THREE DIMENSIONAL PARTITION

AND INFINITE RENORMALIZATION

Yunping Jiang

March 13, 2002

Abstract. We provide a detailed construction of the three dimensional partition forthe Julia set of an infinitely renormalizable quadratic polynomial. We show that for anunbranched infinitely renormalizable quadratic polynomial having complex bounds, thethree-dimensional partition determines points in the Julia set dynamically. Furthermore,we construct a partition in the parameter space about a subset consisting of infinitelyrenormalizabe points. We prove that this subset is dense on the boundary of the Mandel-brot set. We show that every point in this subset is an unbranched quadratic polynomialhaving complex bounds. So its Julia set is locally connected. Moreover, we prove that theMandelbrot set is locally connected at every point in this subset.

Contents

§1 Introduction.

§2 Some basic facts about the dynamics of a quadratic polynomial.

§3 Yoccoz partition and renormalizability.

§4 Construction of a three-dimensional partition following renormalization.§4.1 Infinite renormalization.§4.2 Three-dimensional partition.§4.3 Local connectivity.

§5 A partition in the parameter space.§5.1 Some basic facts about the Mandelbrot set from Douady-Hubbard.§5.2 On partitions around infinitely renormalizable points

Key words and phrases. Yoccoz partition, three dimensional partition, infinite renormalization,quadratic polynomial, the Mandelbrot set, local connectivity.

2000 Mathematics Subject Classification: Primary 37F25, 37F45 and Secondary 37Cxx, 37ExxThis research is supported in part by NSF grants and by PSC-CUNY awards

Typeset by AMS-TEX

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§1. Introduction.

Organizing information provided by a dynamical system is important in the studyof this dynamical system. A very useful method is called Markov partitions, which wasintroduced by Sinai [Si1,Si2] into dynamical systems from probability theory and whichwas generalized by Bowen [Bo1,Bo2]. For example, the dynamical system generated bythe famous horseshoe map constructed by Smale [Sm] has a Markov partition. Actually,Bowen showed that one can always construct a (finite) Markov partition for an Axiom Adynamical system (see [Bo3]). So many interesting questions, in particular, those relateto chaos, of the dynamical system are well understood by using symbolic dynamicalsystems.

When things come to a non Axiom A dynamical system, the situation becomes morecomplicate. For example, when we study the dynamics of a quadratic polynomial whosecritical orbit is recurrent, we can not really find a (finite) Markov partition. This cre-ates a major difficulty in the study of dynamics of a quadratic polynomial. Recently,Yoccoz constructed a partition for a non-renormalizable quadratic polynomial. Usingthis partition and a puzzle rule on the partition, he proved that the Julia set of anon-renormalizable quadratic polynomial is locally connected. Another similar parti-tion for a cubic polynomial whose one critical orbit is bounded and the other criticalorbit in the complex plane tends to infinity is constructed by Branner and Hubbardin [BH1,BH2]. Using this partition and a puzzle rule on the partition, they provedthat the Julia set of such a cubic polynomial is a Cantor set whose Lebesgue measureis zero. Moreover, using the partition constructed by Yoccoz for a non-renormalizablequadratic polynomial, Shishikura [Sk] and Lyubich [Ly1] proved that the Lebesgue mea-sure of the Julia set of this polynomial is zero. We have further developed this kind ofpartitions for an infinitely renormalizable quadratic polynomial in [Ji3] and [Ji2]. Usingthis development, we proved in [Ji3] that any two Feigenbaum-like folding maps are qua-sisymmetrically conjugate. And in [Ji2], we proved that the Julia set of a unbranchedinfinitely renormalizable quadratic polynomial having complex bounds is locally con-nected. By combining the result in [Ji2] and Sullivan’s result [Su] on the existence ofcomplex bounds for the Feigenbaum quadratic polynomial, we concluded in [HJ] thatthe Julia set of the Feigenbaum quadratic polynomial is locally connected. Levins andvan Strien [LS] and Lyubich and Yamploski [LY] further proved that a real infinitelyrenormalizable quadratic polynomial has complex bounds. Thus its Julia set is locallyconnected.

Although the main result in [Ji2] is about the local connectivity of the Julia setof an infinitely renormalizable quadratic polynomial, the construction of a partitionin the study is also interesting. We provide a detailed exposition in §4 about thispartition which we call the three dimensional partition for the Julia set of an infin-itely renormalizable quadratic polynomial. In the same section, we show that for anunbranched infinitely renormalizable quadratic polynomial having complex bounds, thethree-dimensional partition determines points in the Julia set dynamically (Theorem 6).

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Moreover, we transfer the three dimensional partitions into the parameter space andprove in §5.2 that there is a subset which is dense on the boundary of the Mandelbrotset such that every point in this subset is an unbranched quadratic polynomial havingcomplex bounds (Theorems 7 and 8). Furthermore, by applying the Rouche Theorem inclassical complex analysis, we prove in §5.2 that the Mandelbrot set is locally connectedat every point in this subset (Theorems 7 and 8). Another interesting subset on theboundary of the Mandelbrot set is constructed by Douady and Hubbard. Douady andHubbard proved that this subset is generic and the Julia set of every point in this subsetis not locally connected. We provide some information about Douady and Hubbard’sconstruction in §5.1. The reader who is interested in more details could refer to Milnor’sexpository article [Mi2].

We review some basic facts about dynamics of quadratic polynomial in §2. There aretwo text books [Mi1,CG] available in complex dynamics. The reader, who is interested inthese basic facts and who would like to learn more about complex dynamics, could referto them. In §3, we give an outline of the partition constructed by Yoccoz for a quadraticpolynomial and show its relation with the renormalizability of the polynomial. Thereare two expository articles [Hu,Mi2] written by Hubbard and Milnor for the partitionconstructed by Yoccoz and a puzzle rule on this partition. The reader who would like tolearn more may go to there. He may also go to [Ji1, pp. 206-219]. For the reader whois interested in the renormalization theory of quadratic polynomials and related topicsthere are two well written books [Mc1,Mc2] by McMullen.

§2. Some basic facts about the dynamics of quadratic polynomials.

Let C be the complex plane and let C be the extended complex plane (the Riemannsphere). Suppose f is a holomorphic map from a domain Ω ⊂ C into itself. A point z inΩ is called a periodic point of period k ≥ 1 if fk(z) = z but fi(z) 6= z for all 1 ≤ i < k.The number λ = (fk)′(z) is called the multiplier of f at z. A periodic point of period1 is also called a fixed point. A periodic point z of f is said to be super-attracting,attracting, neutral, or repelling if |λ| = 0, 0 < |λ| < 1, |λ| = 1, or |λ| > 1. The proof ofthe following theorem can be found in Milnor’s book [Mi1, pp. 86-88].

Theorem A (Theorem of Bottcher). Suppose p is a super-attracting periodic pointof period k of a holomorphic map f from a domain Ω into itself. There is a neighborhoodW of p, a holomorphic diffeomorphism h : W → h(W ) with h(p) = 0, and a uniqueinteger n > 1 such that

h fk h−1(w) = wn

for w ∈ h(W ). Furthermore, h is unique up to multiplication by an (n − 1)-st root ofunity.

Denote Dr = z ∈ C | |z| < r the disk of radius r centered 0. Let qc(z) = z2 +c be aquadratic polynomial. For r large enough, U = q−1

c (Dr) is a simply connected domain3

and its closure is relatively compact in Dr. Then qc : U → V = Dr is a holomorphic,proper, degree two branched covering map. This is a model of quadratic-like mapsdefined by Douady and Hubbard [DH1] as follows: A quadratic-like map is a triple(f, U, V ) such that U and V are simply connected domains isomorphic to a disc withU ⊂ V and such that f : U → V is a holomorphic, proper, degree two branched coveringmapping. For a quadratic-like map (f, U, V ),

Kf = ∩∞n=0f−n(U)

is called the filled-in Julia set. The Julia set Jf is the boundary of Kf . Both Kf andJf are compact. A quadratic-like map (f, U, V ) has only one critical point which wealways denote as 0. Refer to [Mi1, pp. 91-92] for a proof of the following theorem.

Theorem B. The set Kf (as well as Jf ) is connected if and only if 0 is in Kf . More-over, if 0 is not in Kf , Kf = Jf is a Cantor set.

Consider a quadratic polynomial qc(z) = z2 + c. By the definition, the filled-in Juliaset of qc is the set of all points not going to infinity under forward iterates of qc andthe Julia set is the boundary of the filled-in Julia set. We use Kc and Jc to mean itsfilled-in Julia set and Julia set. For c = 0 and every r > 1, (q0, Dr, Dr2) is a quadratic-like map whose filled-in Julia set is K0 = D1. For any c, ∞ is a super-attracting fixedpoint of qc, applying Theorem A, there is a unique holomorphic diffeomorphism hc

(called the Bottcher change of coordinate) defined on a neighborhood B0 about ∞ withhc(∞) = ∞, h′c(∞) = 1 such that

hc qc h−1c (z) = z2, z ∈ B0.

Assume hc(B0) = C \ Dr (for a fixed large number r > 1). Let Bn = q−nc (B0). If

0 6∈ Bn, thenqc : Bn ∩ C→ Bn−1 ∩ C, n ≥ 1,

is unramified covering map of degree two. So we can inductively extend

hc : Bn → C \Dr

12n

such thathc qc h−1

c (z) = z2 z ∈ Bn.

LetBc(∞) = z ∈ C | qnc (z) →∞ as n →∞

be the basin of ∞. Then Kc = C \Bc(∞). Let

Gc(z) = limn→∞

log+ |qnc (z)|2n

,

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where log+ x = sup0, log x, be the Green function of Kc. It is a proper harmonicfunction whose zero set is Kc and whose critical points are bounded by Gc(0). So theBottcher change of coordinate can be extended up to

hc : Uc = z ∈ C | Gc(z) > Gc(0) → z ∈ C | |z| > Gc(0).such that

hc qc h−1c (z) = z2.

Moreover, Gc = log |hc|. The set G−1c (log r), r > 1, is called an equipotential curve.

If Kc is connected (equivalent to say 0 6∈ Bc(∞)), then Uc = Bc(∞) \ ∞; if Kc isa Cantor set (equivalent to say 0 ∈ Bc(∞)), then Uc is punctured disk bounded by an∞ shape curve passing 0. By the definition, the Mandelbrot set M is defined as the setof parameters c such that Kc is connected.

Suppose c ∈M. Let Sr = z ∈ C | |z| = r be a circle of radius r > 1. Then

sr = h−1c (Sr) = G−1

c (log r)

is an equipotential curve. Note that

qc(sr) = sr2 .

