PII. S0161171202011924 http://ijmms.hindawi.com © Hindawi Publishing Corp.
CONTRACTIVE CURVES
ARACELI BONIFANT and MARIUS DABIJA
Received 6 February 2001 and in revised form 31 August 2001
We discuss the dynamics of the correspondences associated to those plane curves whose local sections contract the Poincaré metric in a hyperbolic planar domain.
2000 Mathematics Subject Classification: 37F05, 37F15, 30C80.
1. Introduction. We consider certain 1-dimensional, holomorphic correspondences of hyperbolic type, which we call “contractive curves.” These are curves whose local sections contract the Poincaré metric of a hyperbolic planar domain. The model for our analysis is given by the hyperbolic Julia sets.
This work has been motivated by the work in Ochs’ recent paper [5]. In the first part, we adapt to our purposes the Schwarz lemma for correspondences that appears in Minda’s paper [3]. In the second part, we discuss the dynamics of a contractive curve, and the properties of the associated attractor. Such curves usually do not have global sections that contract the hyperbolic metric. Nevertheless, the associated dynamical systems have much in common with the iterated function systems of hyperbolic type. The basis for our discussion is the paper [2] by Barnsley and Demko.
2. Preliminaries
2.1. Hyperbolic metric. The hyperbolic metric (infinitesimal length-element) on the unit disk∆= {|z|<1} ⊂Cisds∆(z)=(2/(1−|z|2))|dz|.
LetU⊂Cbe a planar domain (open and connected subset ofC), and assume that the complementC\Ucontains at least two points, so thatUis a hyperbolic Riemann surface; the universal covering ofU is (biholomorphic to)∆. The densityρU of the hyperbolic metric inU,dsU(z)=ρU(z)|dz|, is defined as follows. Fix an unramified covering ∆ φ→U. Givenz∈U, choose anyt∈φ−1(z), and define ρU(z)=2/(1− |t|2)|φ(t)|. The definition ofρU(z)does not depend on the choice of∆ φ→U, nor
on the choice oft∈φ−1(z). Note thatρU is positive and real-analytic.
The hyperbolic distancesUinUis obtained by integrating the infinitesimal length-elementdsU. The metric space(U , sU)is complete.
The (Gaussian) curvature of a metric ρ(z)|dz|is Kρ= −(4/ρ2)(∂2(logρ)/∂z∂z). The hyperbolic metricdsU has constant negative curvature,KρU≡ −1.
2.2. Hausdorff distance. Given two subsetsK≠∅≠Hof a metric space(X, d), let
d(K, H):=inf(k,h)∈K×Hd(k, h), and define the Hausdorff distance
d(K, H):=max
sup
k∈Kd(k, H),suph∈Hd(K, h)
. (2.1)
WhenK1⊂K,d(K1, H)≤d(K, H).
Denote by(X, d):=(X,d)the metric space of nonempty compact subsets ofX.
Clearly,(X, d)is isometrically embedded in(X, d),d(x, y)=d(x, y).
If(X, d)is a complete metric space, then(X, d)is a complete metric space. There-fore, whenUis a hyperbolic domain,(U , sU)is a complete metric space.
Given(X, dX) f→(Y , dY)continuous, let(X, dX) f→(Y , dY)denote the contin-uous mapf(K):=f (K). Let Lip(f )denote the Lipschitz “norm” off,
Lip(f )=sup
dYf (x), f (u)
dX(x, u) :(x, u)∈X×X, x≠u
. (2.2)
We have Lip(f)=Lip(f ).
2.3. Stolz subdomains. Given a subdomainV of a hyperbolic planar domainU,
ρU(z)≤ρV(z)for allz∈V. Equality at some pointz∈V implies thatU=V.
Definition2.1. A subdomainVof a hyperbolic planar domainUis calledStolzif and only if there exists a constantk >1 so thatρV(z)/ρU(z)≥k >1 for allz∈V.
Equivalently,Vis Stolz inUif and only if the inclusion(V , sV) →i (U , sU)is a strict contraction, that is, Lip(i) <1.
IfV⊂U, thenVis Stolz inU. Roughly speaking, ifV⊂Uare planar domains with piecewiseC1-smooth boundaries, thenV is Stolz inUif and only if the boundaries
∂U and∂V have no tangency at any common boundary point.
