Three Dimensional Partition and Infinite Renormalization

THREE DIMENSIONAL PARTITION AND INFINITE RENORMALIZATION

Yunping Jiang March 13, 2002

Abstract. We provide a detailed construction of the three dimensional partition for the Julia set of an infinitely renormalizable quadratic polynomial. We show that for an unbranched infinitely renormalizable quadratic polynomial having complex bounds, the three-dimensional partition determines points in the Julia set dynamically. Furthermore, we construct a partition in the parameter space about a subset consisting of infinitely renormalizabe 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 polynomial having complex bounds. So its Julia set is locally connected. Moreover, we prove that the Mandelbrot 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, 37Exx This research is supported in part by NSF grants and by PSC-CUNY awards

Typeset by AMS-TEX 1 §1. Introduction. Organizing information provided by a dynamical system is important in the study of this dynamical system. A very useful method is called Markov partitions, which was introduced by Sinai [Si1,Si2] into dynamical systems from probability theory and which was generalized by Bowen [Bo1,Bo2]. For example, the dynamical system generated by the 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 A dynamical system (see [Bo3]). So many interesting questions, in particular, those relate to chaos, of the dynamical system are well understood by using symbolic dynamical systems. When things come to a non Axiom A dynamical system, the situation becomes more complicate. For example, when we study the dynamics of a quadratic polynomial whose critical 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. Using this partition and a puzzle rule on the partition, he proved that the Julia set of a non-renormalizable quadratic polynomial is locally connected. Another similar partition for a cubic polynomial whose one critical orbit is bounded and the other critical orbit in the complex plane tends to infinity is constructed by Branner and Hubbard in [BH1,BH2]. Using this partition and a puzzle rule on the partition, they proved that the Julia set of such a cubic polynomial is a Cantor set whose Lebesgue measure is zero. Moreover, using the partition constructed by Yoccoz for a non-renormalizable quadratic polynomial, Shishikura [Sk] and Lyubich [Ly1] proved that the Lebesgue measure of the Julia set of this polynomial is zero. We have further developed this kind of partitions for an infinitely renormalizable quadratic polynomial in [Ji3] and [Ji2]. Using this 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 unbranched infinitely renormalizable quadratic polynomial having complex bounds is locally connected. By combining the result in [Ji2] and Sullivan’s result [Su] on the existence of complex bounds for the Feigenbaum quadratic polynomial, we concluded in [HJ] that the Julia set of the Feigenbaum quadratic polynomial is locally connected. Levins and van Strien [LS] and Lyubich and Yamploski [LY] further proved that a real infinitely renormalizable quadratic polynomial has complex bounds. Thus its Julia set is locally connected. Although the main result in [Ji2] is about the local connectivity of the Julia set of an infinitely renormalizable quadratic polynomial, the construction of a partition in the study is also interesting. We provide a detailed exposition in §4 about this partition which we call the three dimensional partition for the Julia set of an infinitely renormalizable quadratic polynomial. In the same section, we show that for an unbranched infinitely renormalizable quadratic polynomial having complex bounds, the three-dimensional partition determines points in the Julia set dynamically (Theorem 6). 2 Moreover, we transfer the three dimensional partitions into the parameter space and prove in §5.2 that there is a subset which is dense on the boundary of the Mandelbrot set such that every point in this subset is an unbranched quadratic polynomial having complex bounds (Theorems 7 and 8). Furthermore, by applying the RouchéTheorem in classical complex analysis, we prove in §5.2 that the Mandelbrot set is locally connected at every point in this subset (Theorems 7 and 8). Another interesting subset on the boundary of the Mandelbrot set is constructed by Douady and Hubbard. Douady and Hubbard proved that this subset is generic and the Julia set of every point in this subset is not locally connected. We provide some information about Douady and Hubbard’s construction in §5.1. The reader who is interested in more details could refer to Milnor’s expository article [Mi2]. We review some basic facts about dynamics of quadratic polynomial in §2. There are two text books [Mi1,CG] available in complex dynamics. The reader, who is interested in these basic facts and who would like to learn more about complex dynamics, could refer to them. In §3, we give an outline of the partition constructed by Yoccoz for a quadratic polynomial and show its relation with the renormalizability of the polynomial. There are two expository articles [Hu,Mi2] written by Hubbard and Milnor for the partition constructed by Yoccoz and a puzzle rule on this partition. The reader who would like to learn more may go to there. He may also go to [Ji1, pp. 206-219]. For the reader who is interested in the renormalization theory of quadratic polynomials and related topics there 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 Riemann sphere). Suppose f is a holomorphic map from a domain Ω ⊂ C into itself. A point z in Ω is called a periodic point of period k ≥ 1 if f ◦k(z) = z but f ◦i(z) 6= z for all 1 ≤ i < k. The number ‚ = (f ◦k)0(z) is called the multiplier of f at z. A periodic point of period 1 is also called a fixed point. A periodic point z of f is said to be super-attracting, attracting, neutral, or repelling if |‚| = 0, 0 < |‚| < 1, |‚| = 1, or |‚| > 1. The proof of the following theorem can be found in Milnor’s book [Mi1, pp. 86-88]. Theorem A (Theorem of Böttcher). Suppose p is a super-attracting periodic point of period k of a holomorphic map f from a domain Ω into itself. There is a neighborhood W of p, a holomorphic diffeomorphism h : W → h(W ) with h(p) = 0, and a unique integer n > 1 such that h ◦ f ◦k ◦ h−1(w) = wn for w ∈ h(W ). Furthermore, h is unique up to multiplication by an (n − 1)-st root of unity.