For every r > 1, let Ur = h−1c (Dr). Then Ur is a domain bounded by the equipotential

curve sr. Furthermore, (qc, Ur, Ur2) is a quadratic-like map whose filled-in Julia set isalways Kc. In the rest of paper, a quadratic polynomial q(z) = z2 + c could also meana quadratic-like map (q, Ur, Ur2) for some r > 1.

Take Eθ = z ∈ C | |z| > 1, arg(z) = 2πθ for 0 ≤ θ < 1. Let

eθ = h−1c (Eθ).

Then eθ is called an external ray of angle θ and

qc(eθ) = e2θ (mod 1).

An external ray is an integral curve of the gradient vector field ∇G on Bc(∞). Anexternal ray eθ is said to land at Jc if eθ has only one limiting point at Jc. It is periodicwith period m if qic (eθ) ∩ eθ = ∅ for 1 ≤ i < m and if qmc (eθ) = eθ. Refer to [Hu,Mi2]for the following theorem (the reader can also find a proof in [Ji1, pp. 199]).

Theorem C (Douady and Yoccoz). Suppose qc(z) = z2+c is a quadratic polynomialwith a connected filled-in Julia set Kc. Every repelling periodic point of qc is a landingpoint of finitely many periodic external rays with the same period.

Two quadratic-like maps (f, U, V ) and (g, U ′, V ′) are said to be topologically conju-gate if there is a homeomorphism h from a neighborhood Kf ⊂ X ⊂ U to a neighbor-hood Kg ⊂ Y ⊂ U ′ such that

h f(z) = g h(z), z ∈ X.

If h is quasiconformal (see [Ah]) (resp. holomorphic), then they are quasiconformally(resp. holomorphically) conjugate. If h can be chosen such that hz = 0 a.e. on Kf ,then they are hybrid equivalent.

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Theorem D (Douady and Hubbard [DH1]). Suppose (f, U, V ) is a quadratic-likemap whose Julia set Jf is connected. Then there is a unique quadratic polynomialqc(z) = z2 + c which is hybrid equivalent to f .

Therefore, we have the following theorem by referring to [Mi1] and Theorem D.

Theorem E. Let (f, U, V ) be a quadratic-like map.

(1) The Julia set Jf is completely invariant, i.e., f(Jf ) = Jf and f−1(Jf ) = Jf .(2) The Julia set Jf is perfect, i.e., J ′f = Jf , where J ′f means the set of limit points

of Jf .(3) The set of all repelling periodic points Ef is dense in Jf , i.e., Ef = Jf .(4) The limit set of f−n(z)∞n=0 is Jf for every z in V .(5) The Julia set Jf has no interior point.(6) If f has no attracting, super-attracting, and neutral periodic points in V , then

Kf = Jf .

§3. Yoccoz partition and renormalizability.

Consider a quadratic polynomial qc(z) = z2 + c whose Julia set Jc is connected. Theexternal ray e0 is the only one fixed by qc. It lands either at a repelling or parabolicfixed point β of qc. Suppose β is repelling. From Theorem C, e0 is the only external raylanding at β (see Fig. 1). So Jc \β is still connected. We call β the non-separate fixedpoint. Let α 6= β be the other fixed point of qc. If α is attracting or super-attracting,then Jc = Kc \ B(α) where B(α) = z ∈ C | qnc (z) → α as n → ∞ is the basin ofα. In this case, qc is hyperbolic and Jc is a Jordan curve. If α is repelling. Then thereare at least two periodic external rays landing at α (see Fig. 1). All of them are in onecycle of period k ≥ 2 (refer to Theorem H in §5.1). We use Λ to denote the union ofthis cycle of external rays. Then Λ cuts C into finitely many simply connected domains(see Fig. 1)

Ω0, Ω1, . . . , Ωk−1.

Each domain contains points in Jc. This implies that Jc \ α is disconnected. We callα the separate fixed point.

The point 0 is the only finite critical point of qc. Let ci = qic (0), i ≥ 1, be the ith

critical value. We use CO to denote the critical orbit ci∞i=0. The critical point 0 is saidto be recurrent if for any neighborhood W of 0, there is a critical value ci 6= 0 in W . Wewill only consider those quadratic polynomial whose critical point is recurrent and whichhas no neutral periodic points in this section. These are assumed for a quadratic-likemap too. Under these assumptions, qc has only repelling periodic points and Jc = Kc.

Let Ur ⊂ C be the open domain bounded by an equipotential curve sr. For a fixedr > 1, (q, U√r, Ur) is a quadratic-like map. The set Λ cuts Ur into finitely many Jordandomains (see Fig. 1). Let C0 be the closure of the one containing 0, and let B0,i be the

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closure of the domain containing ci for 1 ≤ i < k. We call the collection

η0 = C0, B0,1, . . . , B0,k−1

the original partition about Jc.

Fig. 1

Since Λ is forward invariant under qc, the image qc(C0 ∩ Jc) (resp. q(B0,i ∩ Jc) forevery 1 ≤ i < k) is the union of some sets from

η0 ∩ Jc = C0 ∩ Jc, B0,1 ∩ Jc, . . . , B0,k−1 ∩ Jc.

Each qc|B0,i is degree one, proper, and holomorphic but qc|C0 is degree two, proper,and holomorphic.

For each n > 0, let αn = q−nc (α) and Λn = q−n

c (Λ) (denote α0 = α and Λ0 = Λ).The set Λn is the union of external rays landing at points in αn. It cuts the closure ofthe domain U

r1

2ninto a finite number of closed Jordan domains,

ηn = Cn, Bn,1, . . . , Bn,kn.

Here we use Cn to denote the one containing 0 and Bn,1, . . . , Bn,qn to denote others.Since qc(Λn) = Λn−1, qc(Cn) (resp. qc(Bn,i)) is in ηn−1. Each qc|Bn,i is degree one,holomorphic, proper but qc|Cn is degree two, holomorphic, and proper. We call ηn thenth-partition about Jc. Thus we get a sequence

ξ = ηn∞n=0

of nested partitions, which we call the Yoccoz partition about Jc.Let Λ∞ = ∪∞n=0Λn. For every x ∈ Jc \ Λ∞, there is only one sequence Dn(x)∞n=0

of nested closed Jordan domains such that x ∈ Dn(x) ∈ ηn. For every x in Λ∞, there7

are k sequences Dn,i(x)∞n=0, 1 ≤ i ≤ k, of nested closed Jordan domains such thatx ∈ Dn,i(x) ∈ ηn. In this case, x is on the boundary of each Dn,i(x) but it is an interiorpoint of the closed domain ∪k

i=1Dn,i(x). When there is no confusion, we will only useDn(x)∞n=0 to denote such a sequence even in the case that x ∈ Λ∞. Further, we call

x ∈ · · · ⊆ Dn(x) ⊆ Dn−1(x) ⊆ · · · ⊆ D1(x) ⊆ D0(x)

an x-end. In particular, we call

0 ∈ · · · ⊆ Cn ⊆ Cn−1 ⊆ · · · ⊆ C2 ⊆ C1 ⊆ C0

the critical end. One of the important problems in dynamical systems is whether thesequence ξ provides a complete description of points in Jc dynamically?

Definition 1. We say ξ determines points in Jc dynamically if for any x 6= y ∈ Jc,there is an integer n ≥ 0 and two domains x ∈ D and y ∈ D′ in ηn such that D∩D′ = ∅.

From Theorem D, the sequence ξ can be also constructed similarly for any quadratic-like map whose Julia set is connected. Definition 1 relates to the concept called renor-malizability for quadratic-like maps.

Definition 2. Let (f, U, V ) be a quadratic-like map whose Julia set Jf is connected.We say it is (once) renormalizable if there is an integer n′ ≥ 2 and a domain 0 ∈ U ′ ⊂ Usuch that

f1 = fn′: U ′ → V ′ = f1(U ′) ⊂ V

is a quadratic-like map with connected Julia set Jf1 = J(n′, U ′, V ′). In this situation, wecall (f1, U

′, V ′) a renormalization of (f, U, V ). Otherwise, we call f non-renormalizable.

The relation between ξ and the renormalizability of a quadratic polynomial is that

Theorem F (Yoccoz). The polynomial qc(z) = z2 + c is non-renormalizable if andonly if ξ determines points in Jc dynamically.

A connected set J is called locally connected if for every point x ∈ J and neighbor-hood V about x, there is a neighborhood U ⊂ V about x such that U ∩ J is connected.Theorem F implies that

Corollary A (Yoccoz). The Julia set of a non-renormalizable quadratic polynomialis locally connected.

The reader may refer to [Hu,Mi2] for Yoccoz’ results. (One can also find proofs ofTheorem F and Corollary A in [Ji1, pp.206-219]).

Remark 1. From Theorem D, every quadratic-like map f : U → V whose Julia set isconnected is hybrid equivalent to a unique quadratic polynomial, all arguments in thissection can be applied to a quadratic-like map (by considering induced external raysand equipotential curves from the hybrid equivalent quadratic polynomial).

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§4. Construction of a three-dimensionalpartition following renormalization.

§4.1. Infinite renormalization.A quadratic-like map f : U → V with connected Julia set Jf is said to be twice

renormalizable if(1) it is once renormalizable and(2) a renormalization f1 = fn

′: U ′ → V ′ is again once renormalizable.

Consequently, we have renormalizations

f1 = fn′: U ′ → V ′ and f2 = fn

′′n′1 : U ′′ → V ′′

and their Julia sets Jf1 and Jf2 .Similarly, we can definite a l-times renormalizable quadratic-like map (f, U, V ) and

renormalizations (assume f0 = f , k0 = n′ and k1 = n′′)

fi = fki−1i−1 : Ui → Vil

i=1

with Julia sets Jfili=1. A quadratic-like map f : U → V is infinitely renormalizable if

it is l-times renormalizable for every l > 0.Suppose f : U → V is a renormalizable quadratic-like map. Let f1 = fn

′: U ′ →

V ′ be a renormalization whose the Julia set J1 = Jf1 . Suppose its two fixed pointsα1 and β1 are repelling. Then one fixed point β1 does not disconnect J1, i.e., J1 \β1 is still connected, and the other fixed point α1 disconnects J1, i.e., J1 \ α1 isdisconnected. Similarly α1 (resp. β1) is called the separating (non-separating) fixedpoint. McMullen [Mc1] discovered that different types of renormalizations can occur inthe renormalization theory of quadratic-like maps. Let J(i) = fi(J1), 1 ≤ i < n′. Therenormalization f1 is said to be

(1) α-type if J(i) ∩ J(j) = α1 for some i 6= j, 0 ≤ i, j < n′;(2) β-type if J(i) ∩ J(j) = β1 for some i 6= j, 0 ≤ i, j < n′;(3) disjoint type if J(i) ∩ J(j) = ∅ for all i 6= j, 0 ≤ i, j < n′.

