1

On the topology of wandering Julia components 1

Cui Guizhen, Peng Wenjuan and Tan Lei

October 16, 2009

Abstract

It is known that for a rational map f with a disconnected Julia set, the setof wandering Julia components is uncountable. We prove that all but countablymany of them have a simple topology, namely having one or two complementarycomponents. We show that the remaining countable subset Σ is backward invariant.Conjecturally Σ does not contain an infinite orbit. We give a very strong necessarycondition for Σ to contain an infinite orbit, thus proving the conjecture for manydifferent cases. We provide also two sufficient conditions for a Julia component tobe a point. Finally we construct several examples describing different topologicalstructures of Julia components.

1 Introduction

Let f be a rational map from the Riemann sphere C onto itself of degree d ≥ 2. Bythe classical theory of iterated rational maps initiated by Fatou and Julia around1920, we know that the Riemann sphere C is divided into two sets: one is called theFatou set F := {z ∈ C | the sequence {fn}n≥1 is normal in a neighborhood of z}and the other is called the Julia set J := C\F . For the Fatou set, the following isa well-known theorem of Sullivan:

Theorem A. ([Sul]) Every Fatou domain of the rational map f is preperiodic.

Furthermore, Shishikura ([Shi]) proved that there are at most 2d−2 periodic cyclesof Fatou domains of the rational map f .

Suppose the Julia set J is disconnected. A Julia component of f is a connectedcomponent of J . The map f maps each Julia component onto a Julia component. AJulia component J is periodic if fp(J) = J for some p ≥ 1, preperiodic if fn(J) isperiodic for some n ≥ 0, and wandering otherwise, that is if f i(J)∩f j(J) = ∅ for alli 6= j ≥ 0. Thus a component J is wandering if and only if orb(J) = {fn(J) | n ∈ N}is infinite. Since J is disconnected, there are countably many preperiodic Juliacomponents and uncountably many wandering Julia components (see [Bea], [Mil1],[McM], [PT1]).

Let J be a periodic Julia component with period p containing more than onepoint. Then by a theorem of McMullen (see [McM]), we know that (fp, J) is quasi-conformally conjugate to (g,J (g)) where g is a rational map with a connected Juliaset J (g). More precisely, there exists a quasi-conformal map φ from the Riemannsphere C onto itself such that φ(J) = J (g) and φ ◦ fp = g ◦ φ on J . So the studyof the preperiodic components is in some sense converted to the study to rationalmaps with connected Julia sets.

For wandering Julia components, so far, we don’t know much about them. Weknow some topological properties of them for special rational maps.

Theorem B. ([QY]) Every wandering Julia component of a polynomial is a point.

12000 Mathematics Subject Classification: 37F10, 37F20

2

The result is generalized to a rational map with a completely invariant Fatoudomain, see [Z].

A non-empty connected compact set J ⊂ C is simply (resp. doubly) connectedif C\J has exactly one (resp. two) connected components. We say that a rationalmap is geometrically finite, resp. nice, if every critical point in the Julia set (ifany) is preperiodic, resp. if every Julia component containing a critical point (ifany) is preperiodic. We have

Theorem C. ([PT1]) 1. Every wandering Julia component of a geometrically finiterational map is a Jordan curve or a point.

2. Every wandering Julia component of a nice map is either simply or doublyconnected.

Furthermore, they depicted when a simply connected Julia component of a nicemap is a point.

Theorem D. ([Mil2]) The Julia set of a rational map with two critical points iseither connected or totally disconnected.

In this paper, we will investigate the topological structure of wandering Juliacomponents. First, we define a function

C : {non-empty connected compact sets ( C} → {1, 2, 3}

as follows: for J ( C a connected compact set with J 6= ∅,

C(J) =

1 if J is simply connected2 if J is doubly connected3 otherwise.

= min{3,#{complementary components of J}}.

In case that J is a Julia component of the rational map f , we can show thatn 7→ C(fn(J)) is weakly decreasing (see Lemma 2.1 below). For a wandering J ,define C(orb(J)) as limn→∞C(fn(J)).

For a Julia component of the rational map f , we refer to it as a buried compo-nent if it has no intersection with the boundary of any Fatou domain of f , otherwisewe call it exposed. One can refer to section 7 in [McM], section 2 in [PT1] andsection 4 in this paper for examples of buried components.

Our main result here is the following:

Theorem 1.1. Let f be a rational map with a disconnected Julia set.(a) A wandering Julia component J with C(orb(J)) ≥ 2 is necessarily buried,

that is, it has no intersection with the boundary of any Fatou domain of f .(b) The set Σ := {J | J a Julia component, C(J) = 3} is backward invariant,

and is either empty or countable.(c) If Σ contains the forward orbit of at least one wandering Julia component,

then there are two distinct Julia components J1 6= J2 satisfying simultaneously:

(∗)

they are both wandering and they both contain a critical value;they are both buried;C(J1) = C(J2) = C(orb(J1)) = C(orb(J2)) = 1;orb(Js) accumulates to Jt for any pair (s, t) ∈ {1, 2}2;J1 /∈ orb(J2) and J2 /∈ orb(J1).

3

We will give an example showing that the condition (∗) is not sufficient forhaving a wandering Julia component J with C(orb(J)) = 3.

These results support the conjecture that there is no Julia component J withinfinite orbit and with C(orb(J)) = 3. We mention two easy to check corollaries ofTheorem 1.1 (with the first one containing Theorem C as a particular case):

Corollary 1.2. Let f be a rational map with disconnected Julia set. If all but atmost one Julia components containing critical points are either exposed or preperi-odic, then every wandering Julia component is eventually simply connected or doublyconnected.

Corollary 1.3. Let f be a rational map with disconnected Julia set. If all butone Julia components containing critical points are doubly or multiply connected,then every wandering Julia component is eventually simply connected or doublyconnected.

The paper is organized as follows. In Section 2, we prove Theorem 1.1 and themain tool we used is Theorem A. In Section 3, we provide two sufficient conditions fora Julia component to be a point. Finally we use quasi-conformal surgery to constructseveral examples describing different topological structures of Julia components inSection 4.

2 Proof of Theorem 1.1

2.1 Preliminary results

For a non-empty compact connected set K ⊂ C, we say that an open set W is acomplementary component of K if W is a connected component of C\K.

Lemma 2.1. Let f : C → C be a branched covering. Let K ⊂ C be a non-emptycompact connected set and K ′ be a connected component of f−1(K).

(i) We have C(K ′) ≥ C(K). Furthermore, if K ′ contains no critical points off , then C(K ′) = C(K).

(ii) The map f induces a correspondence

f∗ : {complementary components of K ′} → {complementary components of K}so that f∗(W ′) = W if f maps the prime ends of W ′ onto those of W . Moreovereither f(W ′) = f∗(W ′) or f(W ′) = C. The latter occurs if and only if W ′ ∩f−1(K) 6= ∅.Proof. (i) Set Q = {complementary components of K} and

R = {complementary components of K containing critical values}.Then #R < ∞.

Take a finitely connected domain U ⊂ C with smooth boundary satisfying that

(1) K ⊂ U ,(2) U\K contains no critical values of f ,(3) for qU the number of boundary components of U , we have qU = 1 if #R ≤ 1and qU = #R otherwise.

Then U can be decomposed into three sets: A t K t D, where A is the unionof finitely many disjoint annuli and D is the union of disjoint simply connecteddomains. The sets A and D have the following properties:

4

(a) #{connected components of A} = qU .

(b) A ∪D contain no critical values.

(c) For each component of A, one of its boundary components is contained in∂K while the other one is a component of ∂U . For each component of D, itsboundary is contained in ∂K.