2 Denote Dr = {z ∈ C | |z| < r} the disk of radius r centered 0. Let qc(z) = z +c be a −1 quadratic polynomial. For r large enough, U = qc (Dr) is a simply connected domain 3 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 maps deﬁned 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 with U ⊂ V and such that f : U → V is a holomorphic, proper, degree two branched covering mapping. For a quadratic-like map (f; U; V ),

∞ −n Kf = ∩n=0f (U) is called the ﬁlled-in Julia set. The Julia set Jf is the boundary of Kf . Both Kf and Jf are compact. A quadratic-like map (f; U; V ) has only one critical point which we always 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. 2 Consider a quadratic polynomial qc(z) = z + c. By the definition, the filled-in Julia set of qc is the set of all points not going to infinity under forward iterates of qc and the Julia set is the boundary of the filled-in Julia set. We use Kc and Jc to mean its filled-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 fixed point of qc, applying Theorem A, there is a unique holomorphic diffeomorphism hc (called the Böttcher change of coordinate) defined on a neighborhood B0 about ∞ with 0 hc(∞) = ∞, hc(∞) = 1 such that

−1 2 hc ◦ qc ◦ hc (z) = z ; z ∈ B0:

−n Assume hc(B0) = C \ Dr (for a ﬁxed large number r > 1). Let Bn = qc (B0). If 0 6∈ Bn, then qc : Bn ∩ C → Bn−1 ∩ C; n ≥ 1; is unramiﬁed covering map of degree two. So we can inductively extend

hc : Bn → C \ D 1 r 2n such that −1 2 hc ◦ qc ◦ hc (z) = z z ∈ Bn: Let ◦n Bc(∞) = {z ∈ C | qc (z) → ∞ as n → ∞} be the basin of ∞. Then Kc = C \ Bc(∞). Let

+ ◦n log |qc (z)| Gc(z) = lim ; n→∞ 2n 4 + where log x = sup{0; log x}, be the Green function of Kc. It is a proper harmonic function whose zero set is Kc and whose critical points are bounded by Gc(0). So the B¨ottcher change of coordinate can be extended up to

hc : Uc = {z ∈ C | Gc(z) > Gc(0)} → {z ∈ C | |z| > Gc(0)}: such that −1 2 hc ◦ qc ◦ hc (z) = z : −1 Moreover, Gc = log |hc|. The set Gc (log r), r > 1, is called an equipotential curve. If Kc is connected (equivalent to say 0 6∈ Bc(∞)), then Uc = Bc(∞) \ {∞}; if Kc is a 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 set of parameters c such that Kc is connected. Suppose c ∈ M. Let Sr = {z ∈ C | |z| = r} be a circle of radius r > 1. Then −1 −1 sr = hc (Sr) = Gc (log r) is an equipotential curve. Note that

qc(sr) = sr2 : −1 For every r > 1, let Ur = hc (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 is 2 always Kc. In the rest of paper, a quadratic polynomial q(z) = z + c could also mean a quadratic-like map (q; Ur;Ur2 ) for some r > 1. Take Eθ = {z ∈ C | |z| > 1; arg(z) = 2…} for 0 ≤ < 1. Let −1 eθ = hc (Eθ):

Then eθ is called an external ray of angle and

qc(eθ) = e2θ (mod 1):