The β-type and the disjoint type are also called simple. If f : U → V is infinitelyrenormalizable, i.e., it has an infinite sequence of renormalizations, then it will havean infinite sequence of simple renormalizations [Mc1]. In this subsection, we will con-struct a most natural sequence of simple renormalizations for an infinitely renormalizablequadratic-like map. Before this, we first prove the following modulus inequality (referto [Ji2] and see Fig. 2). A Riemann surface A with π1(A) ∼= Z is called an annulus.It is conformally isomorphic to C∗ = C \ 0, ∆∗ = D1 \ 0, or the round annulusAr = z ∈ C | 1 < |z| < r for some 1 < r < ∞. In case A is isomorphic to Ar, themodulus of A is defined by

mod(A) =log r

2π.

In the other two cases, mod(A) = ∞.9

Theorem 1 (Modulus Inequality). Suppose f : U → V is a renormalizable quadratic-like map and 0 is not periodic. For any n′-renormalization

f1 = fn′: U ′ → V ′, n′ ≥ 2,

we havemod(U \ U ′) ≥ 1

2mod(V \ U).

Fig. 2: Modulus inequality in renormalization.

Proof. Since f1 = fn′: U ′ → V ′ is quadratic-like, it has only critical point 0. So the

first critical value c1 = f(0) of f is not in V ′. (Otherwise, there will be a point x 6= 0 inU ′ such that f(n

′−1)(x) = 0, since c1 has only one preimage 0 under f and 0 is not aperiodic point of f . Then x will be a critical point of f1.) Since V ′ is simply connected,f has two analytic inverse branches;

g0 : V ′ → g0(V ′) ⊂ U and g1 : V ′ → g1(V ′) ⊂ U.

One of them is f(n′−1)(U ′). Therefore, f(U ′) is a domain inside U and containing c1.

Consider the annuli U \ U ′ and V \ f(U ′). Then

f : U \ U ′ → V \ f(U ′)

is a degree two holomorphic covering map. This implies that

mod(U \ U ′) =12mod(V \ f(U ′)).

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Since f(U ′) is a subset of U , the annulus V \U is a sub-annulus of the annulus V \f(U ′),i.e., V \ U ⊆ V \ f(U ′). Thus we have that

mod(U \ U ′) ≥ 12mod(V \ U).

¤We will use N(X, ε) = x ∈ C | d(x,X) < ε to denote the ε-neighborhood of X

in the complex plane. Consider a renormalizable quadratic-like map f : U → V whoseJulia set Jf is connected. Let ξ = ηn∞n=0 be the Yoccoz partitions about Jf . Let

0 ∈ · · · ⊆ Cn ⊆ Cn−1 ⊆ · · · ⊆ C2 ⊆ C1 ⊆ C0

be the critical end. DenoteJ1 = ∩∞i=0Ci.

We have two integers n1 ≥ 0, k1 > 1 such that

f1 = fk1 : Ck1+n1 → Cn1

is a degree two branched covering map and such that Cn1 ⊂ N(J1, 1). We can furthertake domains Ck1+n1 ⊆ U1 ⊂ U and Cn1 ⊆ V1 ⊂ V such that

f1 = fk1 : U1 → V1

is a quadratic-like map whose Julia set is J1. (Note that f1 = fk1 : U1 → V1 is a simplerenormalization of f : U → V .)

Theorem 2. Suppose g = fk1 : U ′ → V ′ is any k1-renormalization of f : U → V .Then its Julia set is always J1.

Proof. The point 0 is in the intersection U ′∩U1. Suppose U ′′ is the connected componentof U ′ ∩ U1 containing 0. Then h = fk1 : U ′′ → V ′′ ⊂ V ′ ∩ V1 is also a renormalizationof f and its Julia set Jh is connected. It is easy to check that Jh ⊆ Jg ∩ Jf1 . Let βf1

and αf1 be the non-separating and separating fixed points of f1 and let βg and αg bethe non-separating and separating fixed points of g. All βf1 , αf1 , βg, and αg are in Jh

since each of g, f1 and h has exactly two fixed points in its domain, and every fixedpoint of h has to be a common fixed point of g and f1. Since f1 = fk1 : U1 → V1 andg = fk1 : U ′ → V ′ are both degree two branched covering maps and βf1 is in bothU ′ and U1, all preimages of βf1 under iterates of g and f1 are in both U ′ and U1. Buteach of Julia sets Jh, J1, and Jg is the closure of the set of all preimages of βf1 underiterates of h, g, and f1. Therefore, Jh = Jg = J1. ¤Remark 2. From Theorem 2, for any k1-renormalization

f1 = fk1 : U ′ → V ′

of f : U → V , there is a critical piece Ck1+n ⊂ U ′ such that

f1 = fk1 : Ck1+n → Cn ⊂ V ′

is a degree two branched covering map.11

§4.2. Three-dimensional partition.Take an infinitely renormalizable quadratic polynomial qc(z) = z2 + c. Let U and V

be domains bounded by equipotential curves such that f0 = qc : U → V is a quadratic-like map. Let α0 = α, β0 = β, η0

n = ηn, ξ0 = ξ, C0n = Cn, Λ0

n = Λn, and Λ0∞ = Λ∞ be

the same as before. Let n1 ≥ 0 and k1 ≥ 2 be two integers such that

f1 = fk10 : C0

k1+n1→ C0

n1⊂ N(J1, 1) where J1 = ∩∞i=0C

0i .

Suppose β1 and α1 are the non-separating and separating fixed points of f1, i.e., J1\β1is still connected and J1 \ α1 is not. The points β1 and α1 are also repelling periodicpoints of qc. There are at least two, but a finite number, external rays of qc landingat α1. Let Λ1

0 be the union of external rays landing at α1. Then Λ10 cuts V 1

0 = C0n1

into a finite number of closed domains. Let η10 be the collection of these domains. Let

Λ1n = f−n

1 (Λ10) for any n > 0. Then Λ1

n cuts V 1n = f−n

1 (V 10 ) into a finite number of

closed domains. Let η1n be the collection of these domains. The sequence ξ1 = η1

n∞n=0

is a sequence of nested partitions about J1. We call it the first partition. (We also callξ0 the 0th partition.) Let Λ1

∞ = ∪∞n=0Λ1n.

The domain C1n ∈ η1

n containing 0 is called the critical piece. It is clear the restrictionf1 to C1

n is degree two branched covering map but to all other domains in η1n are degree

one. LetJ2 = ∩∞n=0C

1n.

There are two integers n2 ≥ 0 and k2 ≥ 2 such that

f2 = fk21 : C1

n2+k2→ C1

n2

is a degree two branched cover map and such that C1n2⊂ N(J2, 1/2).

Inductively, for every i ≥ 2, suppose we have already constructed

fi = fkii−1 : Ci−1

ni+ki→ Ci−1

ni.

Let βi and αi be the non-separating and separating fixed points of fi; i.e., Ji\βi is stillconnected and Ji \ αi is not. The points βi and αi are also repelling periodic pointsof qc. There are at least two, but a finite number, external rays of qc landing at αi. LetΛi

0 be the union of external rays landing at αi. Then Λi0 cuts V i

0 = Ci−1ni

into finitelymany closed domains. Let ηi

0 be the collection of these domains. Let Λin = f−n

i (Λi0) for

any n > 0. Then Λin cuts V i

n = f−ni (V i

0 ) into finitely many closed domains. Let ηin be

the collection of these domains. The domain Cin in ηi

n containing 0 is called the criticalpiece in ηi

n. It is clear that fi restricted to all domains but Cin are bijective and fi|Ci

n

is a degree two branched covering map. Let

Ji+1 = ∩∞n=0Cin.

12

There are two integers ni+1 ≥ 0, ki+1 ≥ 2 such that

fi+1 = fki+1i : Ci

ni+1+ki+1→ Ci

ni+1

is a degree two branched cover map and such that Cini+1

⊂ N(Ji+1, 1/(i + 1)). Letξi = ηi

n∞n=0. It is the sequence of nested partitions about Ji. We call it the ith

partition.

Remark 3. For any ki+1-renormalization fi+1 = fki+1i : U ′ → V ′ of fi : Ui → Vi, we

have an integer n > 0 such that Cin ⊂ V ′ ∩ N(Ji+1, 1/(i + 1)) and fi+1 = f

ki+1i :

Cin+ki+1

→ Cin is a degree two branched covering map. We will still use ξi to mean

ξi ∩ Cin+ki+1

. Therefore, (Ui+1, Vi+1) can be an arbitrary domains such that fi+1 =

fki+1i : Ui+1 → Vi+1 is a ki+1-renormalization of fi : Ui → Vi.

Let mi =∏i

j=1 ki, 1 ≤ i < ∞. We have thus constructed a most natural infinitesequence of simple renormalizations,

fi = qmic : Ui → Vi∞i=1,

and the nested-nested sequence ξi∞i=0 of partitions about Ji∞i=0 (where J0 = Jc).We call Ξ = ξi∞i=0 the first three-dimensional partition about Jc.

Definition 3. We say that the first three-dimensional partition Ξ determines pointsin X ⊆ Jc dynamically if for any z ∈ X, there are integers n, l, i ≥ 0 such that forx = qlc (z) and any z′ 6= x ∈ Jc, there is a domains D ∈ ηi

n such that x ∈ D but z′ 6∈ D′.

Let Orb(z) = qnc (z)∞n=0 be the orbit of z and O(z) be the closure of Orb(z). Let

Γ = z | O(z) ∩ Ji = ∅ for some i ≥ 1

Theorem 3. The first three-dimensional partition Ξ determines points in Γ dynami-cally from all other points in Jc.