(d) For a component Ai of A, each connected component A′i of f−1(Ai) is anannulus disjoint from f−1(K) with one boundary lying in ∂f−1(U) and theother lying in ∂f−1(K). The map f : A′i → Ai is a covering. For a componentDi of D, each connected component D′

i of f−1(Di) is a simply connecteddomain disjoint from f−1(K) with ∂D′

i ⊂ ∂f−1(K) and with f : D′i → Di a

homeomorphism.

There exists a connected components U ′ of f−1(U) containing K ′. Clearlyf−1(K)∩U ′ = U ′\f−1(A∪D). It follows from (d) that f−1(K)∩U ′ is connected (forthis one can use approximations of f−1(K ∪D)∩U ′ by domains with Jordan curveboundaries, and then the Jordan curve theorem). Therefore f−1(K) ∩ U ′ = K ′

and U ′ = A′ t K ′ t D′, where A′ = f−1(A) ∩ U ′, D′ = f−1(D) ∩ U ′. Fur-thermore the boundary curves of U ′ are in pairwise distinct complementary com-ponents of K ′. Denote by qU ′ the number of these boundary curves. We haveqU ′ = #{connected components of A′} ≥ 1.

The restricted map f : U ′ → U is a branched covering. Denote its degree by d.Denote by qD, qD′ the number of connected components of D, D′ respectively (theymay be zero or +∞). Let k = #{critical points in U ′ counted with multiplicity}.We have, by definition, the point (d) above and the Riemann-Hurwitz Formula:

C(K) = min{3, qU + qD} ;C(K ′) = min{3, qU ′ + qD′} ;

qD′ = d · qD ; (2− qU ′) + k = d(2− qU ).

C(K ′) = 1 =⇒ qU ′ = 1, qD′ = 0 =⇒ qU = 1, qD = 0 =⇒ C(K) = 1.

C(K ′) = 2 =⇒

qU ′ = 1, qD′ = 1 =⇒ qU = 1, qD = 1, d = 1 =⇒ C(K) = 2or

qU ′ = 2, qD′ = 0 =⇒

qU = 1 (if k ≥ 1), qD = 0 =⇒ C(K) = 1orqU = 2 (if k = 0), qD = 0 =⇒ C(K) = 2.

C(K ′) = 3, k = 0 =⇒

qU ′ = 1, qD′ ≥ 2 =⇒ qU = 1, d = 1, qD ≥ 2 =⇒ C(K) = 3orqU ′ = 2, qD′ ≥ 1 =⇒ qU = 2, qD ≥ 1 =⇒ C(K) = 3orqU ′ ≥ 3 =⇒ qU ≥ 3 =⇒ C(K) = 3.

Obviously, C(K ′) = 3 ≥ C(K), if k > 0.(ii) Let W ′ be a complementary component of K ′.Assume at first W ′∩f−1(K) = ∅. Then f(W ′) is open, connected, and is disjoint

from K. So f(W ′) is contained in a unique complementary component W of K.Then f−1(W ) has a unique connected component U ′ containing W ′. But U ′ is open,connected, and is disjoint from K ′. So U ′ must be contained in a complementary

5

component of K ′. Consequently U ′ = W ′ and f(W ′) = W . Clearly f∗(W ′) = W aswell.

Assume now W ′ ∩ f−1(K) 6= ∅. Then W ′\f−1(K) has a unique component Ewith ∂E ⊃ ∂W ′. One can show as above that f(E) =: W is a complementarycomponent of K, and f∗(W ′) = W . To show f(W ′) = C we choose a Jordan curveη ⊂ W close to ∂W so that the annular component of W\η, as well as η, does notcontain critical values. We now use the following fact:Fact: let δ ⊂ C be a Jordan curve containing no critical values of f . Then(a) f−1(δ) is the union of finitely many pairwise disjoint Jordan curves;(b) f maps every component of C\f−1(δ) properly onto a complementary componentof δ;(c) any two components of C\f−1(δ) sharing a common boundary curve are mappedonto distinct components of C\δ.

For our curve η, we know that W ′ ∩ f−1(η) has at least two curves, and oneof them (the one close to ∂W ′) separates the others from ∂W ′. This implies thatf(W ′) = C.

Let now I be a subset of C. We define a separating number relative to I

SI : {non-empty connected compact sets in C disjoint from I} → {1, 2, 3}

as follows:

SI(J) =

1 if exactly one complementary component of J intersects I;2 if exactly two complementary components of J intersects I;3 if more than two complementary components of J intersects I.

Clearly SI(J) ≤ C(J).Let f be a rational map with disconnected Julia set J .We fix a normalization (up to conformal conjugacy) so that f(∞) = ∞. Set

I = {∞} ∪ {the critical values of f} .

The set I is finite. It is not difficult to see {J a Julia component | J ∩ I =∅, SI(J) = 3} is a finite set. Let

Jprep = {preperiodic Julia components of f},JI = {comp. of f−n(J) | n ≥ 0, J a Julia component, J ∩ I 6= ∅, }\Jprep,Spre−3 = {comp. of f−n(J) |

n ≥ 0, J a wandering Julia component, J ∩ I = ∅, SI(J) = 3}.

For i = 1, 2,

Si = {J | J a wandering Julia component, J ∩ I = ∅, SI(J) = i}\(Spre−3 ∪ JI).

There are countably many Julia components in Jprep ∪JI ∪Spre−3. But S1 ∪S2

consists of uncountably many Julia components.Set A = S1∪S2. In the following, we will focus on the set A. From the definitions

of these sets, we can see that f(A) ⊂ A (by abuse of notation, we use f to denotethe map induced by f on the set of Julia components).

The set A has the following decomposition. For i = 1, 2, 3, set

Ci = {J ∈ A | C(J) = i},

6

A

V1 V2 · · · VL

A = C1 t C2 t C3

= S1 t S2

V ⊂ S2 ∩ C3

V = V1 t V2 t · · · t VL

F : S2 ∩ C3 → S2 ∩ C3

G : V → V

C1

C2

C3

S2 S1

V

Figure 1: Decompositions of A

Then by Lemma 2.1, f−1(Ci) ∩ A ⊂ Ci and f(Ci) ⊂ Ci. Furthermore C1 ⊂ S1.Definition. (a) For a bounded compact connected set K in C, denote by K, thefill-in, to be the union of K with all the bounded complementary components of K.

(b) For every J ∈ S2, i.e. J is a wandering Julia component with a uniquebounded complementary component containing critical values, denote by V (J) thiscomplementary component .

For J ′, J ′′ two distinct components in S2, there are only three possible configu-rations:

J ′′ ⊂⊂ V (J ′), J ′ ⊂⊂ V (J ′′), or J ′′ ∩ J ′ = ∅ . (1)

Lemma 2.2. (a) For J ∈ S1 and for J ′ a component of f−1(J), the set J ′ ismapped homeomorphically onto J by f . In particular f(W ′) = f∗(W ′) for everycomplementary component W ′ of J ′.

(b) For J ∈ A, if #orb(J) ∩ S2 < ∞, then J ∈ C1.(c) If C3 6= ∅ then S2 ∩ C3 6= ∅, and for any J ∈ C3, #orb(J) ∩ (S2 ∩ C3) = ∞.

Proof. (a) follows from the fact that for J ∈ S1, the set J is simply connected andcontains no critical values.

(b) Take an integer n0 such that orb(fn0(J)) ∩ S2 = ∅. Then f maps fn(J)homeomorphically onto fn+1(J) for n ≥ n0.

