ(z)= L,
z∈S(a,α)
z→a
where S(a,α)denotestheStolzanglewithcenterata andopeningα. Thenumber
L is commonly denoted by 6 limz→aϕ(z). When 6 limz→aϕ(z)=a, the point a
ANALYTICFLOWSONTHEUNITDISK 255
is called a boundary fixed point of ϕ. If we suppose additionally that ϕ(D)⊂D,
theangularlimit
ϕ(z)−a
6 lim
z→a z−a
always exists and belongs to (0,+∞) ∪ {∞}. This value is denoted by ϕ0(a)
and we refer to it as the multiplier of ϕ in a. When ϕ0(a) ∈ (0,+∞), the limit
6 limz→aϕ0(z) exists and is equal to ϕ0(a). Following the more or less standard
usageiniterationtheory,wewillsaythataisattractiveifϕ0(a)∈(0,1];repulsiveif
ϕ0(a)∈(1,+∞)andsuperrepulsiveifϕ0(a)=∞. Thesedefinitionsareconsistent
with the intuitive geometric meaning of a repulsive or attractive point, thanks to
thecelebratedDenjoy–WolffTheorem[Shapiro1993,Section5.1].
Now, we come back to the semigroup 8=(ϕ ) and the associated vector field
t
G with DW-point b ∈ ∂D. A point a ∈ ∂D is a boundary fixed point of 8 if a
is a boundary fixed point for each ϕ , where t ≥ 0. It is a nontrivial fact due to
t
Cowen [1981, Theorem 5.2] that “for each ϕ ” can be replaced by “for some ϕ ”.
t t
Moreover,theDW-pointbisaboundaryfixedpointof8and,indeed,theDenjoy–
Wolff Theorem admits the following version for semigroups (see [Berkson and
Porta1978;Siskakis1985]):
The point b is the unique boundary fixed point of 8 with ϕ0(b) ≤ 1 for some
t
(henceforall)t >0. Moreover,foreveryz∈D,wehavelimt→+∞ϕt(z)=b.
Combining the results of [Siskakis 1985] and [Cowen 1981], it is possible to
deducethatthedynamicalcharacterofthemultipliersofallthefunctionsϕ isthe
t
same. Thatis,apointξ∈∂Disattractive(orrepulsive,orsuperrepulsive)forsome
t ifandonlyifthesamehappensforallt.
Our study of the system w˙ =G(w) will be carried out by considering a model
flow where the trajectories become straight lines but they fill in a more involved
planar domain (cid:127). In other words, our results will explain the dynamics of w˙ =
G(w)intermsofthegeometryof(cid:127). Thisdomain(cid:127)isconstructedusingthetheory
ofunivalentfunctions[BerksonandPorta1978;Heins1981;Siskakis1985]: there
is a unique univalent function h :D→C with h(0)=0 whose image (cid:127):=h(D)
satisfies,
(cid:127)+t ⊂(cid:127) foreacht >0
andsuchthat
ϕ (z)=bh−1(h(b¯z)+t) fort ≥0, z∈D.
t
Wecall(cid:127):=h(D)⊂Ctheassociatedplanardomainof8. Clearly,ifwefixz∈D
and consider the trajectory t → γ (t) = ϕ (z) with respect to the vector field G,
z t
thecorrespondingtrajectoryinthemodelflowist →h(z)+t.
Our dynamical approach to w˙ = G(w) deals with the following topics: types
oftrajectories,theirω-limitsandα-limitsandtheirrelationshipwiththeboundary
256 MANUELD.CONTRERASANDSANTIAGODÍAZ-MADRIGAL
fixedpoints;themultipliersofthesepointsandtheirdynamicmeaning;theslopeof
thetrajectoriesalongwhichtheyarriveatsuchafixedpoint. Werecallthatapoint
ξ∈C∞iscalledanω-limitpointofacurve0:(s1,s2)→C(−∞≤s1<s2≤+∞)
if there exists a strictly increasing sequence (t ) ⊂ (s ,s ) converging to s such
n 1 2 2
that 0(t ) → ξ. The set of all ω-limit points of 0 is called the ω-limit set and
n
denoted by ω(0). The definitions of α-limit and α(0) can be given in a similar
waybutnowtheroleofs isplayedbythepoints .