Proof. Suppose z ∈ Γ. Take n0 = 0, k0 = 1, and C−1n0

= U and C−1n0

= V . We haveCj−1

njfor all j ≥ 1. There is the smallest integer i ≥ 0 such that O(z) ∩ Ci−1

ni6= ∅ and

O(z) ∩ Ji+1 = ∅. Take l ≥ 0 the smallest integer such that x = qlc (z) ∈ Ci−1ni

andconsider the ith partition ξi = ηi

n∞n=0 inside Ci−1ni

. Since Ji+1 = ∩∞n=0Cin, there is an

integer N ≥ 0 such that O(x) ∩ CiN = ∅. Consider

ηiN = Ci

N , BiN,1, B

iN,2, . . . , Bi

N,l

in the ith partition ξi. Assume that BiN,1 contains fi(0). (Note that fi(0) is an interior

point of BiN,1.) Then fi(O(x)) is disjoint with Bi

N,1 since O(x) is disjoint with CiN+1 ⊆

13

CiN and fi(Ci

N+1) = BiN,1. Suppose x ∈ D is a domain in ηi

N+1 ∈ ξi. Suppose D ⊆ BiN,j

and fi(D) = BN,j′ for 2 ≤ j, j′ ≤ r. Then fi : D → BN,j′ has the inverse

gjj′ : BiN,j′ → D.

Therefore for each pair (BiN,j , B

iN,j), 2 ≤ j, j′ ≤ r, satisfying gjj′(BN,j′) ⊆ BN,j ,

we can thicken BN,j′ and BN,j to simply connected domains BN,j′,jj′ and BN,j,jj′

such that BN,j′ ⊂ BN,j,jj′ and BN,j ⊂ BN,j,jj′ and such that gjj′ can be extendedto a schlicht function from BN,j,jj′ into BN,j,jj′ . We still use gjj′ to denote this ex-tended function. Now consider BN,j,jj′ and BN,j,jj′ as hyperbolic Riemann surfaceswith hyperbolic distances dH,j,jj′ and dH,j,jj′ . Then gjj′ on BN,j′ strictly contractsthese hyperbolic distances; more precisely, there is a constant 0 < λjj′ < 1 suchthat dH,j,jj′(gjj′(x), gjj′(y)) < λjj′dH,j,jj′(x, y) for x and y in BN,j′ . Since thereare only a finite number of such pairs, we have a constant 0 < λ < 1 such thatdH,j′,jj′(gjj′(x), gjj′(y)) < λdH,j,jj′(x, y) for all such pairs. Let

x ∈ · · · ⊆ Din(x) ⊆ Di

n−1(x) ⊆ · · · ⊆ Di1(x) ⊆ Di

0(x)

be any x-end in the ith partition ξi, where Din(x) ∈ ηi

n. Then fmi (Din(x)) for n−m > N

is in BiN,j for some 2 ≤ j ≤ q. Therefore, there is a constant C > 0 such that for any

Din(x) and for any n > N ,

d(Din(x)) = max

y,z∈Din(x)

|y − z| ≤ Cλn−N .

Thus d(Din(x)) tends to zero as n goes to infinity. Thus for any z′ 6= x ∈ Jc, we have

an integer n > 0 such that Dn(x) does not contain z′. ¤Definition 4. An infinitely renormalizable quadratic polynomial qc(z) = z2 + c is saidto have complex bounds if there are a constant λ > 0 and an infinite sequence of simplerenormalizations fis = q

misc : Uis → Vis∞s=1 such that the modulus mod(Vis \ U is) is

greater than λ for every s ≥ 1.

Theorem 4. Suppose qc(z) = z2+c is an infinitely renormalizable quadratic polynomialhaving complex bounds. Then the first three-dimensional partition Ξ determinants 0dynamically from other points.

Proof. To make notations simple, we assume

fi = qmic : Ui → Vi∞i=1

is the infinite sequence of simple renormalizations in Definition 4. Let λ > 0 be theconstant in Definition 4. Then Ui∞i=1 is a sequence of nested domains containing 0.From Theorem 1 (the modulus inequality), we have

mod(Ui \ U i+1) ≥ 12mod(Vi \ U i) >

λ

2.

14

Let Ai = Ui \ U i+1 for i ≥ 1 and X = ∩∞i=1Ui. Since U1 \X = ∪∞i=1Ai,

mod(U1 \X) ≥∞∑

i=1

mod(Ai) = ∞.

Thus, X = 0 (following the Grotzsch argument and refer to [BH1,BH2] and [Mi2, pp.11]). This implies that the diameter d(Ui) tends to 0 as i goes to infinity.

Consider the ith partition ξi = ηin∞n=0 for every i ≥ 0. Remember that Ci

n is themember in ηi

n containing 0. Consider the corresponding critical end

0 ∈ · · · ⊆ Cin ⊆ Ci

n−1 ⊆ · · · ⊆ Ci1 ⊆ Ci

0

in ξi. Since Ji+1 = ∩∞j=0Cij is the Julia set of fi+1 : Ui+1 → Vi+1, there is a Ci

l(i)

contained in Ui+1. The diameter d(Cil(i)) of Ci

l(i) tends to zero as i goes to infinity.Thus for any z 6= 0 ∈ Jc, we can find Ci

l(i) does not containing z. Since 0 is aninterior point of Ci

l(i), we have an integer n > l(i) and a domain z ∈ D ∈ ηin such that

D ∩ Cin = ∅. ¤

Following Theorem 4, if qc has complex bounds, then ∩∞i=1Ji = 0 since Ji ⊆ Cil(i).

Now we construct our second three-dimensional partition Υ about Jc. Denote by κ1

the first partition which will be constructed as follows: Consider ξ0 = η0n∞n=0 in Ξ.

Take C0k(0) ∈ η0

k(0) ∈ ξ0 where k(0) = n1 + k1. Put all domains in η0k(0)+1 which are

the preimages of C0k(0) under qc into κ1 and let η0c

k(0)+1 be the rest of the domains.Consider η0

k(0)+2 ∩ η0ck(0)+1 consisting of all domains in η0

k(0)+2 which are subdomains ofthe domains in η0c

k(0)+1. Put all domains in η0k(0)+2 ∩ η0c

k(0)+1 which are the preimages ofC0

k(0) under q2c into κ1 and let η0ck(0)+2 be the rest of the domains. Suppose we already

have η0ck(0)+s for s ≥ 2. Consider η0

k(0)+s+1∩η0ck0+s consisting of all domains in η0

k(0)+s+1

which are subdomains of the domains in η0ck(0)+s. Put all domains in η0

k(0)+s+1 ∩ η0ck(0)+s

which are the preimages of C0k(0) under q

(s+1)c into κ1 and let η0c

k(0)+s+1 be the restof the domains. Thus we can construct the partition κ1 inductively. This partitioncovers points in Jc minus all points not entering the interior of C0

k(0) under all forwarditerations of qc.

Next consider the ξ1 = η1n∞n=0 in Ξ. Take C1

k(1) ∈ η1k(1) ∈ ξ1 where k(1) = n2 + k2.

We can use similar arguments to those in the previous paragraph by considering f1 :C0

k(0) → C0k(0)−k1

(to replacing qc : U → V ) to get a partition κ1,1 in C0k(0). Then we

use all iterations of qc to pull back this partition following κ1 to get a partition κ2. It isa sub-partition of κ1 and covers points in Jc minus all points not entering the interiorof C1

k(1) under iterations of qc.Suppose we have already constructed the (j−1)th partition κj−1 for j ≥ 2. Consider

the partition ξj = ηjn∞n=0 in Ξ. Take Cj

k(j) ∈ ηjk(j) ∈ ξj where k(j) = nj+1 + kj+1.

15

Similarly, by considering fj : Cj−1k(j−1) → Cj−1

k(j−1)−kj, we get a partition κj,1 in Cj−1

k(j−1).Then we use all backward iterations of fj−1 to pull back this partition following κj−1

to get a partition κj,2 in Cj−2k(j−2) and all backward iterations of fj−2 to pull back this

partition following κj−1 to get a partition κj,3 in Cj−3k(j−3), and so on to obtain a partition

κj = κj,j in V . It is a sub-partition of κj−1 and covers points in the Julia set minus allpoints not entering the interior of Cj

k(j) under forward iterations of qc. By induction,we have a sequence of nested partitions

Υ = κj∞j=1

which covers points in Jc \ Γ. We call Υ = κj∞j=1 the second three-dimensionalpartition about Jc.

Definition 5. We say that the second three-dimensional partition Υ determines pointsin X ⊆ Jc dynamically if for any x ∈ X and x′ 6= x ∈ Jc, there is an integer i ≥ 1 anda domain x ∈ D ∈ κi ∈ Υ such that x′ 6∈ D.

Definition 6. We say an infinitely renormalizable quadratic polynomial qc(z) = z2 + cis unbranched if there are Jil

∞l=1, neighborhoods Wl of Jil, and a constant µ > 0 such

that the modulus mod(Wl \ Jil) is greater than µ and Wl \ Jil

contains no point in thecritical orbit CO of qc.

Theorem 5. Suppose qc(z) = z2 + c is an infinitely renormalizable quadratic poly-nomial. If ∩∞i=1Ji = 0 and if qc is unbranched, then the second three-dimensionalpartition Υ determines points in Jc \ Γ dynamically.

Proof. Consider the ith partition ξi = ηin∞n=0 for every i ≥ 0 and the corresponding

critical end0 ∈ · · · ⊆ Ci

n ⊆ Cin−1 ⊆ · · · ⊆ Ci

1 ⊆ Ci0.

Consider fi+1 = fki+1i : Ci

ni+1+ki+1→ Ci

ni+1. Let k(i) = ni+1+ki+1. Since ∩∞i=1Ji = 0

and Cik(i) ⊂ N(Ji+1, 1/(i + 1)),

∩∞i=0Cik(i) = 0.

Let µ > 0 and Wl∞l=1 be a constant and domains satisfying Definition 6. Withoutloss of generality, we assume in Definition 6 that l = i and il = i + 1. Since

mod(Wi \ Ji+1) ≥ µ and Ji+1 = ∩∞n=0Cin,

by choosing k(i) large enough, we can assume

mod(Wi \ Cik(i)) ≥

µ

2.

16

for all i ≥ 1. Also, by modifying Wi, we can assume the diameter diam(Wi) tends tozero as i goes to ∞.

Suppose x 6= 0 in Jc \ Γ. Then the orbit Orb(x) = qnc (x)∞n=0 enters every Cik(i)

infinitely many times. Consider the second three-dimensional partition Υ = κj∞j=1

and the x-end in this partition,

x ∈ · · · ⊆ Dj(x) ⊆ Dj−1(x) ⊆ · · · ⊆ D1(x),

where Dj(x) ∈ κj . Let ij(x) ≥ 0 be the unique integer such that

qij(x)c : Dj+1(x) → Cj

k(j)

is a proper holomorphic diffeomorphism (see Fig. 3). Let

gj,x : Cjk(j) → Dj+1(x)

be its inverse. Since there is no critical values ci = qi(0)∞i=1 in Wj \Cjk(j), gj,x can be

extended to a proper holomorphic diffeomorphism on Wj which we still denote as gj,x.For each i ≥ 1, since the diameter d(Wj) tends to 0 as j goes to ∞, we can find

an integer j = j(i) > i such that Wj ⊂ Cik(i). Let xi = q

ii(x)c (x) ∈ Ci

k(i) and xj =

qij(x)c (x) = q

ij(xi)c (xi) ∈ Cj

k(j).