Example2.2. The disk{|z−R|< R}is not Stolz in the half-plane{(z) >0}. For
a > b≥1, the angleAa= {|arg(z)|< π /2a}is Stolz inAb.
3. Schwarz lemma for correspondences
3.1. Ahlfors’ lemma. Recall the definition of ultra-hyperbolic metricsρ(z)|dz|in planar domains. An upper semicontinuous functionV ρ→[0,∞)is anultra-hyperbolic density functionif and only if, for allz∈V withρ(z) >0, there exists a positiveC2 -smooth functionVz ρz→(0,∞), defined in a neighborhoodVzofz, so thatKρz≤ −1,
ρz≤ρandρz(z)=ρ(z). Recall Ahlfors’ lemma (see [1]).
Proposition3.1. IfU f→V is a holomorphic map from a hyperbolic planar
do-mainUto an ultra-hyperbolic domain(V , ρ), thenρ(f )|f| ≤ρU. In particular,ρ≤ρV.
Definition3.2. LetUandVbe planar domains. A curveC⊂U×Vis calledproper if and only if the following two conditions are satisfied:
(i) the projectionC p→Uis proper;
(ii) no local branch ofChas vertical tangent.
Definition3.3. Alocal sectionof a curveC⊂U×V is a local section of the pro-jectionC p→U, that is, a holomorphic mapU0
f
→Vdefined on a subdomainU0⊂U so that Graph(f )⊂C.
3.3. Schwarz lemma of Nehari and Minda. Using Ahlfors’ lemma, Nehari [4] and then Minda [3] prove a Schwarz-Pick lemma for multivalent functions. We formulate and prove their result, in a more convenient form.
Proposition 3.4. LetU andV be hyperbolic planar domains. IfC ⊂U×V is a
proper curve, thenρV(f )|f| ≤ρU, for every local sectionfofC.
Proof. Given a local branchB=(C0, c)of C, letp(B):=p(c)∈U andq(B):=
q(c)∈V. Let(C,0) νB→Bbe a normalization,νB(t)=(x(t), y(t)). The slope ofBis
s(B)=limt→0(y(t)/x(t)).
Let ˜C ν→Cbe a normalization map of the (possibly reducible) plane curveC. Recall that ˜C is the curve of local branches ofC. Let ˜C νˇ→Cˇbe the dual curve ofC, that is, the curve of tangents of local branches of C. Then ˇC⊂ᏸ, where ᏸC2 is the space of nonvertical lines inC2. Letᏸ σ→Cbe the holomorphic map that associates to a line its slope. The map ˜C s→C,Bs(B), is holomorphic since it factorizes as
˜
C ˇν→Cˇ⊂ᏸ σ→C.
Given x∈U, letC(x) be the set of local branchesB ofC with p(B)=x. Since the projection C p→U is proper, C(x)is finite. Define the function U ρ→[0,∞),
ρ(x):=maxB∈C(x)ρV(q(B))·|s(B)|. By definition,ρV(f )|f| ≤ρfor every local section fofC. Clearly,ρ∈C(U ). We show thatρis ultra-hyperbolic.
LetBbe a local branch ofC withs(B)≠0. Letm(B)denote the multiplicity ofB. SinceBhas nonvertical tangent,m(B)equals the local degree ofB p→U. SinceBhas nonhorizontal tangent,m(B)equals the local degree ofq:B→V. We can define, in a punctured neighborhood ofp(B), the function
ρB(x):=
β∈B(x)
ρVq(β)·s(β)
1/m(B)
. (3.1)
Clearly,ρBis real-analytic, and 0< ρB≤ρin suitable local coordinates in ˜C,U, andV, we see thatρBisC2-smooth atρ(B)withρB(p(B))=ρV(q(B))·|s(B)|. We prove now thatKρB≤ −1.
LetW ρi→(0,∞), 1≤i≤m, beC2-smooth positive functions in a planar domain, and letW g→(0,∞)denote their geometric mean,g=[iρi]1/m. The inequality
be-tween the arithmetic mean and the geometric mean of finitely many positive numbers implies thatifK:=maxi(Kρi) <0, thenKg≤K.