2 1
Fromthepointofviewofω-limits,andinourcontext,theresultofBerksonand
Portacitedabovereadsasfollows:
Let 8 be a semigroup of analytic functions with boundary DW-point b, planar
domain (cid:127) and vector field G. Moreover, let γ be a trajectory of the vector field
z
G. Then γz is defined for every t ≥0, limt→+∞γz(t)=b and its largest possible
intervalofdefinitionis(T,+∞)with−∞≤T <0.
The article contains four sections after this introduction. In Section 2 we state
themainresultsofthepaper. Withtheaimofmakingthepapermorereadable,we
havegroupedinSection3somefactsaboutthehyperbolicdistanceontheunitdisk.
In Section 4 we obtain new results about the relationship among boundary fixed
points,nontangentialconvergenceandangularderivate. Someoftheseresultsmay
havesomeinterestoftheirown,butherewethinkofthemasnecessaryingredients
for the proof of the main results. In Section 5 we give the proofs of all the results
statedinSection2.
2. Statementofthemainresults
Fortheanalysisofthemultipliersofϕ attheDW-pointb,weintroduceageometric
t
quantity associated to the planar domain (cid:127). Namely, let us denote by ν((cid:127)) the
supremumofallpositivenumbersβ suchthatthereisc∈Cwith
(cid:8)c+ti :−1β <t < 1β(cid:9)⊂(cid:127).
2 2
We point out that this is a well-defined concept since (cid:127) is open and nonvoid.
Clearly, ν((cid:127)) ∈ (0,+∞]. It is an exercise to show that, whenever ν((cid:127)) < +∞,
the number ν((cid:127)) coincides with the infimum of all positive β such that, there is
c∈Rwith
(cid:127)⊆(cid:8)z∈C:c− 1β <Imz<c+ 1β(cid:9).
2 2
Bearinginmindtheproperty(cid:127)+t⊂(cid:127),weseethatν((cid:127))representsthe“narrowest”
widthofanopenstripparalleltotherealaxisandcompletelyenclosing(cid:127).
The following theorem explains the relation between ν((cid:127)) and the multipliers
ofϕ intheDW-pointb,whenν((cid:127))<∞.
t
ANALYTICFLOWSONTHEUNITDISK 257
Theorem 2.1. Let 8 = (ϕ ) be a semigroup of analytic functions with boundary
t
DW-pointbandplanardomain(cid:127). Thefollowingstatementsareequivalent.
(1) ν((cid:127))<∞.
(2) Forallt >0,wehaveϕ0(b)<1.
t
(3) Thereist >0suchthatϕ0(b)<1.
t
(cid:16) π (cid:17)
Moreover,whenν((cid:127))<∞,itfollowsthatϕ0(b)=exp − t forallt ≥0.
t ν((cid:127))
FromTheorem2.1andthefactthatϕ0(b)≤1forsome(andthereforeall)t>0,
t
wereadilydeduce:
Corollary 2.2. Let 8=(ϕ ) be a semigroup of analytic functions with boundary
t
DW-pointbandplanardomain(cid:127). Thefollowingstatementsareequivalent.
(1) ν((cid:127))=∞.
(2) Forallt >0,wehaveϕ0(b)=1.
t
(3) Thereist >0suchthatϕ0(b)=1.
t
The relation between boundary fixed points and the backward evolution of the
trajectoriesisexpressedinthenextproposition.
Proposition2.3. Let8=(ϕ )beasemigroupofanalyticfunctionswithboundary
t
DW-point b, planar domain h(D) = (cid:127) and vector field G. Moreover, let γ be a
z
trajectory of the vector field G with the maximum possible interval of definition
(T,+∞)being−∞≤T <0.
(1) If−∞<T,thelimit
a:= lim γ (s)∈∂D,
z
s→T
exists,a isnotaboundaryfixedpointof8,andh(z)+T ∈∂(cid:127).
(2) IfT =−∞,thelimit
a:= lim γ (s)∈∂D
z
s→−∞
existsanda isaboundaryfixedpointof8.