Consider the xi-end in the second three-dimensional partition Υ,

xi ∈ · · · ⊆ Dl(xi) ⊆ Dl−1(xi) ⊆ · · · ⊆ D1(xi),

where Dl(xi) ∈ κl. Then qij+1−ii(x)(xi)c : Dj+1−ii(x)(xi) → Cj

k(j) is a proper holomor-phic diffeomorphism (see Fig. 3). Let gij be its inverse. Then gij can be extended to Wj

because of the unbranched condition. We still use gij to denote this extension. SinceCi

k(i) is bounded by external rays landing at some pre-images of αi under iterations ofqc and by equipotential curves of qc, Wij = gij(Wj) is contained in Ci

k(i). Thus

mod(Wi \Wij) ≥ µ

2.

Consider Xi = gi,x(Wi) and Xj = gj,x(Wj) = gi,x(Wij). Then

mod(Xi \Xj) ≥ µ

2

since gi,x is conformal.17

Fig. 3: The second three-dimensional partition. Here q = qc.

Therefore, inductively, we can find an infinite sequence of nested domains Xit∞t=1

such thatmod(Xit \Xit+1) ≥

µ

2for t ≥ 1. Let X = ∩∞t=1Xit . Then Xi1 = ∪∞t=1(Xit \Xit+1) and

mod(Xi1 \X) ≥∞∑

t=1

mod(Xit \Xit+1) = ∞.

Thus, X = x (following the Grotzsch argument and refer to [BH1,BH2] and [Mi2, pp.11]). This implies that the diameter of Xit tends to zero as t goes to infinity. SinceDit+1(x) = gi,x(Cit

k(it)), we have that

Dit+1(x) ⊆ Xit .

So the diameter d(Dit+1(x)) tends to zero as t goes to infinity. Thus ∩∞i=0Di(x) = x.This implies that the second three-dimensional partition Υ determines points in Jc \ Γdynamically from other points in Jc. ¤

Our three-dimensional partition Σ is constructed from Ξ and Υ as follows. Firstconsider the second three-dimensional partition Υ = κj∞j=1. For each domain D ∈ κj ,

18

there is a unique integer s ≥ 0 such that qsc : D → Cjk(j) is diffeomorphic. Now consider

ξj+1 ∈ Ξ. Thenξj+1(C

jk(j)) = ξj+1 ∩ Cj

k(j)

is a partition insider Cjk(j). Pull back ξj+1(C

jk(j)) by qsc : D → Cj

k(j) to D, we get apartition inside D, which we denote as ξj+1(D). Let

σj = tD∈κj ξj+1(D).

It is a partition about Jc. Furthermore, σj+1 is a sub-partition of σj . We call

Σ = σj∞j=1

the three-dimensional partition about Jc.

Definition 7. We say that the three-dimensional partition Σ determines points in Jc

dynamically if for any z 6= z′ ∈ Jc, there is an integer j ≥ 1 and domains z ∈ D, z′ ∈D′ ∈ σj ∈ Σ such that D ∩D′ = ∅.

Combining Theorems 3 and 5, we have that

Theorem 6. Suppose qc(z) = z2+c is a unbranched infinitely renormalizable quadraticpolynomial having complex bounds. Then the three-dimensional partition Σ determinespoints in Jc dynamically.

Proof. If one of z and z′ is in Jc \ Γ, it follows directly from the proofs of Theorems 4and 5. If z ∈ Γ, from the proof of Theorem 3, there are the smallest integer l ≥ 0 andthe x = qlc (z)-end

x ∈ · · · ⊆ Di+1n (x) ⊆ Di+1

n−1(x) ⊆ · · · ⊆ Di+11 (x) ⊆ Di+1

0 (x)

in the (i + 1)th partition ξi+1 ∈ Ξ such that the diameter d(Di+1n (x)) tends to zero

as n goes to infinity. Take z ∈ D ∈ κi. There is a smallest integer s ≥ 0 such thatqs : D → Ci

k(i) is diffeomorphic. Therefore, there is a z-end in ξi+1(D) ∈ Σ,

z ∈ · · · ⊆ Di+1n (z) ⊆ Di+1

n−1(z) ⊆ · · · ⊆ Di+11 (z) ⊆ Di+1

0 (z),

such that the diameter d(Di+1n (z)) tends to zero as n goes to infinity. This implies that

if one of z and z′ in Γ, then the theorem still holds. ¤§4.3 Local connectivity.

The three-dimensional partition Σ for an infinitely renormalizable quadratic poly-nomial qc(z) = z2 + c provides a nice dynamical description of points in the Juliaset Jc. One can expect that it is useful in the study of topological, geometrical, andmeasure-theoretical properties of qc. In this subsection, we show one application.

19

Lemma 1. For any domain D in Σ, D ∩ Jc is connected.

Proof. Since the domain D is bounded by finitely many external rays Π = eθjm

j=1

and some equipotential curve for qc, then ∂D ∩ Jc consists of a finite number of pointspim′

j=1. Every pi is a landing point of two external rays in Π. Suppose D ∩ Jc isnot connected for some D. Then there are two disjoint open sets X and Y such thatD∩Jc = (D∩Jc∩X)∪(D∩Jc∩Y ). Suppose that p1, . . . , pm′′ are in X and that pm′′+1,. . . , pm′ are in Y . The two external rays in Π landing at pj cut C into two domains.One of them, denoted as Zj , is disjoint with D, i.e., D∩Zj = ∅. Then U ′ = U ∪∪m′′

j=1Zj

and V ′ = V ∪ ∪m′j=m′′+1Zj are two disjoint open sets, and Jc = (U ′ ∩ Jc) ∪ (V ′ ∩ Jc).

This contradicts the fact that Jc is connected. ¤

Corollary B (Jiang [Ji2]). If qc(z) = z2 + c is a unbranched infinitely renormalizablequadratic polynomial having complex bounds, then its Julia set Jc is locally connected.

Proof. . Take a point z ∈ Jc. From Theorem 6, we have a sequence of domainsDn(z) from the three-dimensional partition Σ such that z ∈ Dn(z) ⊆ Dn−1(z) and thediameter d(Dn(z)) tends to 0 as n goes to infinity. This fact with Lemma 1 implies thatDn(z)∞n=0 is a basis of connected neighborhoods about z. So Jc is locally connectedat z. ¤

Corollary B indicates that unbranched and complex bounds conditions are importantin the study of local connectivity of the Julia set of an infinitely renormalizable quadraticpolynomial. A real infinitely renormalizable quadratic polynomial is always unbranchedif it has complex bounds. Many people have worked out some results about complexbounds for real infinitely renormalizable quadratic polynomials. The Feigenbaum poly-nomial is a real infinitely renormalizable quadratic polynomial q∞(z) = z2 + t∞, t∞real, such that

fi(z) = q2i

∞ (z) : Ui → Vi

is a sequence of simple renormalizations. Sullivan [Su] (see also [MS,Ji1]) proved thatq∞ has the complex bounds. In the study of Sullivan’s result, we concluded from [Ji2]that

Corollary C (Hu-Jiang [HJ]). The Julia set of the Feigenbaum polynomial is locallyconnected.

Furthermore, Levins and van Strien, and Lyubich and Yampolsky showed that anyreal infinitely renormalizable quadratic polynomial has complex bounds. Therefore,

Corollary D (Levins-van Strien [LS] and Lyubich-Yampolsky [LY). A realinfinitely renormalizable quadratic polynomial has complex bounds. Therefore, its Juliaset is locally connected.

20

§5. A partition in the parameter space.

In this section, we use regular letters to denote sets in the dynamical plane (z-plane)and curly letters to denote sets in the parameter plane (c-plane).

§5.1 Some basic facts about the Mandelbrot set from Douady-Hubbard.The following basic facts about the Mandelbrot set is mostly from [DH1,HD2,HD3].

The reader may also refer to [Mi2,Hu,Ro,Sc].Let f(z) = az2 + bz +d be a quadratic polynomial. Conjugating f by an appropriate

linear map h(z) = ez + s, we get h f h−1(z) = z2 + c. We call qc(z) = z2 + c theDouady-Hubbard family of quadratic polynomials.

The Mandelbrot set M is the compact set of all parameters c in C such that the Juliaset Jc of qc is connected. From Theorem B, M consists of all parameters c such that0 has bounded orbit. We will often identity c with the corresponding polynomial qc. Apoint c ∈M (really means qc) is called hyperbolic if and only if it has a unique attractingperiodic orbit. Let z0, · · · , zp−1 be the attracting periodic orbit for a hyperbolic c andlet λp(c) = qpc (z0). The hyperbolic maps form an open subset in C. When c changes inone connected component H of this open set, the period p and the combinatorial typeof the attracting periodic orbit are fixed. Here p is called the period of H. Moreover,λH(c) maps H to D1 = z ∈ C | |z| < 1 conformally and can be extends uniquely to ahomeomorphism

λH(c) : H → D1.

Note that H has a unique center cH = λ−1H (0). The point rH = λ−1

H (1) is called theroot of H. For example, the cardioid H0 is the hyperbolic component of c such that qc

has an attracting fixed point. Then λH0(c) = 1−√1− 4c. The center is 0, the root is1/4, and λ−1

H0(e2πiθ) = e2πiθ(2− e2πiθ)/4.

Consider a hyperbolic component H of period p ≥ 1 and the corresponding λH(c).For each r ∈ ∂H such that λH(r) = e

2πi mp′ , (m, p′) = 1, it is the root of a hyperbolic

component H′ of period pp′. (Note that in the case m = 0 and p′ = 1, H′ = H, and inall other cases, H′ 6= H.) Here H′ is called the satellite of H with internal angle m/p′

and denoted as H′ = H ∗H(m/p′).For each c ∈ C, let

hc : Uc = z ∈ C | Gc(z) > Gc(0) → z ∈ C | |z| > Gc(0)be the Bottcher change of coordinate from §2, where Gc is the Green function for Kc.Then

hc qc h−1c (z) = z2.

and Gc = log |hc|. For c ∈ C \M, c ∈ Uc and thus we have hc(c).