1≤i≤m(B). LetW ρi→(0,∞),ρi:=ρV(fi)|fi|. The restriction toWofρBis the
geo-metric mean of the functionsρi. The conformal invariance of the curvature implies thatKρi=KρV= −1. Therefore,KρB≤ −1 inW. It follows thatKρB≤ −1 in a punctured neighborhood ofq(B). By continuity,KρB(q(B))≤ −1.
Now, fixx∈Uwithρ(x) >0, and chooseB∈C(x)withρ(x)=ρV(q(B))· |s(B)|. ThenρB is a support function ofρof x. In conclusion,ρis ultra-hyperbolic inU. Ahlfors’ lemma implies thatρ≤ρU, and the proof is finished.
Remark 3.5. Let C ⊂U×V (x, y)be a curve, with projections C p→U and
C q→V. There existsF∈ᏻ(U×V )so thatC= {F=0}. Let Reg(C)denote the smooth locus ofC. DefineC =supReg(C)[(ρV(q)/ρU(p))·((∂F /∂x)/(∂F /∂y))]. The
right-hand side is independent ofF. For every local sectionfofC,f∗dsV≤ CdsU. When
pis proper,Proposition 3.4says that eitherC = ∞(whenC has vertical tangents), orC ≤1 (whenChas no vertical tangent).
Remark3.6. Given an integern≥1 and a holomorphic∆ g→∆, consider the curve
C= {yn=g(x)} ⊂∆×∆. Letm(g)denote the minimum of the orders of the zeros
ofg. ClearlyC = ∞if and only ifm(g) < n. Assume thatm(g)≥n. ThenC ≤1, and the Schwarz-Pick lemma implies, for allz∈∆,
g(z)≤ng(z) 1−1/n
1−g(z)2/n
1−|z|2 . (3.2)
It is amusing to see this directly, as follows. Rewrite the inequality as(1−|z|2)|g(z)| ≤ |g(z)|·un(|g(z)|), with(0,1] un→(0,∞),un(t)=n(t−1/n−t1/n). Using the
confor-mal invariance of the quantity(1−|z|2)|g(z)|, we may assume thatz=0 andg(0)≠
0. We need to show that|g/g|(0)≤un(|g(0)|). Note two properties ofun: (i) un(s)+un(t)≤un(st)fors,tin(0,1];
(ii) unu∞, withu∞(t)= −2 log(t).
Writeg=Bh, whereB(z)=j((z−zj)/(1−zjz))nj is the Blaschke product
asso-ciated tog,nj≥n. Note that∆ h→∆is nonvanishing, hence, by the classical Schwarz lemma,|h/h|(0)≤u∞(|h(0)|). Also,
B
B
(0)= j
nj1−aj
2
aj =
j
unjajnj
≤
j
unajnj≤un
j ajnj
=unB(0),
(3.3)
that is,
Therefore,
g
g
(0)≤B
B
(0)+h
h (0) ≤unB(0)+u∞h(0) ≤unB(0)+unh(0) ≤ung(0),
(3.5)
and the inequality (3.2) is shown.
LetU,V be planar domains, andC⊂U×V a curve, with projectionsC p→Uand
C q→V. Assume thatpis proper. IfKis compact inU, thenp−1(K)is compact inC, henceq(p−1(K))is compact in V. We have therefore a map
U C→V, C(K)= q(p−1(K)).
Corollary3.7. LetU,V be hyperbolic planar domains. IfC⊂U×V is a proper
curve, thenLip(C)≤1.
Proof. GivenK,H inU, fixy0∈C(K), x0∈Kwith(x0, y0)∈C, andu0∈H withsU(x0, u0)=sU(x0, H). It suffices to findv0∈C(u0)withsV(y0, v0)≤sU(x0, u0). To show that suchv0exists, we use an analytic continuation argument.
Any local branch ofC at(x0, y0)has a normalization of formt(x, y)=(x0+
tm, ψ(t)). Fix a local sectionD
0
f0
→V ofC that extends continuously to a section
D0
f0
→V, withx0∈∂D0andf0(x0)=y0.