We want to show that fn(J) ∩ fm(J) = ∅ for all n 6= m, n,m ≥ n0.Otherwise one is nested in the other.If fn+p(J) ⊂ fn(J) for some n ≥ n0 and p ≥ 1, then by Schwarz Lemma, fp

has an attracting fixed point in intfn(J) and intfn(J) is contained in the attractingbasin which contradicts with the fact that fn+p(J) ⊂ intfn(J). If fn+p(J) ⊃ fn(J),then int fn+p(J) is an open set intersecting J . So int fn+lp(J) ⊃ J for some l > 1which contradicts with the fact fn+lp(J) ⊂ J . Hence we know that fn(J)∩fm(J) =∅ for all n 6= m ≥ n0. We conclude that fn0(J) ∈ C1 for otherwise we could get awandering Fatou domain in intfn0(J).

(c) Assume there is J ∈ C3. Then from f(C3) ⊂ C3 we have orb(J) ⊂ C3. By(b), #orb(J) ∩ S2 = ∞. So #orb(J) ∩ (S2 ∩ C3) = ∞, and S2 ∩ C3 6= ∅.

Assume J ∈ S2 ∩ C3. Let J ′ be a connected component of f−1(J). Take anannulus A containing J so that A∩I = ∅. In particular A contains no critical value.

7

Then there exists a connected component A′ of f−1(A) which is also an annulus andwhich contains J ′. Denote the inner boundary of A′ by γ′ and the complementarycomponent of J ′ containing γ′ by U ′.

Denote by γ± the outer/inner boundary of A. The complementary componentof J containing γ− is simply V (J).

We want to determine f(W ′) explicitly for any bounded complementary compo-nent W ′ of J ′ :

Lemma 2.3. (0) f(γ′) is one of γ±.(1) Assume f(γ′) = γ−. Then f∗(U ′) = V (J). If int(γ′) ∩ f−1(γ) = ∅, then

f(γ′) = γ and f(U ′) = V (J). Otherwise f(γ′) = f(U ′) = C.(2) Assume f(γ′) = γ+. If int(γ′) ∩ f−1(γ+) = ∅, then f(γ′) = C\int(γ+),

and f(U ′) is equal to the unbounded complementary component of J . Otherwise,f(γ′) = f(U ′) = C.

(3) U ′ is the unique bounded complementary component of J ′ intersecting f−1(I).(4) For any bounded complementary component W ′ of J ′ with W ′ 6= U ′, we

have f(W ′) = f∗(W ′), that f(W ′) is a bounded complementary component of J , isdistinct from V (J), and f : W ′ → f(W ′) is a homeomorphism.

Proof. This lemma follows directly from Lemma 2.1.

2.2 Proof of Theorem 1.1.(a).

Notice that the set of buried (resp. exposed) Julia components is fully invariant byf .

Let J be an exposed wandering Julia component. Then J ∩ ∂U 6= ∅ for someFatou domain U . Note that by Sullivan’s theorem (see Theorem D) U is eventuallyperiodic and fn(J) ∩ ∂fn(U) 6= ∅ for all n ≥ n0. So replacing f by an iterate of fif necessary, we may assume f(U) = U . This implies that U is infinitely connected.By Fatou’s classification of periodic Fatou domains, we have U is either attractingor parabolic (see [Mil1], [Bea]).

We now normalize f so that ∞ is the fixed point with U as a basin.For this normalization we define I and the sets J∗,S∗ as above. Define the

filled-in set J accordingly.We claim that fn(J) is a connected component of C\U and fn(J) ∩ fm(J) = ∅

for all n 6= m, n,m ≥ n0. In fact,

fn(J) , fn(J) ∪ (the bounded complementary components of fn(J))= C\(the unbounded component of C\fn(J))= C\(the unbounded component of C\Bn)⊂ C\U,

where Bn is a connected component of ∂U . Note that all the boundary componentsare pairwise disjoint, thus the claim holds. Combining with the fact that the set I isfinite, we know orb(fn0(J)) ∩ S2 is a finite set and then by Lemma 2.2, fn(J) ∈ C1

for n ≥ n1 for some n1 ≥ n0. That is, C(orb(J)) = 1.

2.3 Proof of Theorem 1.1.(b).

Assume that the Julia set J of f is disconnected.

8

The fact that the set Σ := {J | J a Julia component, C(J) = 3} is backwardinvariant follows from Lemma 2.1.(i).

Set Q[i] = {a + ib ∈ C | a, b ∈ Q}. This is a countable dense subset of C. Definea map

φ : Σ → (Q[i])3, J 7→ (z1, z2, z3) ∈ (Q[i])3

so that zi and zj are in distinct complementary components of J whenever i 6= j.It is easily seen that φ is injective. So Σ is at most countable.Assume Σ 6= ∅ and that Σ is finite. As Σ is backward invariant, one can define

a multivalued map f−1 : Σ → Σ mapping J to a component of f−1(J). As Σ isfinite, this map is actually single valued, and is bijective. This implies that for anyJ ∈ Σ we have f(J) ∈ Σ and that the grand orbit J of J consists of finitely manyJulia components. It follows that the iterative sequence {fn}n≥1 is normal outsideJ and consequently J = J .

We then reach a contradiction as J , being disconnected by assumption, musthave uncountably many connected components.

It follows that Σ is either empty or countable.

2.4 Preliminary results for the proof of Theorem 1.1.(c).

Assume for this entire subsection that C3 6= ∅. It then follows from Lemma 2.2.(c)that

C3 6= ∅ =⇒ S2 ∩ C3 6= ∅ . (2)

Fix a choice of a normalization so that ∞ is a fixed point. Then the Juliacomponent containing ∞ (if any) is a fixed component. In any case every wanderingJulia component is bounded.

Define now I,J∗,S∗,A, J as above for this normalization.By Lemma 2.2.(c), every Julia component J in S2 ∩ C3 returns eventually to

S2 ∩ C3 under the iterations of f . Denote by s(J) > 0 the minimal return time.This induces a first-return time function together with a first-return map:

s : S2 ∩ C3 → N∗, J → s(J);F : S2 ∩ C3 → S2 ∩ C3, J → fs(J)(J).

Notice that {fn(J) | fn(J) ∈ S2} = orb(J) ∩ S2 = orbF (J) and F is injective onorbF (J), thus F is injective on orb(J) ∩ S2.

Let now J ∈ S2 ∩ C3. Applying Lemma 2.3 we know that there exists only onebounded complementary component of fs(J)−1(J) intersecting f−1(I). Notice that

fs(J)−1 maps J onto fs(J)−1(J) homeomorphically. Hence we conclude that thereexists only one bounded complementary component of J intersecting f−s(J)(I). Set

U(J) := the unique bounded complementary component of J intersecting f−s(J)(I) .

It may or may not occur that U(J) = V (J).

Lemma 2.4. Let J ∈ S2 ∩ C3. Let P be a bounded complementary component of Jwith P 6= U(J). Then there is a minimal integer k such that P is mapped by someiterate of f onto U(F k(J)). Moreover for this k, U(F k(J)) 6= V (F k(J)).

Proof. Set Ji = F i(J) for i ≥ 1.As P 6= U(J), fs(J)(P ) is a bounded complementary component of J1, and is

distinct from V (J1).

9

If fs(J)(P ) 6= U(J1), then fs(J)+s(J1)(P ) is a bounded complementary com-ponent of J2, and is distinct from V (J2). If again fs(J)+s(J1)(P ) 6= U(J2), thenfs(J)+s(J1)+s(J2)(P ) is a bounded complementary component of J3, and is distinctfrom V (J3), and so on.