Theorem G (Douady-Hubbard [DH3]). The map

ΦM(c) = hc(c) : C \M → C \D1

21

is a conformal map. Thus M is connected.

The equipotential curve of radius r > 1 of M is

Sr = Φ−1M (c ∈ C | |c| = r)

and the external ray of angle 0 ≤ θ < 1 of M is

Eθ = Φ−1M (c ∈ C | |c| > 1 and arg(c) = 2πθ).

For example, E0 = (1/4,∞) and E1/2 = (−∞,−2). An external ray Eθ is said to landat M if it has only one limiting point at M. Both of E0 and E1/2 land at M (E0 landsat 0 and E1/2 lands at −2). Furthermore,

(1) every external ray Eθ of rational angle θ = m/p lands at a point in M.(2) If θ = m/p with (m, p) = 1 and p = 2p′, then Eθ lands at a point c ∈ M such

that the critical orbit CO of qc is preperiodic (such a c is called a Misiurewiczpoint). Conversely, every Misiurewicz point is a landing point of an external rayEθ of angle θ = m/p with (m, p) = 1 and p = 2p′.

(3) For θ = m/p with (m, p) = 1 and p = 2p′, let c be the landing (Misiurewicz)point of Eθ, the external ray eθ in the z-plane for qc lands at its critical valueof c. (Tan Lei [Ta] showed some similarity between M around a Misiurewicz cand corresponding Jc around c.)

(4) Let H be a hyperbolic component of period p > 1, there are exactly two externalrays Eθ− and Eθ+ land at the root rH where θ− = m−/(2p − 1) and θ+ =m+/(2p − 1) for 1 ≤ m− < m+ < 2p − 1.

Consider the cardioid H0 and a satellite H(m/p) = H ∗ H0(m/p) of angle m/p,(m, p) = 1. Let Eθ− and Eθ+ be two external rays landing at the root rH(m/p). Thenthe closure of Eθ− ∪ E+

θ cuts C into two connected components. Let Wm/p be the onecontaining H(m/p).

Theorem H (Goldberg-Milnor [GM]). For any c ∈ Wm/p, consider qc. There areexactly p external rays in the z-plane that land at the separating fixed point αc of qc.These p external rays have angles 2iθ+, i = 0, · · · , p− 1. Moreover, c is in the domainbounded by two external rays of angle θ− and θ+.

The main conjecture in this direction is that the Mandelbrot set M is locally con-nected. From the Caratheodory theorem, if M is locally connected, then ΦM can beextended to a homeomorphism from C \ M to C \D1, thus every external ray lands ata unique point in M. Yoccoz proved that M is locally connected at those points c suchthat qc are not infinitely renormalizable. Therefore, the study of the above conjecture isconcentrated at all infinitely renormalizable points. Consider a monic polynomial P (z)of degree d whose Julia set is connected. Let p be a repelling fixed point of P and letλ = P ′(p). Suppose there are totally m′ external rays of P landing at p. Label these

22

external rays in cyclical order. Suppose the ith external ray is mapped to the (i+k′)th(mod m′) external ray. Let r = gcd(m′, k′). Then there are r cycles of external rayslanding at p. Let m′ = rm and k′ = rk, (m, k) = 1. The Yoccoz inequality says thatthere is a branch τ of log λ satisfying

<τ

|τ − 2πi km |2

≥ rm

2 log d.

In other words, τ belongs to the closed disc of radius (log d)/(rm) tangent to the imagi-nary axis at 2πik/m. Using this inequality, Yoccoz showed that M is locally connectedat every point c such that qc has an irrational indifferent periodic point (refer to [Hu]).Furthermore, by applying the Yoccoz inequality, Douady-Hubbard constructed a genericsubset of infinitely renormalizable quadratic polynomials whose Julia set is non-locallyconnected but M is locally connected at every point in this subset. They used a methodcalled tuning in this construction. The tuning can be described roughly as follows. Con-sider a hyperbolic component H of period p. There are two external rays Eθ−(H) andEθ+(H) land at its root rH. The set Eθ−(H) ∪ Eθ+(H) ∪ rH cuts C into two domains.Denote the one containing H as W(H). Then W(H) contains a small copy M(H) of theMandelbrot set M such that H is the image of the cardioid H0. For every c ∈ M(H),qc is renormalizable and, more precisely, there is a simple renormalization

qm(H)c : U → V.

Let H′ be a satellite of H of period p′. We can similarly constructW(H′) andM(H′) forH′ but we will denote them as W(H∗H′) and M(H∗H′). (Without having confusion,we also denote them as W(p ∗ p′) and M(p ∗ p′)). For every c ∈ W(p ∗ p′),

qm(p)c : U → V

is renormalizable and has a simple renormalization

qm(p)p′c : U ′ ⊂ U → V ′ ⊂ V.

This processing is called tuning. Starting from the cardioid and a sequence of positiveintegers, p0, p1, · · · , Douady and Hubbard constructed a sequence of nested tuning sets

Wp0∗p1∗···∗pn= W(p0 ∗ p1 ∗ · · · ∗ pn) ∩M.

Using the Yoccoz inequality, they showed that if pn tends to ∞ fast enough (i.e., pn >>

pn−1), the diameter d(Wp0∗p1∗···∗pn) tends to zero as n goes to infinity, therefore, M islocally connected at the intersection point

c ∈ ∩∞n=0Wp0∗p1∗···∗pn .

Furthermore, qc is infinitely renormalizable whose Julia set is not locally connected(see [Mi2]).

In the next subsection, we will construct another subset of M such that this subset isdense on the boundary of M and consists of unbranched infinitely renormalizable pointsc having complex bounds (therefore, the Julia set Jc of qc is locally connected accordingto Corollary B) such that M is locally connected at every point in this subset.

23

§5.2. On partitions around infinitely renormalizable points.In this section, we prove that

Theorem 7. Suppose c is a Misiurewicz point. Then there is a subset A(c) ⊂M suchthat

(1) c is a limit point of A(c),(2) for every c′ ∈ A(c), qc′(z) = z2 + c′ is an unbranched infinitely renormalizable

quadratic polynomial having complex bounds, (thus, the corresponding Julia setis locally connected according to Corollary B), and

(3) the Mandelbrot set M is locally connected at every point c′ ∈ A(c).

Since the set B of Misiurewicz points is dense on the boundary of M, we have

Corollary 1. The set A = ∪c∈BA(c) is dense on the boundary ∂M.

We will use the following well-known result in the classical complex analysis (see [La]).

Theorem I (Rouche Theorem). Suppose D is a domain in the complex plane C.Let γ ⊂ D be a closed path homologous to 0 and assume that γ has an interior. Let fand g be analytic on D, and

|f(z)− g(z)| < |f(z)|, z ∈ γ.

Then f and g have the same number of zeros in the interior of γ.

To have a better explanation to our idea, we first prove a special case of Theorem 7.

Theorem 8. There is a subset A(−2) ⊂M such that −2 is a limit point of A(−2) andA(−2) consists of unbranched infinitely renormalizable points having complex bounds atwhich M is locally connected. (Therefore, the corresponding Julia sets are all locallyconnected according to Corollary B).

Basic construction. Let q−2(z) = z2− 2. Its Julia set J2 is [−2, 2]. It has the non-separate fixed point β(−2) = 2 and the separate fixed point α(−2), i.e., J−2 \ β(−2)is still connected and J−2 \ α(2) is disconnected. Let G0,l(−2) be the closure of theunion of two external rays landing at α(−2). Let h−2 be the Botteche coordinate forq−2, i.e.,

h−2 q−2 h−1−2(z) = z2.

Then h−2 maps G0,l(−2) to two straight rays in C \ D1 (which have angles 1/3 and2/3). We use Λ0,l to denote the closure of the union of these two straight rays.

Let

LH = z = x + yi ∈ C | x < 0 and RH = z = x + yi ∈ C | x > 024

be the left and right half planes. The restriction q0|(C\(−∞, 0]) has two inverse branches

gl,0 : C \ (−∞, 0] → LH and gr,0 : C \ (−∞, 0] → RH.

Let Λn,r = gn+1r,0 (Λ0,l) and Λn+1,l = gn+1

l,0 (Λn,r) for n = 0, 1, · · · . It is easy to see thatΛn,r tends to the ray of angle 0 and Λn,l tends to the ray of angle 1/2. Let

Λ = [∪∞n=0(Λn,r ∪ Λn,l)] ∪ ((−∞,−1] ∪ [1,∞)).

Let G(−2) = h−1−2(Λ). Then it is the union of external rays for q−2 landing at the set

A(−2) = [∪∞n=0(gnr,−2(α(−2)) ∪ gn

l,−2(α(−2)))] ∪ −2, 2.

We also denote Gn,r(−2) = h−1−2(Λn,r) and Gn,l(−2) = h−1

−2(Λn,l).For c ∈ W1/2, let G0,l(c) be the closure of union of two external rays landing at its

separating fixed point α(c). Let α(c) be another preimage of α(c). Let G0,r(c) be theclosure of union of two external rays landing at α(c). Then G0,l(c) ∪ C0,r(c) cuts Cinto three domains. One we call Er(c) contains the non-separating fixed point β(c) ofqc. The other we call El(c) contains the other preimage β(c) of β(c) under qc. Therestriction qc|(El,c(c) ∪ Er,c)(c) has two inverse branches

gl,c : E → El(c) and gr,c : E → Er(c).

Let hc be the Bottcher changes of coordinate for qc(z) = z2 + c, i.e.,

hc qc h−1c = z2.

Let G(c) = h−1c (Λ). Then it is the union of external rays for qc landing at the set

A(c) = [∪∞n=0(gnr,c(α(c)) ∪ gn

l,c(α(c)))] ∪ β(c), β(c).

We also denote Gn,r(c) = h−1c (Λn,r) and Gn,l(c) = h−1

c (Λn,l).Let

φc = h−1c h−2.

When restricted to G(−2), φc conjugates q−2|G(−2) to qc|G(c). (The other way toconstruct the graph G(c) from G(−2) for c close to −2 is by using the structure stabilitytheorem. Since the graph G−2 does not contain 0, and since α(−2), β(2) and ∞ areonly periodic points of q−2 on G(−2), for a small neighborhood Ω of G(−2), q−2|Ωis a hyperbolic endomorphism. From the structural stability theorem (see [Sh,Pr]),any C1-perturbation f of q−2|Ω is topologically conjugate to q−2|Ω, i.e., there is ahomeomorphism φ : Ω → φ(Ω) (close to the identity) such that f φ = φ q−2 onq−1−2(Ω). Therefore, the graph G(−2) is preserved for qc as long as c is close to −2. Let

25

φc be the corresponding homeomorphism. Then G(c) = φc(G(−2)) is the correspondinggraph.)