Let B(p)⊂U denote the discrete set of branch points ofC p→U. Choose x∈ D0\B(p)and u∈U\B(p). Let y=f0(x). Let D⊂U\B(p) be a simply connected domain with{x, u} ⊂D. The section-germ(f0, x)can be analytically extended to a sectionD f→U. If we putv=s(u), we havesV(y, v)=sV(f (x), f (u))≤sU(x, u).
Now, letD0\B(p)xn→x0andU\B(p)un→u0. LetDn⊂U\B(p)be a simply connected domain with{xn, un} ⊂Dn, and letDn fn→Ube the analytic extension of the section-germ(f0, xn). Then
yn:=fnxn=f0
xn=f0
xn→f0
x0
=f0
x0
=y0. (3.6)
Letvn=fn(un). Sincep is proper, we may assume (extracting a subsequence), that
vn→v0, for somev0∈V. Then(u0, v0)∈C, that is,v0∈C(u0). Therefore,
sVy0, v0
≤sVy0, yn
+sVyn, vn+sVvn, v0
≤sVy0, yn
+sUxn, un+sVvn, v0
→sUx0, u0
. (3.7)
The proof is complete.
4. Attractor associated to a contractive curve
4.1. Contractive curves. GivenV ⊂U, letV i→U denote the inclusion map.
Given a proper curveC⊂U×V,U
˜ C
→Udenotes the self-map ˜C=iC.
Remark 4.2. The proof of Corollary 3.7shows that contractive curves have the followingcoherence property:there is a constant k=k(U , V ) >1such that, for all
((x, y), u)∈C×U, there existsv∈V with(u, v)∈CandksU(y, v)≤sU(x, u).
Theorem4.3. IfC⊂U×V is a contractive curve, then(U , sU) ˜ C
→(U , sU)is a strict contraction. Consequently,˜C has a unique fixed point AC. For allK∈U,
limr→∞˜rC(K)=AC in(U , sU).
Proof. Indeed, Lip(˜C)≤Lip(i)Lip(C)≤Lip(i) <1. Since (U , sU)is com-plete, Banach’s principle finishes the proof.
The hyperbolic metric is locally equivalent to the Euclidean metricdeU= |dz|, hence limr→∞˜rC(K)=AC in(U , eU), for allK∈U. We callAC theattractorassociated to
the contractive curveC.
4.2. Orbits and limit sets. LetUbe a planar domain,V⊂Ua subdomain,C⊂U×V
a curve withC p→Uproper. A pointx∈Uisfixedif and only if(x, x)∈C. The pointx
isperiodic if and only ifx∈˜r
C(x)for somer≥1. We denote by Fix(C)the set of
fixed points ofC, and by Per(C)the set of periodic points ofC. Aweak orbitof a pointx∈U is a sequence of pointsxr∈˜r
C(x),r≥0. Anorbit
ofxis a sequence of pointsxr∈U,r≥0, withx0=xand(xr, xr+1)∈Cfor allr≥0. A (weak)suborbit ofxis a subsequence of a (weak) orbit.
The limit setAwo(x)is the set of pointsu∈Usuch that some weak orbit ofx con-verges tou;Aso(x)is the set of pointsu∈Usuch that some suborbit ofxconverges tou. Similarly, define the limit setsAwso(x)andAo(x).
Thetotal orbit X+ofX⊂Uis the union of orbits of the points ofX.
Corollary4.4. IfC⊂U×Vis a contractive curve, then, for all pointsx∈U,
Awo(x)=Awso(x)=Aso(x)=AC, Ao(x)=Fix(C). (4.1)
Proof. Given arbitrary pointsx∈Uanda∈AC, limrs U(˜
r
C(x), AC)=0, hence
limrsU(˜rC(x), a)=0. We get a sequence xr ∈˜rC(x)with limrxr =a, so thata∈ Awo(x). Therefore,AC ⊂Awo(x). Fixu∈Awso(x), and let(xr
j)jbe a weak suborbit ofx that converges tou. Since limjsU(˜
rj
C(x), AC)=0, we get limjsU(xrj, AC)=0, and thensU(u, AC)=0. SinceAC is compact,u∈AC. Therefore,Awso(x)⊂AC.