In our analysis of backward trajectories, a crucial step will be to develop tools
in order to detect when different trajectories go to the same boundary fixed point.
Forthis,weintroducetheplanarsubset
(cid:18) (cid:19)
\
V((cid:127)):=int ((cid:127)+t) .
t≥0
We will check that V((cid:127))+t = V((cid:127)), for every t ∈R, so we call it the invariant
setassociatedto(cid:127). Itisworthmentioningthat V((cid:127))canbeempty. But,if V((cid:127))
258 MANUELD.CONTRERASANDSANTIAGODÍAZ-MADRIGAL
is nonvoid and since it is obviously open, we can write V((cid:127)) as the union of the
countable (finite or infinite) family of its different connected components. These
componentswillbedenotedby V ((cid:127))(j ∈ J)and,dependingonthecase, J =N
j
or J ={1,...,n}forsomen∈N.
It can be proved that every component V ((cid:127)) of the invariant set V((cid:127)) is an
j
openstriporahalf-planeparalleltotherealaxisinbothcases.
Theorem 2.4. Let 8 = (ϕ ) be a semigroup of analytic functions with boundary
t
DW-pointbandplanardomain(cid:127)=h(D).
(1) IfacomponentV ((cid:127))ofV((cid:127))isahalf-plane,thenforeveryz∈h−1(V ((cid:127))),
j j
thecorrespondingtrajectoryγ isdefinedforallt ∈(−∞,0]andtheα-limit
z
α(γ )={b}.
z
(2) If there is z ∈ D such that the corresponding trajectory γ is defined for all
z
t ∈ (−∞,0] and α(γ ) = {b}, then there exists a component V ((cid:127)) of the
z j
invariantsetof(cid:127)whichisahalf-planeandh(z)∈V ((cid:127)).
j
There can be one or two half-open components of V((cid:127)) related to b. In either
case,wehaveν((cid:127))=∞and,therefore,byCorollary2.2,ϕ0(b)=1,forallt >0.
t
Thosecomponentsoftheinvariantsetwhicharestripsarecloselyrelatedtothe
collection of the repulsive boundary fixed points of 8=(ϕ ), that is, those points
t
a∈∂Dsuchthatϕ0(a)∈(1,+∞),foreveryt >0.
t
Theorem 2.5. Let 8 = (ϕ ) be a semigroup of analytic functions with boundary
t
DW-pointbandplanardomain(cid:127)=h(D).
(1) Letthecomponent V ((cid:127))of V((cid:127))beanopenstrip. Thereisauniquerepul-
j
siveboundaryfixedpointξ(V ((cid:127)))of8suchthat,foreveryz∈h−1(V ((cid:127))),
j j
the corresponding trajectory γ is defined for all t ∈ (−∞,0] and α(γ ) =
z z
{ξ(V ((cid:127)))}.
j
(2) The map ξ thus defined between open strip components of the invariant set
andrepulsiveboundaryfixedpointsisbijective.
(3) If there is z ∈ D such that the corresponding trajectory γ is defined for all
z
t ∈ (−∞,0] and α(γ ) is a repulsive boundary fixed point ξ(V ((cid:127))), then
z j
h(z)∈V ((cid:127)).
j
Now, we study the multipliers of the functions ϕ at boundary fixed points ξ
t
whichareα-limitsoftrajectoriesofthesystemw˙ =G(w). Clearly,wehavethree
possibilities: ξ istheDW-point,ξ isrepulsiveorξ issuperrepulsive. Therepulsive
casewillbeanalyzedagainbymeansofthestripcomponentsoftheinvariantset.
ANALYTICFLOWSONTHEUNITDISK 259
Theorem 2.6. Let 8 = (ϕ ) be a semigroup of analytic functions with planar
t
domain(cid:127). Ifξ ∈∂Disarepulsiveboundaryfixedpointof8,then
(cid:18) π (cid:19)
ϕ0(ξ)=exp t , forallt ≥0,
t β ((cid:127))
j
whereβ ((cid:127))isthewidth(necessarilyfinite)ofthestrip V ((cid:127))oftheinvariantset
j j
of(cid:127)associatedtoξ.