Let U(−2) be a fixed domain bounded by an equipotential curve s(−2) for q−2. Thenq−2 : U(−2) → V (−2) is a quadratic-like map where V (−2) = q−2(U(−2)). The setG0,l(−2) ∪ G0,r(−2) cuts U(−2) into three disjoint domains. Denote D0(−2) be theclosure of the one containing 0 (see Fig. 4).

Fig. 4: In the figure, γ = s(−2) and Γ = G0,l(−2).

Let Dn(−2) = gnr,−2(D0(−2)) and let Bn(−2) = gnl,−2(Dn−1(−2)) for n ≥ 1 (seeFig. 4). Since 2 is an expanding fixed point of q−2 and q(−2) = 2, the diameterdiam(Bn) of Bn tends to zero exponentially as n goes to infinity. Let 0 6∈ U0 be a smallneighborhood about −2 in the parameter space such that diam(U0) ≤ 1 and such thatthe corresponding graph G(c) for qc exists for c in U0. For c in U0, let φc : G(−2) → G(c)be the conjugacy from q−2 to qc. Let sc = φc(s) be the corresponding equipotential curvefor qc. Let U(c) be the closure of the domain bounded by sc. Similarly, G0,l(c)∪G0,r(c)cuts U(c) into three disjoint domains. Denote D0(c) be the closure of the one containing0. Let Dn(c) = gnr,c(D0(c)) and let Bn(c) = gl,c(Dn−1(c)) for n ≥ 1. Note that β(c) andα(c) are the non-separate and separate fixed points of qc and β(c) is the other inverseimage of β(c) under qc. Then Bn(c) and Dn(c) tend to β(c) and β(c), respectively,as n goes to infinity. Since β(c) is an expanding fixed point of qc and since there is aconstant µ > 1 such that |q′c(β(c))| ≥ µ for all c in U0, the diameter diam(Bn(c)) tendsto 0 uniformly on U0 and the set Bn(c) approaches to β(c) uniformly on U0 as n goesto infinity. Let

Wn = c | c ∈ Bn(c).Then Wn, for n = 1, 2, · · · , give a partition in the parameter space.

Let f(c) = qc(0) − β(c) and g(c) = qc(0) − x for x ∈ Bn(c). Suppose γ = ∂U0 is aclosed path homologous to 0 in U0 such that

m = minc∈∂U0

= minc∈∂U0

|qc(0)− β(c)| > 0.

26

Because Bn(c) approaches β(c) uniformly on U0 as n goes to infinity, there is an integerN0 > 0 such that for n ≥ N0,

|f(c)− g(c)| = |β(c)− x| < m ≤ |f(c)|, c ∈ ∂U0.

Since the equation f(c) = qc(0)− β(c) = 0 has a unique solution −2 in U0, Theorem I(the Rouche Theorem) implies that g(c) = qc(0)−x = 0 has a unique solution, which isin U0, for any x in Bn(c) for n ≥ N0. Therefore Wn ⊆ U0 for n ≥ N0. So diam(Wn) ≤ 1for n ≥ N0 (see Fig. 5).

Fig. 5

Lemma 2. The intersection Mn = Wn ∩M for n ≥ N0 is connected.

Proof. The boundary Bn(c) consists of four curves Gn,l(c), Gn−1,l(c), and some piecesof the equipotential curve sn(c) = q−n

c (s(c)). Note that q(n+1)c (Gn,l(c)) = G0,l(c) and

qnc (Gn−1,r(c)) = G0,l(c). The domain Wn is bounded by the closure of two externalrays

Gn = c | c ∈ Gn,l(c), Gn−1 = c | c ∈ Gn−1(c)and some pieces of the equipotential curve

Sn = c | c ∈ sn(c).

Each of intersections Gn ∩M and Gn−1 ∩M consists only one point, denoted as an andan−1. The closure of Gn (or Gn−1) in the extended complex plane is a topological curveisomorphic to a circle and it intersects M only at an (or an−1). Let us still denote theseextended curves as Gn and Gn−1. Then Gn and Gn−1 cut C into three domains. Theone on an side is called Xn and the one on an−1 side is called Xn−1.

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If Mn is disconnected. Let U and V be two non-empty domains such that U ∩V = ∅and such that

(U ∩ Mn) ∪ (V ∩ Mn) = Mn.

Assume an ∈ U and an−1 ∈ V (other cases can be proved similarly). Let U = U ∪ Xn

and V = V ∪Xn−1. Then U and V are two non-empty domains such that U ∩ V = ∅ andsuch that

(U ∩M) ∪ (V ∩M) = M.

This would say that M is disconnected. This contradiction implies that Mn must beconnected. ¤Proof of Theorem 8. For each Wn where n ≥ N0, since c ∈ Bn(c), Cn(c) = q−1

c (Bn(c))is the closure of a connected and simply connected domain which contains 0 and whichis a sub-domain of D0(c). Let

fn,c = q(n+1)c : Cn(c) → D0(c).

Then it is a quadratic-like map (see Fig. 6). Furthermore, since diam(Bn(c)) tends tozero as n goes to infinity uniformly on U0, for N0 large enough we can have that themodulus mod(An(c)) ≥ 1 where An(c) = D0(c) \ Cn(c). Moreover, the family

fn,c : Cn(c) → D0(c) ; c ∈ Wn

is a full family, so Wn contains a copy Mn of the Mandelbrot set M (refer to [Mi2,pp. 102-106]). For c ∈ Mn, the Julia set Jfn,c of fn,c : Cn(c) → D0(c) is connected.Therefore, for c ∈ A1 = ∪∞n≥N0

Mn, qc is once renormalizable.

Fig. 6

For a fixed integer i0 ≥ N0, consider Wi0 and Mi0 ; there is a parameter ci0 ∈ Mi0

such that fi0 = fi0,ci0: Ci0 = Ci0(ci0) → Di0 = D0(ci0) is hybrid equivalent to

q−2(z) = z2 − 2. The quadratic-like map fi0 : Ci0 → Di0 has the non-separate fixed28

point and the separate fixed point, which we simple denote as β and α. Let β be anotherpre-image of β under fi0 .

Let G be the closure of the union of two external rays for qci0landing at α. Then

fi0(G) = G and G∩Jfi0= α. Let G = f−1

i0(G). Then G cuts Ci0 into three domains.

Denote Di00 be the one containing 0. Let β ∈ Ei00 and β ∈ Ei01 be the components ofthe closure of Ci0 \Di00. Let gi00 and gi01 be the inverses of fi0 |Ei00 and fi0 |Ei01. Let

Di0n = gni01(Di00) and Bi0n = gi00(Di0(n−1))

for n ≥ 1. Again by the Bottcher changes of coordinate hci0and hc (or the structure

stability theorem), we can similarly prove that β, β, α, and G, G, and ∂Ci0 are preservedfor c close to ci0 . Similar to the argument in the above, we can find a small neighborhoodUi0 about ci0 with diam(Ui0) ≤ 1/2 such that the corresponding domains Bi0(c) andDi0(c) can be constructed for qc for c ∈ Ui0 . Let

Wi0n = c ∈ C | fi0,c(0) ∈ Bi0n(c).

Then Wi0ni0≥1,n≥1 give a sub-partition of Wnn≥1.The diameter diam(Bi0n(c)) tends to zero uniformly on Ui0 and the set Bi0n(c)

approaches to β(c) uniformly on Ui0 as n goes to infinity. Suppose Ui0 is simply con-nected and the boundary curve γ = ∂Ui0 is a closed path homologous to 0 in Ui0 .Since the equation fi0,c(0)− βi0(c) = 0 has a unique solution ci0 (following Thurston’stheorem for critically finite rational maps (see [DH4]) and also refer to [DH3]), m =minc∈∂Ui0

|fi0,c(0)− βi0(c)| > 0. Let f(c) = fi0,c(0)− βi0(c) and g(c) = fi0,c(0)− x forx ∈ Bi0n(c). There is an integer Ni0 > 0 such that for n ≥ Ni0 ,

|f(c)− g(c)| = |βi0(c)− x| < m ≤ |f(c)|, c ∈ ∂Ui0 .

Theorem I (Rouche Theorem) now implies that g(c) = fi0,c(0) − x = 0 has a uniquesolution in Ui0 for any x in Bi0n(c) for n ≥ Ni0 . Therefore Wi0n ⊆ Ui0 for n ≥ Ni0 . Sodiam(Wi0n) ≤ 1/2 for n ≥ Ni0 .

Similar to the proof of Lemma 2, we have Mi0n = Wi0 ∩M is connected for n ≥ Ni0 .For each c in Wi0n, n ≥ Ni0 , let Ci0n(c) = f−1

i0,c(Bi0n(c)). Then

fi0n,c = f(n+1)i0,c : Ci0n(c) → Di00(c)

is a quadratic-like map. Furthermore, since diam(Bi0n(c)) tends to zero as n goes toinfinity uniformly on Ui0 , for Ni0 large enough we can have the modulus mod(Ai0n(c)) ≥1 where Ai0n(c) = Di00(c) \ Ci0n(c). Moreover,

fi0n,c : Ci0n(c) → Di00(c) | c ∈ Wi0n29

is a full family. ThusWi0n contains a copyMi0n of the Mandelbrot setM (refer to [Mi2,pp 102-106]). For c ∈ A2 = ∪i0≥N0 ∪i1≥Ni0

Mi0i1 , the Julia set of fi0n,c : Ci0n(c) →Di00(c) is connected, so qc is twice renormalizable.

We use the induction to complete the construction of our subset A(−2) around −2.Suppose we have constructed Ww where w = i0i1 . . . ik−1 and i0 ≥ N0, i1 ≥ Ni1 , . . . ,ik−1 ≥ Ni0i1...ik−2 . Let v = i0 . . . ik−2. There is a parameter cw ∈ Mw such thatfw = fw,cw

: Cw = Cw(cw) → Dw = Dv0(cw) is hybrid equivalent to q−2(z) = z2 − 2.The quadratic-like map fw : Cw → Dw has the non-separate fixed point βw and theseparate fixed point αw. Let Gw be the closure of the union of two external rays forqcw which land at αw. Let Gw = f−1

w (Gw). Then Gw cuts Cw into three domains. LetDw0 be the one containing 0. Denote βw be another preimage of βw under fw. Letβw ∈ Ew0 and βw ∈ Ew1 be the components of the closure of Cw \Dw0. Let gw0 andgw1 be the inverses of fw|Ew0 and fw|Ew1. Let

Dwn = gnw1(Dw0) and Bwn = gw0(Dw(n−1))

for n ≥ 1. By the Bottcher changes of coordinate hcw and hc (or the structure stabilitytheorem), βw, βw, αw, and Gw, Gw, and ∂Cw are preserved for c close to cw. We can finda small neighborhood Uw about cw with diam(Uw) ≤ 1/2k such that the correspondingdomains Dwn(c) and Bwn(c) can be constructed for qc, c ∈ Uw. Let

Wwn = c ∈ C | fw,c(0) ∈ Bwn(c), n = 1, 2, · · · .