Clearly,Awo(x)⊂Awso(x), and we obtainAC=Awo(x)=Awso(x).
Let( j)j be a sequence that decreases to 0. Sincea∈Awo(x), there existsx 1∈ ˜
r1
C (x)withsU(a, x1)≤ 1. Sincea∈Awo(x1), there existsx2∈˜rC2(x1)withsU(a, x2) ≤ 2. Repeating this procedure, we get a suborbitxj ofx withsU(a, xj)≤ j, hence limjxj=a. It follows thatAC⊂Aso(x). SinceAso(x)⊂Awso(x)=AC, we getAso(x)= AC.
Fix an orbit(xr)r ofxthat converges tou. SinceC is closed,(xr, xr+1)∈C, and limr(xr, xr+1)=(u, u), we get(u, u)∈C. Therefore,Ao(x)⊂Fix(C).
Fixu∈Fix(C). Since(u, u)∈C, there existsx1∈Uwith(x, x1)∈CandsU(u, x1)≤
(1/k)sU(u, x). Since(u, u)∈C, there existsx2∈Uwith(x1, x2)∈CandsU(u, x2)≤
(1/k)sU(u, x1). Repeating the procedure, we get an orbitxr of x with sU(u, xr)≤
(1/k)rsU(u, x), hence lim
4.3. Discrete points. Let K denote the set of isolated points of a topological
spaceK. Given a curveC⊂U×V, letL:= {v∈V:U×{v} ⊂C}.
Corollary4.5. AC⊂Per(C)⊂AC for all contractive curvesC⊂U×V. WhenAC
is infinite,A
C⊂L+⊂AC. WhenL= ∅,AC is either finite, or perfect.
Proof. Clearly,L⊂Fix(C)⊂Per(C)⊂AC.
Fixu∈AC andv∈AC. There is a weak orbit(ur)rofuthat converges tov. Since
˜
C(AC)=AC,ur∈AC. Sincev∈AC,ur=vfor allr≥r0. Therefore, everyu∈AC is
pre-periodic to anyv∈A
C. In particular,AC⊂Per(C).
If(u, v)∈C,u∈AC\AC, andv∈AC, thenv∈L. Indeed, choose inAC distinct
pointsun with limnun=u. Constructvn∈AC with (un, vn)∈C and sU(vn, v)≤ sU(un, u). Then limnvn=v. Sincev∈AC,vn=vfor alln≥n0. Therefore,(un, v)∈C
for alln≥n0. SinceCis closed inU×V,C∩(U× {v})is analytic in this line. Since limn(un, v)=(u, v)andUis connected, we getv∈L.
Assume thatv∈A
C\L+. If(u, v)∈C andu∈AC, then u∈AC\L+. Since every u∈AC is pre-periodic tov, we getAC⊂AC, henceAC=AC. SinceAC is compact, it
is finite.
4.4. Periodic points. We prove the density inAC of the periodic points of a con-tractive curveCwithout singular branches.
Corollary4.6. If a contractive curveC⊂U×V has no singular branches, then
AC=Per(C).
Proof. Clearly, Per(C)⊂AC. To prove the other inclusion, note that, under the assumption thatChas no singular branches, every local branchBofC has a section defined in a hyperbolic disk∆U(p(c), ), for some >0. Let (B)be the supremum of such . The function ˜C B (B)∈(0,∞]is lower semicontinuous. Let C := min{ (B):B∈ν−1p−1AC}>0. Then, for allx∈AC and all branchesB ofC atx,B has a section defined on∆U(x, C). In particular, for allx∈AC and ally∈˜C(x), there exists a section∆U(x, C)
f
→VofCwithf (x)=y.
Fixx∈AC. LetO=(x=x0, x1, . . . , xr−1, xr=u)be a finite orbit. Sincexj−1∈AC and (xj−1, xj)∈C for all 1≤j ≤r, there is a section ∆U(x, C)
fj
→V ofC with
fj(xj−1)=xj. Since Lip(fj)≤1/k <1, fj(∆U(xj−1, C))⊂∆U(xj, C), so that f =
fr◦fr−1◦···◦f1is well defined,∆U(x, C)
f
→V, withf (x)=uand Lip(f )≤1/kr.