These results can be translated to the context of the angular derivative of an
analytic function. Cowen and Pommerenke [1982, Theorem 6.1] showed that if
ϕ is an analytic and univalent function in D with ϕ(D)⊂D, Denjoy–Wolff point
b∈∂D, ϕ0(b)<1 and ξ ,ξ ,...,ξ are distinct fixed points of ϕ (different from
1 2 n
b),then
n
X 1 1
≤−
logϕ0(ξ ) logϕ0(b)
j
j=1
and,ifequalityholds,ϕ canbeembeddedinasemigroupofanalyticfunctionsand
D\ϕ(D)consistsofn−1analyticarcs. Indeed,somegeometricmotivationofthis
inequalitycanbereadinthatpaper.
Now,ifwesupposethatϕcanbeembeddedinasemigroupofanalyticfunctions,
byTheorems2.1and2.6,wecanguaranteethat,forsomet >0,
(cid:18) π (cid:19) (cid:18) π (cid:19)
ϕ0(b)=exp − t and ϕ0(ξ )=exp t .
ν((cid:127)) j β ((cid:127))
j
Therefore,wecanalwaysrewritetheinequalityaboveas
n
X
β ((cid:127))≤ν((cid:127)).
j
j=1
Thisinequalityreflectsthegeometricfactthatthenarroweststripincluding(cid:127)must
containthefamilyofthedisjointstrips V ((cid:127)).
j
Finally,wetreatthesuperrepulsivecase.
Theorem 2.7. Let 8 = (ϕ ) be a semigroup of analytic functions with boundary
t
DW-pointbandplanardomain(cid:127)=h(D)asintheintroduction. Ifthereisapoint
z∈Dsuchthat
(cid:18) (cid:19) (cid:18) (cid:19)
\ [
h(z)∈ ((cid:127)+t) \ V ((cid:127))
j
t≥0 j∈J
andthecorrespondingtrajectoryγ isdefinedforallt∈(−∞,0]andα(γ )={ξ},
z z
thenξ isasuperrepulsiveboundaryfixedpointof8andγ istheuniquetrajectory
z
oftheplanardynamicalsystemwithα(γ )={ξ}.
z
260 MANUELD.CONTRERASANDSANTIAGODÍAZ-MADRIGAL
This does not necessarily account for all superrepulsive fixed points: there are
semigroups with a superrepulsive point ξ where there is no trajectory γ with
z
α(γ )={ξ}. Oneexampleisthefollowing: Considertheplanarsubset
z
(cid:127):=(cid:8)z∈C:−1 <Imz<exp(Rez)− 1(cid:9),
2 2
andtakeaRiemannmaph fromDonto(cid:127)withh(0)=0. Now,thesemigroupwe
arelookingforis
ϕ (z):=h−1(h(z)+t), z∈D.
t
The inequality of Cowen and Pommerenke mentioned above implies that the
numberofrepulsivefixedpointsontheboundaryoftheunitdiskisdenumerable.
However, the number of boundary fixed points can be uncountable (of course,
most ofthem must besuperrepulsive). To buildsuch an example, considerthe set
ofrationalnumbersoftheinterval(−1,1),say{α(n):n∈N},andtheplanarsubset
(cid:18) (cid:19)
[
(cid:127):={z∈C:−1<Imz<1}\ ((−∞,−n]+α(n)i)
n∈N
and take a Riemann map h from D onto (cid:127) with h(0)=0. Now, by Theorem 2.7,
thesemigroupofanalyticfunctionsgivenby
ϕ (z):=h−1(h(z)+t), z∈D
t
hasanondenumerablesetofboundaryfixedpoints. Thisexamplewassuggestedto
theauthorsbyRicardoPe´rez-Marco. Wethankhimforthisexampleandotherin-
terestingremarksconcerningthispaper. Additionalinformationabouttheamount
offixedpointsofanalyticself-mapsonDcanbereadin[CowenandPommerenke
1982,Section2].
Our next step is the analysis of the slopes of the trajectories of the system.