They give a sub-partition of Ww.The diameter diam(Bwn(c)) tends to zero uniformly on Uw and the set Bwn(c) ap-

proaches to βw(c) uniformly on Uw as n goes to infinity. Suppose Uw is simply connectedand the boundary curve ∂Uw is a closed path homologous to 0. Since the equationfw,c(0) − βw(c) = 0 has a unique solution cw (following from Thurston’s theorem forcritically finite rational maps (see [DH4]) and also refer to [DH3]),

m = minc∈γ

|fw,c(0)− βw(c)| > 0.

Let f(c) = fw,c(0)− βw(c) and g(c) = fw,c(0)− x for x ∈ Bwn(c). There is an integerNw > 0 such that for n ≥ Nw,

|f(c)− g(c)| = |βw(c)− x| < m ≤ |f(c)|, c ∈ ∂Uw.

Now Theorem I (the Rouche Theorem) implies that g(c) = fw,c(0)−x = 0 has a uniquesolution in Uw for any x in Bwn(c) and n ≥ Nw. Therefore Wwn ⊆ Uw for n ≥ Nw. Sodiam(Wwn) ≤ 1/2k for n ≥ Nw.

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Similar to the proof of Lemma 2, we have Mwn = Wwn∩M for n ≥ Nw is connected.For each c in Wwn where n ≥ Nw, let Cwn(c) = f−1

w,c(Bwn(c)). Then

fwn,c = f(n+1)w,c : Cwn(c) → Dw0(c)

is a quadratic-like map. Furthermore, since diam(Bwn(c)) tends to zero as n goes to in-finity uniformly on Uw, for Nw large enough we can have that the modulus mod(Awn(c)) ≥1 where Awn(c) = Dw0(c) \ Cwn(c). Moreover,

fwn,c : Cwn(c) → Dw0(c) | c ∈ Wwn

is a full family. Therefore, Wwn contains a copy Mwn ⊆ Mwn of M (refer to [Mi2, pp,102-106]). For c ∈ Ak+1 = ∪w ∪ik≥Nw Mwik

, qc is (k + 1)-times renormalizable wherew = i0i1 . . . ik−1 runs over all sequences of integers of length k. We thus construct athree-dimensional partition

Wi0...ik∞k=0

about the subset A(−2) = ∩∞k=1Ak.For each c ∈ A(−2), qc is infinitely renormalizable. Furthermore, −2 is a limit point

of A(−2). For each c ∈ A(−2), there is a corresponding sequence w∞ = i0i1 . . . ik . . .

of integers such that c = ∩∞k=0Wi0...ik. Since Mi0...ik

= Wi0...ik∩M is connected,

Wi0...ik∞k=0 is a basis of connected neighborhoods of M at c. In other words, M

is locally connected at c. Moreover, from our construction, qc is unbranched and hascomplex bounds. Therefore the Julia set Jc is locally connected from Corollary B. ¤Proof of Theorem 7. Let c0 ∈ M be a Misiurewicz point and Jc0 be its Julia set.Then there is the smallest integer m ≥ 1 such that p = qmc0

(0) is a repelling periodicpoint of qc0 of period k ≥ 1. We start the construction of our subset A(c0) and athree-dimensional partition Ww | w = i0i1 · · · in, n = 0, 1, · · · around it as follows.

Let α be the separating fixed point of qc0 . Let G be the closure of the union ofexternal rays landing at α. Let ξ = ηn∞n=0 be the Yoccoz partition about Jc0 (see §3).Let

p ∈ · · · ⊆ Dn(p) ⊆ Dn−1(p) ⊆ · · · ⊆ D1(p) ⊆ D0(p)

be a p-end, that means that p ∈ Dn(p) ⊆ Dn−1(p) and Dn(p) ∈ ηn. Let

c0 ∈ · · · ⊆ En(c0) ⊆ En−1(c0) ⊆ · · · ⊆ E1(c0) ⊆ E0(c0)

be a c0-end, this means that c0 ∈ En(p) ⊆ En−1(p) and En(p) ∈ ηn. We haveq(m−1)c0 (En+m−1(c0)) = Dn(p). Since the diameter diam(Dn(p)) tends to zero as

n → ∞ and since p is a repelling periodic point, we can find an integer l ≥ m suchthat |(qkc0

)′(x)| ≥ λ > 1 for all x ∈ Dl(p) and such that q(m−1)c0 : El+m−1(c0) → Dl(p)

is a homeomorphism. Let t ≥ 0 be the integer such that f = qtc0: Dl(p) → Cr0 is

31

a homeomorphism, where Cr0 is the domain containing 0 in ηr0 , r0 ≥ 0. There is aninteger r > r0 such that r + t > l and such that B0 = f−1(Cr ∩Dl(p)) does not containp. Thus qtc0

: B0 → Cr is a homeomorphism. Define

Bn =(qnkc0

|Dl+nk(p))−1

(B0) ⊆ Dl+nk(p)

for n ≥ 1. Note that Bn is in ηr+q+nk. Then q(q+nk)c0 : Bn → Cr is a homeomorphism.

By the Bottcher changes of coordinate hc0 and hc (or the structure stability theorem),α, G, p, Cr, Dn, and Bn, for n ≥ 0, are all preserved for c close to c0. Thereforethey can be constructed for qc as long as c close to c0 as we did for −2. Let U0 bea neighborhood about c0 with diam(U0) ≤ 1 such that the corresponding α(c), G0(c),p(c), Cr(c), Dn(c), and Bn(c), for n ≥ 0, are all preserved for c ∈ U0. As n goes toinfinity, the diameter diam(Bn(c)) tends to zero uniformly on U0 and the set Bn(c)approaches to p(c) uniformly on U0. Let

Wn = Wn(c0) = c ∈ C | qmc (0) ∈ Bn(c), n ≥ 1.

They give a partition in the parameter space.Suppose U0 is simply connected and the boundary γ is a closed path homologous to

0 in U0. Since the equation qmc (0) − p(c) = 0 has a unique solution c0 in U0 (fol-

lowing Thurston’s Theorem for critically finite rational maps (see [DH4]) and alsorefer to [DH3]), m = minc∈γ |qmc (0) − p(c)| > 0. Let f(c) = qmc (0) − p(c) andg(c) = qmc (0) − x for x ∈ Bn(c). Since Bn(c) approaches p(c) uniformly on U0 asn goes to infinity, there is an integer N0 = N0(c0) > 0 such that for n ≥ N0,

|f(c)− g(c)| = |p(c)− x| < m ≤ |f(c)|, x ∈ ∂U0.

Theorem I (the Rouche Theorem) now implies that g(c) = qmc (0)− x = 0 has a uniquesolution, which is in U0, for any x in Bn(c) and n ≥ N0. Therefore Wn ⊆ U0 for n ≥ N0.So diam(Wn) ≤ 1 for n ≥ N0. A similar argument to the proof of Lemma 2 impliesthat, for n ≥ N0, Mn = M∩Wn is connected (see Fig. 7).

For any c ∈ Wn, n ≥ N0, let Rn(c) be the pre-image of Bn(c) under the map

q(m−1)c0

: El+m−1(c) → Dl(p, c)

and let Cm+r+q+nk(c) = q−1c (Rn(c)). Then Cm+r+q+nk(c) is the closure of a domain

which contains 0 and which is in ηm+r+q+nk. Hence

fn,c = q(q+nk+m)c : Cm+r+q+nk(c) → Cr(c)

32

is a quadratic-like map. Furthermore, since diam(Bn(c)) tends to zero as n goes toinfinity uniformly on U0, for N0 large enough we can have the modulus mod(An(c)) ≥ 1where An(c) = Cr(c) \ Cm+r+q+nk(c). Moreover,

fn,c : Cm+r+q+nk(c) → Cr(c) | c ∈ Wn

is a full family. Thus Wn contains a copy Mn = Mn(c0) of the Mandelbrot set Mwhere Mn ⊆ Mn (refer to [Mi2, pp 102-106]). For any c ∈ A1(c0) = ∪n≥N0Mn, qc isonce renormalizable.

Fig. 7: A small copy of the Mandelbrot set and hairs around it.

Now following almost the same arguments in the proof of Theorem 8, We can con-struct a subset A(c0) and a three-dimensional partition

Ww(c0) | w = i0 · · · in, n = 0, 1, · · ·

about it such that c0 is a limit point of A(c0) and such that every c ∈ A(c0) is infinitelyrenormalizable at which M is locally connected. Furthermore, qc is unbranched and hascomplex bounds from our construction. Therefore the Julia set Jc is locally connectedfollowing Corollary B. It completes the proof. ¤

Remark 4. Eckmann and Epstein [EE] and Douady-Hubbard [DH3] have estimated thesize of Mn. Since Mn \ Mn contains hairs (see Fig. 7) which may destroy the localconnectivity of M, we must estimate the size of Mn. This is the key point in the proofof Theorems 7 and 8. Lyubich [Ly2] announced another subset consisting of infinitelyrenormalizable points such that M is locally connected at every point in this subset. Apoint in his subset must satisfies several complicate conditions. His idea is more closeto Douady and Hubbard’s tuning idea (see §5.1). Our idea is different and simple. Apoint in our subset may not satisfy those conditions. In our proof, the use of the Rouche

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Theorem is interesting. Actually, the three-dimensional partition in the parameter spacecan be constructed more generally following the construction of the three-dimensionalpartition in the phase space for every infinitely renormalization quadratic polynomial.However, finding a proper tool to replace the Rouche Theorem in the proof of Theorems7 and 8 is an interesting problem.

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Department of Mathematics, Queens College of CUNY, Flushing, NY 11367 and De-partment of Mathematics, CUNY Graduate Center, 365 Fifth Avenue New York, NY10016

E-mail address: yunqc@jiang.math.qc.edu

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