Fix 0< < C. Since x∈Awo(x), we can choose the orbit O so that r ≥1 and
sU(u, x)≤ ((k−1)/2k). Then
f∆U(x, )⊂∆U
u, kr
⊂∆U
x,
1
kr+ k−1
2k
⊂∆U
x, k+1
2k
. (4.2)
Therefore,f (∆U(x, ))⊂∆U(x, ), andfmust have a fixed pointz∈∆U(x, ). Then
z∈˜r
C(z), so thatz∈Per(C).
4.5. Continuity. LetV i U be a Stolz subdomain of a hyperbolic domainU⊂C,
k(U , V )=1/Lip(i) >1. Define onU×Vthe distance
Corollary4.7. Let(Ꮿ(U , V ),s)be the space of contractive curves inU×V, and
denote by
Ꮿ(U , V ),s Ꮽ→U , sU (4.4)
the mapCAC. ThenLip(Ꮽ)≤2/(k(U , V )−1).
Proof. FixC, D∈Ꮿ(U , V ), withδ:=s(C, D). Given(x, y)∈C andu∈U, there existsv∈V with (u, v)∈DandsU(y, v)≤2δ/k+sU(x, u)/k. Indeed, there exists
(a, b)∈Dwith sU(x, a)≤δandsU(y, b)≤sV(y, b)/k≤δ/k. Also, there existsv∈ ˜
V(u)withsU(b, v)≤sU(a, u)/k. We get
sU(y, v)≤sU(y, b)+sU(b, v)≤δk+sU(a, u)k
≤δk+sU(x, a)+ksU(x, u)≤2kδ+sU(x, u)k .
(4.5)
Fixx0∈U. Since limr˜rC(x0)=AC and limr˜rD(x0)=AD, it suffices to show that sU(˜
r
C(x0),˜rD(x0))≤2δ/(k−1). Given y∈˜rC(x0), we have to findv∈˜Dr(x0) withsU(y, v)≤2δ/(k−1). Fix aC-orbit(x0, . . . , xr=y). Recursively, construct aD -orbit(x0=u0, . . . , ur=:v)so that, for all 0≤j≤r,sU(xj, uj)≤2dji=1(1/ki). In particular,sU(y, v)≤2d/(k−1).
4.6. Balanced measure of Barnsley-Demko. Syma(U )denotes the quotient ofUa
by the action of the symmetric groupSa. In other words, Syma(U )is the space of ef-fective divisors onUof degreea. Define a complete distanceδon Syma(U )as follows. GivenA=a
j=1(xj)andB=
a
j=1(uj),δ(A, B):=(1/a)minσ∈Sa(
a
j=1sU(xj, uσ (j))). Clearly, the map(U , sU) I→(Syma(U ), δ), whereI(x)=a(x), is an isometric embed-ding. Let Sym(U ):= ⊕aSyma(U ).
LetV i U be a Stolz subdomain of a hyperbolic planar domain. Recall thatk=
1/Lip(i) >1. Let C⊂ U×V be a contractive curve, with projections C p→U and
C q→V. Letdbe the topological degree ofp. Define Syma(U ) →Symda(U ),(D):= i∗q∗p∗D, and Syma(U ) γ→[0,∞),γ(D):=δ(dD,D).
Lemma4.8. With the notations above,Lip()≤1/k. Consequently,γ()≤(1/k)γ. Proof. FixA, B∈Syma(U ). Choose a simply connected domainW⊂Uwith∂W⊃ supp(A)∪supp(B)⊃B(p)∩W, and such that all the sectionsW f→l VofC, 1≤l≤d, extend continuously toW. We haveδ(A, B)=(1/a)aj=1sU(xj, uj), with
A= a
j=1
xj, B= a
j=1
uj, A= a
j=1
d
l=1
flxj, B= a
j=1
d
l=1
fluj. (4.6)
Then
δ(A,B)≤ad1 a
j=1
d
l=1
sUflxj, fluj≤kad1 a
j=1
d
l=1
Consequently,
γ(A)=δ(dA,A)=δ((dA),A)≤1kδ(dA,A)=k1γ(A). (4.8)
Lemma 4.9. Given a continuous function φ∈Ꮿ(U ), let Syma(U ) Φ→C, Φ(D):=
(1/a)x∈Dφ(x).ThenLip(Φ)=Lip(φ)and|Φ()−Φ| ≤Lip(φ)γ.