As before, this requires recalling some notation first and distinguishing between
several cases. These cases are related to the evolution of the distance of a fixed
point to the boundary of the domain (cid:127) and it suggests to introduce the following
notation:
δ(cid:127)(w)=inf{|w−z|:z∈∂(cid:127)}, w∈(cid:127).
Givenacurve0:t∈(s ,s )→0(t)∈Dwithasingletonω-limitω(0)={ζ}⊂
1 2
∂D, the set of the slopes of 0 when arriving at ζ will be denoted by Slope+(0)
anditisdefinedastheω-limitofthecurve
t ∈(s ,s )→Arg(1−ζ¯0(t))∈(cid:0)−π, π(cid:1),
1 2 2 2
whereArg(z)denotestheprincipalargumentofz. Itiswell-knownthatSlope+(0)
is a nonempty, compact, and connected subset of (cid:2)−π, π(cid:3) and we talk about the
2 2
ANALYTICFLOWSONTHEUNITDISK 261
slopeof0 atζ ifSlope+(0)isasinglepoint. Somethingsimilarcanbedonewith
α-limitsandthedefinitionofSlope−(0)isself-explanatory.
InviewofthecontinuousversionoftheDenjoy–WolffTheorem,weknowthat
limt→+∞γz(t) → b, so we may ask about the slope of the forward trajectory γz
when arriving at b. Our next theorem answers precisely this question when ν((cid:127))
isfinite.
Theorem 2.8. Let 8 = (ϕ ) be a semigroup of analytic functions with boundary
t
DW-point b and planar domain (cid:127). If ν((cid:127)) < ∞, then, for each z ∈ D, the set
Slope+(γ ) is a single point and it belongs to (cid:0)−π, π(cid:1). That is, the (forward)
z 2 2
trajectoryγ tendstobwithafixedslopeandnevertangentially. Moreover,given
z
α∈(cid:0)−π, π(cid:1),thereisz∈DsuchthatSlope+(γ )={α}.
2 2 z
Thenexttheoremtreatsthecaseν((cid:127))=+∞. Wenoticethat,givenw∈(cid:127),the
limitlims→+∞δ(cid:127)(w+s)alwaysexistsandbelongsto(0,+∞].
Theorem 2.9. Let 8 = (ϕ ) be a semigroup of analytic functions with boundary
t
DW-pointb,planardomain(cid:127)andν((cid:127))=+∞.
(1) If there is w ∈ (cid:127) such that lims→+∞δ(cid:127)(w +s) = +∞, then all the sets
Slope+(γ )areidentical,whenz runsthewholeD.
z
(2) Ifthereisw∈(cid:127)suchthatlims→+∞δ(cid:127)(w+s)∈(0,+∞),thenallthetrajec-
toriesγ (z ∈D)tendstangentiallytob. Thatis,Slope+(γ )isasinglepoint
z z
anditisequalto−π or π.
2 2
We conjecture that, in case (1) of this theorem, all the subsets Slope+(γ ) are
z
indeedsingletons,soitwouldbepossibletospeakaboutthecommonslopeofthe
trajectoriesofthesystem.
We now study the slopes of backward trajectories. The analysis of the sets
Slope−(γ )willbedonefirst,whentheα-limitofγ istheDW-point.
z z
Theorem 2.10. Let 8=(ϕ ) be a semigroup of analytic functions with boundary
t
DW-pointbandplanardomain(cid:127).
(1) If the α-limit of a trajectory γ is b, then the set Slope−(γ ) is a single point
z z
anditisequalto−π or π. Thatis,thebackwardtrajectoryγ tendstangen-
2 2 z
tiallytob.
(2) Giventwotrajectoriesγ andγ wherebistheirα-limitandsuchthath(z )
z1 z2 1
and h(z ) belong to the closure of the same half-plane component of V((cid:127)),
2
thenSlope−(γ )=Slope−(γ ).
z1 z2
Finally, we study the slopes of the backward trajectories reaching a repulsive
boundaryfixedpoint.
Theorem 2.11. Let 8=(ϕ ) be a semigroup of analytic functions with boundary
t
DW-pointbandplanardomain(cid:127)andξ(V ((cid:127)))arepulsiveboundaryfixedpoint.
j