J J1 J2 · · · Jk−1 · · ·P

fs(J)

−→ ∗ fs(J1)

−→ ∗ −→ · · · −→ ∗ −→ · · ·6= V (J1) 6= V (J2) · · · 6= V (Jk−1) · · ·

6= U(J) 6= U(J1) 6= U(J2) · · · 6= U(Jk−1) · · ·

Assume that the lemma is not true. Then for any n ≥ 1, fs(J)+s(J1)+s(J2)+···+s(Jn−1)(P )is a bounded complementary component of Jn, and is distinct from U(Jn) and fromV (Jn). If follows from (1) and the fact that Jn 6= Jm if n 6= m, that the opensets fs(J)+s(J1)+s(J2)+···+s(Jn−1)(P ), n ≥ 1 are pairwise disjoint. Consequently P isa wandering Fatou domain. Impossible.

There is therefore a minimal k such that fs(J)+s(J1)+s(J2)+···+s(Jk−1)(P ) = U(Jk).Due the minimality of k and the above diagram, we see that U(Jk) 6= V (Jk) for thisk.

SetV = {J ∈ S2 ∩ C3 | V (J) 6= U(J)}.

The above lemma proves:

S2 ∩ C3 6= ∅ =⇒ V 6= ∅ . (3)

Lemma 2.5. Every F -orbit visits V infinitely many times.

Proof. Recall that F maps S2 ∩ C3 into S2 ∩ C3. Let J ∈ S2 ∩ C3. Then there is abounded complementary component P of J with P 6= U(J). By above there is aminimal k such that P is mapped onto U(F k(J)), and for this k, F k(J) ∈ V.

Start now from a J ∈ V. We will try to follow the orbit of V (J). By definitionof V, we know that V (J) 6= U(J) so one can apply Lemma 2.4 to P = V (J). Letnow k to be the minimal integer so that V (J) is mapped onto U(F k(J)). We knowalso F k(J) ∈ V. Set Ji = F i(J). Here is a schematic picture:

J J1 J2 · · · Jk−1 Jk = G(J)

V (J)fs(J)

−→ ∗ fs(J1)

−→ ∗ −→ · · · −→ ∗ fs(Jk−1)

−→ U(Jk)6= V (J1) 6= V (J2) · · · 6= V (Jk−1) 6= V (Jk)

6= U(J) 6= U(J1) 6= U(J2) · · · 6= U(Jk−1)(4)

This induces a pair of maps:

n : V → N∗, J → n(J) := s(J) + s(J1) + · · ·+ s(Jk−1);G : V → V, J → F k(J) = fn(J)(J) .

Therefore we have the following commutative diagram:

V 3 JG−→ G(J) ∈ V

V ↓ ↓ U

U(J) 6= V (J) −→fn(J)

U(G(J)) 6= V (G(J)).

(5)

10

We define an equivalence relation on V: two distinct components J, J ′ of V aresaid to be equivalent if V (J) and V (J ′) contain the same set critical values.

Since there are only finitely many critical values, the set V is decomposed intofinitely many equivalence classes: V = ∪L

i=1Vi.

V1 Vi Vm VL

fn(J)(V (J)) = U(G(J))

E1 Ei Em EL

J

V (J)

G(J)

Figure 2: The equivalence classes in V

Definition (ordering). We define a nesting ordering in each class Vi by:

J ≺ J ′ if J ′ is more deeply nested, that is, V (J) ⊃⊃ J ′.

Lemma 2.6. Consider any equivalence class Vi.(a) The relation ≺ is a total ordering.(b) Both maps s : Vi → N∗ and n : Vi → N∗ are strictly increasing.(c) The set Vi is at most countable.(d) Every infinite sequence of distinct elements in Vi (if any) admits an increas-

ing subsequence with respect to ≺.(e) For any J ∈ Vi, either G(J) ∈ Vl for some l 6= i or G(J) ∈ Vi and G(J) ≺ J .

Proof. (a) is evident by the definition of the equivalence relation in V.(b) Fix a pair J, J ′ ∈ Vi such that J ′ Â J , we will prove a stronger statement:

n(J ′) ≥ s(J ′) > n(J) ≥ s(J). (6)

The fact n(J) ≥ s(J) follows directly from the equation n(J) = s(J) + s(J1) +· · ·+s(Jk−1). Now we show that s(J ′) > n(J) holds. Notice that f i(J ′) ⊂ f i(V (J))for 1 ≤ i ≤ n(J). So by above f i(J ′) does not separate I for i ≤ n(J). Hences(J ′) > n(J). This proves (6) as well as (b).

(c) follows from the injectivity of s : Vi → N∗ (as s is strictly increasing).Alternatively, one can obtain this from the fact that Vi ⊂ C3 ⊂ Σ and by part (b)of Theorem 1.1 the set Σ is at most countable.

(d) By the injectivity of s, we have

∀J ∈ Vi, #{J ′ ∈ Vi | J ′ ≺ J} < ∞ . (7)

Property (d) follows.(e) Assume the contrary, i.e. G(J) ∈ Vi and J ≺ G(J). Then V (J) ⊃⊃ G(J).

But thenfn(J)(V (J))

(5)= U(G(J)) ⊂ G(J) ⊂⊂ V (J) .

By Schwarz Lemma fn(J) has an attracting fixed point in V (J) and every point inV (J) is in the attracting basin. This contradicts that V (J) contains Julia points,for instance G(J). Property (e) follows.

11

For each i ∈ {1, · · · , L}, set

Ei = ∩{V (J ′) | J ′ ∈ Vi}.

Clearly Ei contains critical values and is simply connected. It follows from abovethat for every infinite sequence J1, J2, · · · of distinct elements of Vi (if any), we haveEi = ∩∞m=1V (Jm).

Lemma 2.7. If an equivalence class Vi receives infinitely many visits of a singleorbit orbG(J), J ∈ V, then(i) Ei is a Julia component that is wandering, buried, simply connected, and thatcontains critical values,(ii) Ei 6∈ orbf (Ei)\{Ei}, and for any other Vj such that orbG(J) visits Vj infinitelymany times, Ej 6∈ orbf (Ei).

Proof. (i) For any fk(J) ∈ Vi, set sk = s(fk(J)). Take an infinite increasingsequence k1 < k2 < · · · such that

{fk1(J), fk2(J), · · · } ⊂ Vi .

By Lemma 2.6.(d) and (b), passing to a subsequence if necessary (denoted again byk1, k2, · · · ), we may assume sk1 < sk2 < · · · .

Then Ei =⋂∞

m=1 V (fkm(J)). From Ei ⊂ V (fk(J)) for k = k1, k2, · · · and therelation (4) , we have

fsk(Ei) ⊂ fsk(V (fk(J))) 6= V (fsk+k(J)) for each k = k1, k2, · · · .

Since F is injective on orbF (J), we have {F (fkm(J))}m≥1 are pairwise distinct.It follows from the relation (1) in section 2.1 that the open sets fsk1 (V (fk1(J))),

fsk2 (V (fk2(J))), · · · are pairwise disjoint. Consequently fsk1 (Ei), f sk2 (Ei), · · · arepairwise disjoint. Hence the set Ei contains no points in the Fatou set for otherwisewe would get a wandering Fatou domain in Ei. So it must be contained in a Juliacomponent J with C(J) = 1.

If Ei 6= J , then J will intersect some J ′ ∈ Vi. Note that J ′ and J are both Juliacomponents, so J = J ′. But Ei is contained in a complementary component of J ′.A contradiction. So Ei = J , it is a wandering Julia component, and is buried.

(ii) Notice that for each 1 ≤ l ≤ sk, f l(Ei) ⊂ f l(V (fk(J))) and f l(V (fk(J))) ∩I = ∅. On the other hand, Ei ∩ I 6= ∅ and Ej ∩ I 6= ∅. Hence f l(Ei) 6= Ei, Ej foreach 1 ≤ l ≤ sk. Since limm→∞ skm = +∞, we conclude that (ii) holds.