Proof. Fix two divisors in Syma(U ),A=aj=1(xj)andB=aj=1(uj), withδ(A, B)=
(1/a)aj=1sU(xj, uj). We have
|ΦA−ΦB| =1 a
a
i=1
φxj−φuj≤Lip(φ) a
a
i=1
sUxj, uj=Lip(φ)δ(A, B). (4.9)
SinceΦI=φ, Lip(φ)=Lip(Φ). Also,
Φ(A)−ΦA=Φ(A)−Φ(dA)≤Lip(Φ)δ(A, dA)=Lip(φ)γ(A). (4.10)
For a compactK,·Kdenotes theL∞-norm onᏯ(K).
Corollary4.10. Givenφ∈Ꮿ(U )andD∈Sym(U ), µφ:=limr→∞Φ(rD) exists
inCand does not depend onD. The linear functionalᏯ(U )φµφ∈Cis real and nonnegative, withµ1=1. Also,|µφ| ≤ φAC for allφ∈Ꮿ(U ).
Proof. Assume that Lip(φ) <∞. Then
Φr+1D−ΦrD≤Lip(φ)γ(rD)≤Lip(φ)γ(D)
kr , (4.11)
so the limit exists inC. ForA∈Syma(U )andB∈Symb(U ),
ΦrA−ΦrB=Φr(bA)−Φr(aB)≤LipΦrδ(bA, aB)
=Lip(φ)δ(bA, aB)kr →0, (4.12)
so the limit is independent of the divisor.
LetK=r˜rC(supp(D)). ThenKis compact inU, and ˜C(K)⊂K. Givenφ1andφ2 inᏯ(K),Φ1(rD)andΦ2(rD)are well defined. Moreover, for allr≥0,|Φ1(rD)−
Φ2(rD)| ≤ φ
1−φ2K. The subspace ofᏯ(K)formed by the functions that admit
a Lipschitz extension toUis dense inᏯ(K). It follows that, for arbitraryφ∈Ꮿ(U ), limrΦ(rD)exists inC. To see that this limit is independent of the divisor, takeA, B∈
Sym(U ), putK=r˜rC(supp(A+B)), and approximateφinᏯ(K)with functions that
admit Lipschitz extensions toU.
By definition,µφ∈Rwhenφis real,µφ≥0 whenφ≥0, andµ1=1. ChoosingD with supp(D)⊂AC, we get|µφ| ≤ φAC.
Proposition 4.11. There exists a (unique) probability measure µC on AC with limr→∞Φ(rD)=ACφdµC for allφ∈Ꮿ(U )and allD∈Sym(U ).
By definition, the measureµC describes the frequency with which Borel sets are visited by the total orbit of any point.
Remark4.12. The operatorᏯ(AC) T→Ꮿ(AC),(T φ)(x)=Φ((x))is linear and continuous, with T =1. With ψ := T φ, we get Ψ(r(x)) =Φ(r+1(x)), hence
µψ= µφ. In other words,µC satisfies thefunctional equationT∗µC =µC, whereT∗
denotes the dual ofT.
Remark4.13. Corollary 4.4andRemark 4.12imply that supp(µC)=AC.
5. Examples
5.1. Mixed iteration. LetVbe Stolz in a hyperbolic planar domainU. Given finitely many holomorphic functions U fj→V, 1≤ j ≤n, the union C(f1, . . . , fn) of their graphs is a contractive curve inU×V. LetA(f1, . . . , fn)denote the associated attractor. For a word w =(w1, . . . , wl)∈ {1, . . . , n}r of lengthr (w) =r, let U
fw
→V, fw = fwr◦···◦fw1. Since Lip(fw)≤(1/k)
l(w), withk=k(U , V ) >1,fw has a unique fixed
pointxw∈U. We getA(f1, . . . , fn)=w{xw}.