An external ray is an integral curve of the gradient vector field ∇G on Bc(∞). An external ray eθ is said to land at Jc if eθ has only one limiting point at Jc. It is periodic ◦i ◦m with period m if qc (eθ) ∩ eθ = ∅ for 1 ≤ i < m and if qc (eθ) = eθ. Refer to [Hu,Mi2] for the following theorem (the reader can also find a proof in [Ji1, pp. 199]). 2 Theorem C (Douady and Yoccoz). Suppose qc(z) = z +c is a quadratic polynomial with a connected filled-in Julia set Kc. Every repelling periodic point of qc is a landing point of finitely many periodic external rays with the same period. Two quadratic-like maps (f; U; V ) and (g; U 0;V 0) are said to be topologically conjugate if there is a homeomorphism h from a neighborhood Kf ⊂ X ⊂ U to a neighbor- 0 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. 5 Theorem D (Douady and Hubbard [DH1]). Suppose (f; U; V ) is a quadratic-like map whose Julia set Jf is connected. Then there is a unique quadratic polynomial 2 qc(z) = z + 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 (1) The Julia set Jf is completely invariant, i.e., f(Jf ) = Jf and f (Jf ) = Jf . 0 0 (2) The Julia set Jf is perfect, i.e., Jf = Jf , where Jf 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 . −n ∞ (4) The limit set of {f (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.

2 Consider a quadratic polynomial qc(z) = z + c whose Julia set Jc is connected. The external ray e0 is the only one fixed by qc. It lands either at a repelling or parabolic fixed point fl of qc. Suppose fl is repelling. From Theorem C, e0 is the only external ray landing at fl (see Fig. 1). So Jc \{fl} is still connected. We call fl the non-separate fixed point. Let fi 6= fl be the other fixed point of qc. If fi is attracting or super-attracting, ◦n then Jc = Kc \ B(fi) where B(fi) = {z ∈ C | qc (z) → fi as n → ∞} is the basin of fi. In this case, qc is hyperbolic and Jc is a Jordan curve. If fi is repelling. Then there are at least two periodic external rays landing at fi (see Fig. 1). All of them are in one cycle of period k ≥ 2 (refer to Theorem H in §5.1). We use Λ to denote the union of this 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 \{fi} is disconnected. We call fi the separate fixed point. ◦i th The point 0 is the only finite critical point of qc. Let ci = qc (0), i ≥ 1, be the i ∞ critical value. We use CO to denote the critical orbit {ci}i=0. The critical point 0 is said to be recurrent if for any neighborhood W of 0, there is a critical value ci 6= 0 in W . We will only consider those quadratic polynomial whose critical point is recurrent and which has no neutral periodic points in this section. These are assumed for a quadratic-like map 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 fixed √ r > 1, (q; U r;Ur) is a quadratic-like map. The set Λ cuts Ur into finitely many Jordan domains (see Fig. 1). Let C0 be the closure of the one containing 0, and let B0,i be the 6 closure of the domain containing ci for 1 ≤ i < k. We call the collection

·0 = {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) for every 1 ≤ i < k) is the union of some sets from

·0 ∩ 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. −n −n For each n > 0, let fin = qc (fi) and Λn = qc (Λ) (denote fi0 = {fi} and Λ0 = Λ). The set Λn is the union of external rays landing at points in fin. It cuts the closure of the domain U 1 into a finite number of closed Jordan domains, r 2n

·n = {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 ·n−1. Each qc|Bn,i is degree one, holomorphic, proper but qc|Cn is degree two, holomorphic, and proper. We call ·n the th n -partition about Jc. Thus we get a sequence

∞ » = {·n}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) ∈ ·n. For every x in Λ∞, there 7 ∞ are k sequences {Dn,i(x)}n=0, 1 ≤ i ≤ k, of nested closed Jordan domains such that x ∈ Dn,i(x) ∈ ·n. In this case, x is on the boundary of each Dn,i(x) but it is an interior k point of the closed domain ∪i=1Dn,i(x). When there is no confusion, we will only use ∞ {Dn(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 the sequence » provides a complete description of points in Jc dynamically?