Definition. For a pair Ei, Em (they may be equal), we say that Ei accumulatesto Em, written as Ei → Em, if any neighborhood of Em contains infinitely many offk(Ei), k > 0.

Lemma 2.8. Fix J ∈ V, and i such that the equivalence class Vi receives infinitelymany visits of orbG(J) (such class exists always).(a) There is then m 6= i satisfying the following two properties:

• Vm receives infinitely many visits of orbG(J);• the set Ei accumulates to Em, i.e. Ei → Em.

(b) There is q 6= m (but q may be equal to i) such that Em → Eq and Ei → Eq.

Proof. (a). By Lemma 2.6.(e) and formula (7) , we can find an infinite sequence

t1 < t2 < · · ·

12

such that, for any l, f tl(J) ∈ Vi ∩ orbG(J) and G(f tl(J)) ∈ Vjlfor some jl 6= i.

As there are only finitely many choices for jl, there are m 6= i so that jl = m forinfinitely many l.

Fix any such m. By passing to a subsequence of {tl}, denoted still by {tl}, wemay assume jl ≡ m, i.e. G(f tl(J)) ∈ Vm for all l. In particular Vm receives infinitelymany visits of orbG(J).

We apply now Lemma 2.6.(d) to the infinite sequence {f tl(J), l ∈ N} ⊂ Vi. Thusafter passing to a subsequence of {tl}, denoted still by {tl}, we have

f t1(J) ≺ f t2(J) ≺ · · · .

For any f t(J) ∈ Vi, set nt = n(f t(J)). As the map n is increasing with respectto ≺ (Lemma 2.6.(b)), we have

nt1 < nt2 < · · · .

Note that G is injective on the orbit orbG(J). So {G(f tl(J)), l ∈ N} are pairwisedistinct.

We apply now Lemma 2.6.(d) to the infinite sequence {G(f tl(J)), l ∈ N} ⊂ Vm.Thus after passing to subsequences of {tl}, denoted still by {tl}, we have

G(f t1(J)) ≺ G(f t2(J)) ≺ · · · .

In other words G restricted to this sequence preserves the nesting order from Vi toVm.

On the other hand, for any l ≥ 1, we have Ei ⊂ V (f tl(J)) and fntl (V (f tl(J))) =U(G(f tl(J))) by (5) . Therefore, for any l ≥ 2,

fntl (Ei) ⊂ fntl (V (f tl(J))) = U(G(f tl(J))) ⊂ G(f tl(J))\V (G(f tl(J)))⊂⊂ V (G(f tl−1(J)))\Em .

But⋂

l V (G(f tl−1(J))) = Em. Hence Ei accumulates to Em.(b) Applying part (a) but with Vi replaced by Vm, we find Vq for q 6= m so that

Vq contains infinitely many G(fv(J)), fv(J) ∈ orb(J) ∩ Vm, and Em → Eq.We want to show Ei → Eq. In fact, for any J∗ ∈ Vq, there exists fv(J) ∈ Vm

such thatG(V (fv(J))) = U(G(fv(J))) ⊂ J∗.

Since Ei → Em, there exists f ξ(Ei) ⊂ V (fv(J)). Then G(f ξ(Ei)) ⊂ G(V (fv(J))) ⊂J∗ and therefore Ei → Eq.

2.5 Proof of Theorem 1.1.(c).

Assume that a rational map f has a wandering Julia component J so that C(orb(J)) =3, in other words C(fn(J)) = 3 for any n ≥ 0.

Replacing J by a forward iterate of it if necessary, we may assume that for anyn ≥ 0, fn(J) ∩ I = ∅ and SI(fn(J)) ≤ 2. Therefore J ∈ S1 ∪ S2. As C(J) = 3 itfollows from the definition that J ∈ C3, and therefore C3 6= ∅.

It follows then from (2) and (3) that S2 ∩ C3 6= ∅ and V 6= ∅. So V isdecomposed into finitely many (non-empty) equivalence classes V1, · · · ,VL. Themap G : V → V is also well defined.

13

Fix J ∈ V. Then there is a subset Λ of the index set {1, · · · , L} such that eachVi, i ∈ Λ receives infinitely many visits of orbG(J). By Lemma 2.8 and the finitenessof Λ one can find a cycle Vi1 = Vip+1 ,Vi2 , · · · ,Vip such that

Ei1 → Ei2 → · · · → Eip → Ei1 ,

and Ei1 À Eip . Set J1 = Ei1 and J2 = Eip . By Lemmas 2.7 and 2.8 they satisfy(∗) required by Theorem 1.1.(c).

This ends the proof of Theorem 1.1.

Corollary 2.9. Each equivalence class Vi can be represented by a Jordan curve γi

in the same homotopic class (rel I) of a Julia component in Vi. Then Γ = {γi | i =1, · · · , L} is a multicurve. If γk ∈ Γ such that both complementary components ofγk contains curves of Γ up to homotopy, then the corresponding class Vk can onlybe visited at most finitely many times by an orbit in C3.

3 Point Julia components

In this section, we state two sufficient conditions for a Julia component to be apoint. Let f be a rational map with disconnected Julia component J .

The following proposition follows directly from Shrinking Lemma of Fatou.

Proposition 3.1. Suppose W is a complementary component of a Julia componentsuch that W ∩ J 6= ∅ and W is disjoint from the postcritical set. Then any Juliacomponent J with the f-orbit of J visiting W infinitely many times is a point.

Let J1, J2, · · · , JL be preperiodic Julia components and V1, V2, · · · , VL be pair-wise disjoint complementary components of J1, J2 · · · , JL respectively. Denote V =∪L

i=1Vi. Let U = {z ∈ V | ∃ l ≥ 1 such that f l(z) ∈ V}. Suppose U 6= ∅. Definethe first-return map

R : U → V, z → fu(z)(z),

where u(z) ≥ 1 is the smallest integer with fu(z)(z) ∈ V. Let W be a componentof U. Then there exists an integer u such that u = u(z) for all z ∈ W and fu(W )is some Vi. Choose U to be the union of finitely many components of U which aresimply connected and compactly contained in V. Define R : U → V as R = R|U.Define the filled-in Julia set of the map R

KR := {z ∈ U | Rn(z) ∈ U, n ≥ 1}.

We have the following proposition.

Proposition 3.2. A component K of KR is a point if and only if the R-orbit of Kcontains no periodic components with critical points of R.

Proof. The ”only if” part is easy: suppose K is a point, then so is each Rn(K), n ≥ 1.If Rn0(K) is a periodic critical point for some n0 ≥ 1, that is it is a superattractingperiodic point of R, then it must contain a superattracting Fatou domain of R. Itis impossible.

For the ”if” part, we will only give a sketch of the proof. One can consult [QY],[PQRTY] for details.

14

The connected components of R−n(U) are called puzzle pieces of depth n. For acomponent KR(x) of KR containing x such that the R-orbit of it contains no periodiccomponents with critical points of R, we shall find a nested sequence containing it

Pn1(x) ⊃ Pn′1(x) ⊃ Pn2(x) ⊃ Pn′2(x) ⊃ · · ·

such that∑∞

i=1(mod(Pni(x)\Pn′i(x))) = ∞. Then by Grotzsch inequality,

mod(Pn1(x)\KR(x)) = ∞and hence KR(x) = {x} is a point.

For any x ∈ KR, the definition of tableau T (x) can be found in [QY], [PQRTY].Briefly speaking, it is a two dimensional array Pn(Rl(x)) associated with the R-orbit of x. For x, y ∈ KR, we say the R-orbit of x combinatorially accumulatesto y, written as x → y, if for any n ≥ 0, there exists an integer j > 0 such thatf j(x) ∈ Pn(y) (this definition follows Definition 1 in [QY] ). One can find thedefinitions that the tableau T (c) for a critical point c is non-critical or periodic orreluctantly recurrent or persistently recurrent in [QY] (ref. Definitions 1 and 2 in[QY]).