Remark5.1. It is important thatUbe connected: letf (x)=x2(x−2)2=g(x)/2, and A= {0,1,2}. We have f (A)∪g(A)=A, f|A =0=g|A, and f (U )∪g(U )⊂U, whereUis the union of the three disjoint disks of radius 0.1 and centers 0, 1, 2. Thus,
Acan be viewed as the attractor associated to the mixed iteration off andgonU. But the point 2 is not periodic. Of course, nodomaincontaining the points 0 and 1 can bef-invariant.
Remark 5.2. Let C ⊂U×V be a contractive curve, and assume thatβ(V )=V, whereβ(x+iy)=x−iy. Letβ(C):= {(u, β(v)):(u, v)∈C}. IfU0
f
→V is a section ofC, thenU0
βf
→Vis a section ofβ(C), and all sections ofβ(C)are obtained in this way.(V , dsV) β→(V , dsV)is an isometry, hence Lip(βf )=Lip(f ). Therefore,given two contractive curvesCandDinU×V, withβ(V )=V, the previous results still hold for the “mixed curve”C∪β(D).
Example5.3. Given finitely manyaj∈C, letm=min(|aj|)andM=max(|aj|). If m2−m > M, then
j(y2=x−aj) is contractive in∆(0, m)×∆(0, (m+M)1/2).
The “cross” inFigure 5.1ais the associated attractor, witha1=2.01= −a2. The color intensity indicates the density of the Barnsley-Demko measure.
Example5.4. Givenk,ain∆, the Möbius mapkMa(z)=k((z−a)/(1−az)) con-tracts the Poincaré metric. Fixk1,k2,k,ain∆, and consider the mixed iteration in∆ ofk1M0,k2M0, andkMa. The point(0,0)is a node of the corresponding curve, and 0∈A(k1M0, k2M0, kMa).Figure 5.2shows the associated attractor whena=0.9=k andk1=0.9 exp(i(π /r ))= −k2, withr=4 (Figure 5.2a), andr=9 (Figure 5.2b).
5.2. Contractive Blaschke maps. Given an effective divisorα=dj(aj)in∆, let
(a) (b)
Figure5.1. Mixed Julia set (a), and Ochs’ example (b).
(a) (b)
Figure5.2. Mixed iteration of contractive Möbius maps.
(a) (b)
Example5.5. If|k|<1 ande≤min(dj), the curve(ye=kMα(x))is contractive in∆×∆(0,|k|1/e).Figure 5.3shows the associated attractor, with parameterse=2, α=3(0.7i), k= 0.73 (Figure 5.3a), ande= 2, α= 2(0.9)+3(0.3+i0.5), k=0.75 (Figure 5.3b).
5.3. An example by Ochs. The following example is taken from [5]. Consider the curveC=(ye=kd
i=1(x−ai)), withe > d; max(|ai|)=1; and|k|> ee/dd(e−d)e−d. ThenCis contractive in a suitable product of annuli centered at 0.Figure 5.1bshows the corresponding attractor, forC=(y5=29i(x−1)2).
Acknowledgment.The first author was partially supported by Proyecto CONACYT-Mexico, Sistemas Dinamicos 27958E.
References
[1] L. V. Ahlfors,An extension of Schwarz’s lemma, Trans. Amer. Math. Soc.43(1938), no. 3, 359–364.
[2] M. F. Barnsley and S. Demko,Iterated function systems and the global construction of frac-tals, Proc. Roy. Soc. London Ser. A399(1985), no. 1817, 243–275.
[3] D. Minda,Lower bounds for the hyperbolic metric in convex regions, Rocky Mountain J. Math.13(1983), no. 1, 61–69.
[4] Z. Nehari,A generalization of Schwarz’ lemma, Duke Math. J.14(1947), 1035–1049. [5] G. Ochs,Hyperbolic repellers for algebraic functions, preprint, 2000.
Araceli Bonifant: Instituto de Matematicas, Universidad Nacional Autónoma de
México, Av. Universidad s/n, col. Lomas de Chamilpa, CP62210Cuernavaca, Morelos, Mexico
E-mail address:bonifant@matcuer.unam.mx
Marius Dabija: Institute of Mathematics of the Romanian Academy, P.O. Box1-764, Bucharest70700, Romania
Current address:Department of Mathematics, The Pennsylvania State University,
215McAllister Building, University Park, PA16802, USA