LetCrit(x) = {c ∈ KR | x → c}.

There are the following five possibilities.Case a. Crit(x) = ∅.Case b. ∃ c ∈ Crit(x) such that T (c) is non-critical.Case c. ∃ c ∈ Crit(x) such that T (c) is reluctantly recurrent.Case d. ∃ c ∈ Crit(x) such that T (c) is periodic.Case e. ∃ c ∈ Crit(x) such that T (c) is persistently recurrent but not periodic.In Cases a, b, c, d, we can easily construct the desired sequence (see Propositions1 and 2 in [QY]). In Case e, applying Theorem 2.1 in [PQRTY] together with theproof of Main Proposition in [QY], we can find the desired sequence.

Corollary 3.3. A wandering Julia component J with Rn(J) ⊂ U for all n ≥ 0 isa point.

4 Examples

Wandering Julia components are classified either by their topology (simply, doublyor multiply connected), or by their positions relative to the Fatou components (ex-posed or buried). They may or may not contain critical orbits. One may questionabout the existence and the co-existence of various types of such Julia components.The examples here are built to answer some of these questions.

4.1 Preliminary results

We have shown in Theorem 1.1(a) that if a wandering Julia component is exposed,it must be eventually simply connected.

We have also:

Lemma 4.1. Let f be a rational map. There are eventually simply connected andexposed wandering Julia components if and only if there is an infinitely connectedFatou domain.

15

Proof. Let U be an infinitely connected Fatou domain. Then it must contain un-countably many boundary components. Note that each boundary component iscontained in a unique Julia component and there are only countably many prepe-riodic Julia components. Thus there exists a wandering exposed Julia component.By Theorem 1.1(a), it is eventually simply connected.

Now let J be an eventually simply connected and exposed wandering Julia com-ponent. Then there exists a Fatou domain U such that fn(J) ∩ ∂fn(U) 6= ∅ for alln ≥ 0, and there exists an integer n0 ≥ 0 such that fn(U) is periodic and fn(J) issimply connected for n ≥ n0. So we can assume that J itself is simply connectedand f(U) = U . Each fn(J) contains a unique boundary component for n ≥ 0 be-cause fn(J) ∩ ∂U 6= ∅. Combining with the condition that J is wandering and thefact that f maps each boundary component onto another, we can find a boundarycomponent B with f i(B) ∩ f j(B) = ∅ and hence U has infinitely many boundarycomponents.

Lemma 4.2. ([PT1]). Let f be a nice rational map with the Julia set Jf and thepostcritical set Pf . Let E be the union of Julia components J of f so that eitherJ ∩Pf 6= ∅, or SPf

(J) ≥ 3, or SPf(J) = 2 but J does not separate any pair of Julia

components in the same homotopy class. If C\E has no annular components thatare disjoint from Pf , then all wandering components of Jf are simply connected.

4.2 The disc-annulus surgery

The following surgery exposed in [PT2] will be our essential tool to construct variousexamples.

We say that V ⊂ C is a smooth disc if ∂V is a real-analytic Jordan curve. Bya branched covering we mean a proper C1 map between smooth, oriented (real) 2-manifolds, possibly with boundary, such that the boundary map is a covering mapof (real) 1-manifolds, and such that on the interior, the map is given in appropriatelocal (complex) coordinates by z 7→ zd for some d.

The following two lemmas are the technical ingredients for the definition of ourdisc-annulus surgery.

Lemma 4.3. Let A ⊂ C be an open annulus bounded by two C1 Jordan curves γ±,and let W be an open disc bounded by a C1 Jordan curve η. Give orientations tothe curves such that A and W lie to the left of their boundaries. Let f± : γ± → ηbe two orientation-preserving C1-coverings with degree d± ≥ 1. Then there exists abranched covering a : A → W satisfying the following properties:

1. a|γ± = f±

2. a(A) = W and the degree of a is d+ + d−

3. a can be chosen to be C1 in A and holomorphic and proper in a union of anycollection of finitely many disjoint smooth discs.

We call the map a a covering extension of the boundary maps f±. In practice,we will take a to be holomorphic in a neighborhood of its critical points.

Lemma 4.4. Let D, D′ be two smooths discs in C. Then a holomorphic propermapping F : D → D′ extends to a holomorphic map in a neighborhood of D. Inparticular F : ∂D → ∂D′ is a C1 covering.

A branched covering F : C → C is quasi-regular if F = h ◦ f ◦ g where f is arational map and h, g are quasi-conformal homeomorphisms. A branched covering

16

F is quasi-rational if it is quasi-conformally conjugate to a rational map. The Juliaset of a quasi-rational map is thus well defined, and has the same qualitative metricand measure-theoretic properties as the Julia set of a rational map. Denote theJulia set of F by JF .

Given a rational map or branched covering f , let

Pf = the postcritical set = ∪n>0fn(Cf )

where Cf is the set of critical points of f .

V

V

f(V ) f(V )

f(V ) f(V )

H

F (H)

fF

F

before surgery after surgeryFigure 3: surgery

The following is a particular case of results in [PT2].

Theorem 4.5. Disc-annulus surgery. Let f be a rational map with Julia setJf . Let z0 be a point in the Fatou set such that z0 is neither a periodic point norcontained in a rotation domain. Let (V, H, h, a) satisfy the following conditions:

• V is a smooth disc containing z0 in the Fatou set such that

– ∂V contains no critical points;– f : V → f(V ) is proper;– f j(V ) ∩ V = ∅ for 0 < j < ∞,

• H is a smooth disc with H ⊂ V ;

• h : H → C\f(V ) is bi-holomorphic, and

• a : V \H → f(V ) is a covering extension of the boundary maps such that a isholomorphic near the critical points.

Then the map

F :

C\V → f(C\V ) F (z) = f(z)H → C\f(V ) F (z) = h(z)V \H → f(V ) F (z) = a(z)

is quasi-rational.

We refer to F as a disc-annulus surgery of f supported on a neighborhood of z0.Note that F (V ) = C and deg(F ) = deg(f) + deg(h).

Corollary 4.6. In the above setting, denote by Wf , resp. WF the Fatou componentsof f . resp. F , containing ∂V . If Wf is periodic, then WF is periodic and infinitelyconnected. If Wf is strictly preperiodic, then the connectivity of WF is equal tom0 + m1, where m0 and m1 are the connectivity’s of Wf and f(Wf ). Furthermore

17

1. Jf ⊂ JF , JF is disconnected, and every connected component of Jf is aconnected component of JF .

2. If H ∩ Pf = ∅, then also H ∩ PF = ∅. In this case,

• every Julia component of F passing through H infinitely many times is apoint, and

• every other Julia component of F is conformally homeomorphic to a Juliacomponent of f .

3. H ∩ JF 6= ∅. If H ∩ Pf = ∅ and if Wf is not fixed under the iteration of f ,then there is a cantor set L contained in H with each point of L being a buriedJulia component for F .

Proof. The proofs of 1 and 2 can be found in [PT2]. Here we just prove 3.3. As h is bi-holomorphic, one may define ξ : C\f(V ) → H as the inverse map

of h. It is also a univalent branch of F−1 on C\f(V ). By Schwarz Lemma ξ has aunique attracting fixed point w0 in H. Consequently w0 is a repelling fixed pointof F , thus is contained in JF .

Assume now H ∩ Pf = ∅ and f(Wf ) 6= Wf . Consequently H ∩ PF = ∅ andf(Wf ) ∩ Wf = ∅. There is therefore a Julia component J ′ separating Wf andf(Wf ) (i.e. with Wf and f(Wf ) contained in distinct complementary componentsof J). As f(V ) ⊂ f(Wf ), the component J ′ separates V from f(V ). In other words,J ′ is essentially contained in the annulus C\(H ∪ f(V )).

Set Ui = ξi(H), i = 1, 2. They are Jordan domains. The annulus U1\U2 containsJulia points as it contains ξ2(J ′). By classical properties of the Julia set, the set ofpreimages of w0 is dense in JF . In particular one can find a point w′ ∈ U1\U2 withF i0(w′) = w0 for some i0 ≥ 1.

Let k ≥ 1 be the first time such that F k(w′) ∈ H. Let W be the componentof F−k(H) containing w′. As H is a Jordan domain with H ∩ PF = ∅, we knowthat W is also a Jordan domain and F k maps W univalently onto H. From thechoice of w′ we know that W ∩ (U1\U2) 6= ∅. If there is z ∈ (∂U1 ∪ ∂U2) ∩W , thenF j(z) ⊂ f(V ) for j = 2 or 3. Since fn(V ) ∩ V = ∅, W can not be mapped into Hby the forward iterations of F . It is impossible. Hence W ⊂ U1\U2.

Define G : U2 ∪W → H by G|U2 = F 2 and G|W = F k. Then G is a conformalrepellor. Denote by L its non-escaping set. It is a Cantor set by Schwarz Lemma,and is a subset of JF .

Now ξ(J ′) is a Julia component for F and is contained essentially in the annulusH\U1. But U2 ∪W ⊂ U1. It follows that ξ(J ′) is contained essentially in the annuliH\U2 and H\W .

As every pullback by G of ξ(J ′) is again a Julia component for F , we concludethat every point of L is a buried Julia component for F .

4.3 Examples of Julia components with different topo-logical structures

In this part, we will first construct examples (examples 1-6) where the Julia setsare hyperbolic and there can be no exposed doubly wandering Julia components inthese examples. Hence there are seven possible combinations of ”empty set” and”uncountably many” to be filled in our scheme, and we will give concrete examples

18

of all (examples 1-6) but one which is shown as below:

{wandering Julia comp.} {simply connected} {doubly connected}{exposed} uncountably many ∅{buried} ∅ uncountably many

Here are the six examples:Example 1, a polynomial with a disconnected Julia set satisfying that the filled-inJulia set contains no critical points.

{wandering Julia comp.} {simply connected} {doubly connected}{exposed} uncountably many ∅{buried} ∅ ∅

Example 2, McMullen, resp. Pilgrim-Tan, f(z) =z2

1 + bz2+

11011z3

; with b = 0,

resp. b = −1. See [McM, PT1].

{wandering Julia comp.} {simply connected} {doubly connected}{exposed} ∅ ∅{buried} ∅ uncountably many

Example 3, disc-annulus surgery in a non periodic Fatou domain.

{wandering Julia comp.} {simply connected} {doubly connected}{exposed} ∅ ∅{buried} uncountably many ∅

Let f(z) = z2 − 1. Set z0 = +1. Perform a univalent disc-annulus surgery in aneighborhood V of z0 to get a quasi-rational map F . Apply Lemma 4.1 to F weknow that F has no exposed wandering Julia components. We apply then Lemma4.2 to prove that F has no doubly connected wandering Julia components. As JF isnot connected, it has uncountably many buried simply connected wandering Juliacomponents. All of them are point components by [PT1].Example 4, disc-annulus surgery in a periodic Fatou domain.

{wandering Julia comp.} {simply connected} {doubly connected}{exposed} uncountably many ∅{buried} uncountably many ∅

Let f(z) = z2 − 1. Set z0 = 0.25. Perform a univalent disc-annulus surgery inV = D(0.25, 0.01) to get a quasi-rational map F . Apply Lemma 4.1 to F we knowthat F has uncountably many exposed wandering Julia components. We apply then

19

−1 0 1

Figure 4: The Julia set of z 7→ z2 − 1

Lemma 4.2 to prove that F has no doubly connected wandering Julia components.From Corollary 4.6.3, we get a Cantor set of buried point Julia components for F .

Example 5, disc-annulus surgery in a non periodic Fatou domain of a map inExample 2.

{wandering Julia comp.} {simply connected} {doubly connected}{exposed} ∅ ∅{buried} uncountably many uncountably many

Let f(z) = z2 +1

1011z3. Set z0 = 0. It is mapped by f to the superattracting fixed

point ∞. Perform a univalent disc-annulus surgery in V = D(0, ε) to get a quasi-rational map F . Apply Lemma 4.1 to F we know that F has no exposed wanderingJulia components. We then apply Corollary 4.6 to prove that F has uncountablydoubly connected wandering Julia components (all quasi-circles). From Corollary4.6.3, we get a Cantor set of buried point Julia components of F .

Example 6, double disc-annulus surgery of a map in Example 2.

{wandering Julia comp.} {simply connected} {doubly connected}{exposed} uncountably many ∅{buried} uncountably many uncountably many

Let f(z) = z2 +1

1011z3. Set z0 to be a point in the Fatou component ∞, disjoint

from Pf . Perform a univalent disc-annulus surgery on a small neighborhood V ofz0 disjoint from Pf to get a quasi-rational map F . Apply Lemma 4.1 to F we knowthat F has uncountably many exposed wandering Julia components. We then applyCorollary 4.6.1 to prove that F has uncountably doubly connected wandering Juliacomponents (all quasi-circles).

Perform a second disc-annulus surgery, but this time on a small Jordan domainin the Fatou component of 0 for F , disjoint from PF . Denote the resulting map by F .Applying then Corollary 4.6.1 and 3, we conclude that F has now also uncountablymany buried point Julia components.

20

The above examples are hyperbolic ones. In the end of this subsection, we willconstruct a non-hyperbolic example.Example 7, critical values on buried wandering Julia components.

Let g be a rational map quasi-conformally conjugate to the quasi-rational mapF constructed in Example 6. Assume that the quasi-conformal conjugacy maps ∞to ∞.

Then the immediate basin of ∞ of g contains three distinct critical values∞, v1, v2.

Deform slightly g if necessary so that v1, v2 are in distinct grand orbits.Now make a double disc-annulus surgery on g as follows:For i = 1, 2 choose a smooth neighborhood Vi of vi such that for all j > 0,

gj(V i) ∩ Vi = ∅, and that V1 ∩ gj(V 2) = ∅, V2 ∩ gj(V 1) = ∅.Choose then a smooth disc Hi 3 vi so that Vi ⊃ H i.Choose two distinct points w1, w2 each in a wandering Julia component (of any

type: exposed or buried, simply connected or doubly connected) outside V1 ∪ V2.Choose univalent maps hi : Hi → C\g(Vi) so that hi(vi) = wi, i = 1, 2.Choose covering extension of the boundary maps ai : Vi\Hi → g(Vi), i = 1, 2

such that each ai is holomorphic near the critical points.Then the map

G :

C\(V1 ∪ V2) → g(C\(V1 ∪ V2)) G(z) = g(z)Hi → C\g(Vi) G(z) = hi(z), i = 1, 2Vi\Hi → g(Vi) G(z) = ai(z), i = 1, 2

is quasi-rational.Note that for i = 1, 2, the point vi is a critical value of G, is contained in a

wandering Julia component of the same type as wi, and is non recurrent.By choosing w1, w2 buried we can construct an example showing that our con-

dition in Theorem 1.1.(c) is not sufficient to have wandering Julia components withC(orb(J)) = 3.

By choosing w1 on a Jordan curve wandering Julia component, we obtain anexample that, for c1 a critical point so that G(c1) = v1 (and thus to G2(c1) = w1),the point c1 is contained in a wandering Julia component J with C(J) = 3 (butC(orb(J)) = 2)).

For this example, the set Σ is countable (in particular non-empty) but it doesnot contain any infinite Julia orbit.

4.4 Recurrent critical points in buried simply connectedwandering Julia components

Example 8. We will construct in four steps an example of a rational map witha recurrent critical point in a buried and wandering Julia component. We thenexplain how to construct examples with several such critical points.Step 1. It is known from the work of Branner-Hubbard that there are (many) cubicpolynomials with a Cantor Julia set containing a recurrent critical point.

Let f be one of them. For this map, there is a Jordan disc V ′ bounded byan equipotential, so that f−1(V ′) has two connected components U ′

1 and U ′2 with

disjoint closure and compactly contained in V ′, and that f : U ′i → V ′ is proper of

degree i for i = 1, 2.

21

Step 2. Let g be a degree 5 polynomial with one escaping critical point satisfying thefollowing properties: there is a Jordan disc V , so that g−1(V ) has three connectedcomponents U1 and U2 and U3, all compactly contained in V and have disjointclosures, and that g : U1 ∪U2 → V is quasi-conformally conjugate to f : U ′

1 ∪U ′2 →

V ′, and g : U3 → V is hybrid equivalent to z → z2. Thus two critical points of gescape to ∞, one other critical point is in a point Julia component and is recurrent,while the fourth critical point is a fixed point. We may furthermore assume that 0is the fixed critical point.

The construction of such g is easy: Take a point w ∈ V ′\U ′1 ∪ U ′

2, and a conformalrepresentation φ : V ′ → D (where D denotes the unit disc) so that φ(w) = 0. LetR > 1 so that {|z| ≤ 1/R} ∩ φ(U ′

1 ∪ U ′2) = ∅. Set Φ(z) = R2 · φ(z). Then Φ has an

C1 extension to the boundary, with Φ(V ′) = {|z| ≤ R2} and Φ(U ′1 ∪ U ′

2) ⊂ {R <|z| < R2}.

We can then define

G :

Φ(U ′1 ∪ U ′

2) → {|z| ≤ R2} by G(z) = Φ ◦ f ◦ Φ−1(z){|z| ≤ R} → {|z| ≤ R2} by G(z) = z2

{|z| ≥ R2} → {|z| ≥ R10} by G(z) = z5

{R ≤ |z| ≤ R2}\Φ(U ′1 ∪ U ′

2) → {R2 ≤ |z| ≤ R10} by quasi-regular interpolation.

The last line above is similar to the construction in Lemma 4.3.This G is clearly quasi-rational, and is conjugate to a degree 5 polynomial g

by a quasi-conformal change of coordinates ψ so that ψ(0) = 0, ψ(∞) = ∞. Thisis our desired polynomial. We may then set V = ψ ◦ Φ(V ′) = ψ({|z| < R2}),U3 = ψ({|z| < R}) and Ui = ψ ◦ Φ(U ′

i) for i = 1, 2.Step 3 (folding). This is a technique that has been used to construct examplessuch as that of McMullen and of Pilgrim-Tan, shown by formula in Example 2above.

For K, K ′ ⊂ C two continua we use the notation A(K,K ′) to denote the emptyset in case K ∩K ′ 6= ∅ and to be the unique annular complementary component of(K ∪K ′) if K ∩K ′ = ∅.

For L a Jordan domain in C denote by γL the boundary curve of L.For our polynomial g, choose a small disc neighborhood D of the superattracting

fixed point 0 so that g(D) ⊂⊂ D.Thus γD, γV and γg(V ) denote the corresponding boundary curves.Notice that the escaping critical values v1, v2 of g are contained in A(γV , γg(V )).

Choose now a smooth essential Jordan curve η ⊂ A(γV , γg(V )) so that v1, v2 ∈A(γV , η) and modA(γD, γg(V )) = d ·modA(η, γg(V )) for some integer d. Denote byD∞ the disc containing ∞ bounded by γg(V ).

Now define

H :

V → g(V ) by H(z) = g(z);A(η, γg(V )) → A(γD, γg(V )), η → γg(V ) by a covering of degree d

holomorphic in the interior;D∞ → D by quasi-regular interpolation;A(γV , η) → D∞ by quasi-regular interpolation.

One can prove then this H is quasi-rational, with only one periodic Fatou com-ponent: the immediate basin of 0 (it is simply connected), with one recurrent criticalpoint as a Julia component, that is buried and wandering, and with all the othercritical points attracted by 0. This H has no exposed wandering Julia components.

22

0 • D

γD

γV

A

•v1

•v2

η

A′

γg(V )

D∞ • ∞

Hholomorphic covering

• D =H(D∞)

γD = H(γg(V ))

H(A′)

γg(V ) = H(η)

D∞=H(A) •

Figure 5: folding

Step 4. Get a rational map h from H through a quasi-conformal conjugacy. TheJulia set of h has the same topological properties as that of H. Furthermore ourTheorem 1.1 proves that h has no wandering Julia component J with C(orb(J)) = 3.Further examples. Using a similar folding surgery on a more complicated polyno-mial g one can get a rational map with several recurrent critical points each beinga buried and wandering Julia component.

By adding a disc-annulus surgery in the immediate basin of 0 one can get a mapwith exposed wandering Julia components as well.

References

[Bea] A. F. Beardon, Iteration of Rational Functions, Graduate text in Mathemat-ics, Vol. 132, Springer-Verlag, New York, 1993.

[McM] C. T. McMullen, Automorphisms of rational maps, In: Holomorphic Func-tions and Moduli, edited by Drasin, Earle, Gehring, Kra and Marden, 31-60,Springer, 1988.

[Mil1] J. Milnor, Dynamics in one complex variable, third edition, Princeton Uni-versity Press, 2006.

[Mil2] J. Milnor, On rational maps with two critical points, Experimental Mathe-matics, Vol. 9(2000), No. 4, 481-522.

[PQRTY] Peng, W. J., Qiu, W. Y., P. Roesch, Tan Lei and Yin, Y. C. A tableauapproach of the KSS nest, accepted by Conformal Geometry and Dynamics(AMS), 2009.

[PT1] K. Pilgrim and Tan Lei, Rational maps with disconnected Julia set,Asterique, 261(2000), 349-384.

[PT2] K. Pilgrim and Tan Lei, On disc-annulus surgery of rational maps, Pro-ceedings of the International Conference in Dynamical Systems in Honor ofProfessor Liao Shan-tao 1998, ed. Y. Jiang et L. Wen, World Scientific 1999,pp. 237-250.

[QY] Qiu, W. Y. and Yin, Y. C., Proof of the Branner-Hubbard conjecture onCantor Julia sets, Science in China, Series A, Vol. 52(2009), No. 1, 45-65.

23

[Shi] M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci.Ec. Norm. Sup., 20(1987), 1-29.

[Sul] D. Sullivan, Quasiconformal homeomorphisms and dynamics I: solution of theFatou-Julia problem on wandering domains, Ann.Math., 122(1985), 401-418.

[Z] Zhai, Y. , A generalized version of Branner-Hubbard conjecture for rationalfunctions, to appear in Acta Mathematica Sinica.

Addresses:Cui Guizhen, Academy of Mathematics and Systems Science, Chinese Academy of Sci-

ences, Beijing, 100190, P.R.China, gzcui@math.ac.cnPeng Wenjuan, School of Mathematical Sciences, Peking University, Beijing, 100871,

P.R.China, pengwj@math.pku.edu.cnTan Lei, Universite d’Angers, Faculte des Sciences, LAREMA, 2, Boulevard Lavoisier,

49045 Angers cedex 01, France, Lei.Tan@univ-angers.fr