Deﬁnition 1. We say » determines points in Jc dynamically if for any x 6= y ∈ Jc, 0 0 there is an integer n ≥ 0 and two domains x ∈ D and y ∈ D in ·n such that D∩D = ∅. From Theorem D, the sequence » can be also constructed similarly for any quadratic- like map whose Julia set is connected. Deﬁnition 1 relates to the concept called renormalizability 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 n0 ≥ 2 and a domain 0 ∈ U 0 ⊂ U such that ◦n0 0 0 0 f1 = f : U → V = f1(U ) ⊂ V 0 0 0 is a quadratic-like map with connected Julia set Jf1 = J(n ;U ;V ). In this situation, we 0 0 call (f1;U ;V ) a renormalization of (f; U; V ). Otherwise, we call f non-renormalizable. The relation between » and the renormalizability of a quadratic polynomial is that 2 Theorem F (Yoccoz). The polynomial qc(z) = z + c is non-renormalizable if and only if » determines points in Jc dynamically. A connected set J is called locally connected if for every point x ∈ J and neighborhood 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 polynomial is locally connected. The reader may refer to [Hu,Mi2] for Yoccoz’ results. (One can also find proofs of Theorem F and Corollary A in [Ji1, pp.206-219]). Remark 1. From Theorem D, every quadratic-like map f : U → V whose Julia set is connected is hybrid equivalent to a unique quadratic polynomial, all arguments in this section can be applied to a quadratic-like map (by considering induced external rays and equipotential curves from the hybrid equivalent quadratic polynomial). 8 §4. Construction of a three-dimensional partition 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 ◦n0 0 0 (2) a renormalization f1 = f : U → V is again once renormalizable. Consequently, we have renormalizations

◦n0 0 0 ◦n00n0 00 00 f1 = f : U → V and f2 = f1 : U → V and their Julia sets Jf1 and Jf2 . Similarly, we can deﬁnite a l-times renormalizable quadratic-like map (f; U; V ) and 0 00 renormalizations (assume f0 = f, k0 = n and k1 = n )

◦ki−1 l {fi = fi−1 : Ui → Vi}i=1 l with Julia sets {Jfi }i=1. A quadratic-like map f : U → V is infinitely renormalizable if it is l-times renormalizable for every l > 0. ◦n0 0 Suppose f : U → V is a renormalizable quadratic-like map. Let f1 = f : U → 0 V be a renormalization whose the Julia set J1 = Jf1 . Suppose its two fixed points fi1 and fl1 are repelling. Then one fixed point fl1 does not disconnect J1, i.e., J1 \ {fl1} is still connected, and the other fixed point fi1 disconnects J1, i.e., J1 \{fi1} is disconnected. Similarly fi1 (resp. fl1) is called the separating (non-separating) fixed point. McMullen [Mc1] discovered that different types of renormalizations can occur in ◦i 0 the renormalization theory of quadratic-like maps. Let J(i) = f (J1), 1 ≤ i < n . The renormalization f1 is said to be 0 (1) fi-type if J(i) ∩ J(j) = {fi1} for some i 6= j, 0 ≤ i; j < n ; 0 (2) fl-type if J(i) ∩ J(j) = {fl1} 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 < n0. The fl-type and the disjoint type are also called simple. If f : U → V is infinitely renormalizable, i.e., it has an infinite sequence of renormalizations, then it will have an infinite sequence of simple renormalizations [Mc1]. In this subsection, we will construct a most natural sequence of simple renormalizations for an infinitely renormalizable quadratic-like map. Before this, we first prove the following modulus inequality (refer ∼ to [Ji2] and see Fig. 2). A Riemann surface A with …1(A) = Z is called an annulus. ∗ ∗ It is conformally isomorphic to C = C \{0}, ∆ = D1 \{0}, or the round annulus Ar = {z ∈ C | 1 < |z| < r} for some 1 < r < ∞. In case A is isomorphic to Ar, the modulus of A is defined by log r mod(A) = : 2… 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 n0-renormalization

◦n0 0 0 0 f1 = f : U → V ; n ≥ 2; we have 1 mod(U \ U 0) ≥ mod(V \ U): 2

Fig. 2: Modulus inequality in renormalization.

◦n0 0 0 Proof. Since f1 = f : U → V is quadratic-like, it has only critical point 0. So the 0 ﬁrst critical value c1 = f(0) of f is not in V . (Otherwise, there will be a point x 6= 0 in 0 ◦(n0−1) U such that f (x) = 0, since c1 has only one preimage 0 under f and 0 is not a 0 periodic point of f. Then x will be a critical point of f1.) Since V is simply connected, f has two analytic inverse branches;

0 0 0 0 g0 : V → g0(V ) ⊂ U and g1 : V → g1(V ) ⊂ U:

◦(n0−1) 0 0 One of them is f (U ). Therefore, f(U ) is a domain inside U and containing c1. Consider the annuli U \ U 0 and V \ f(U 0). Then

f : U \ U 0 → V \ f(U 0) is a degree two holomorphic covering map. This implies that

1 mod(U \ U 0) = mod(V \ f(U 0)): 2 10 Since f(U 0) is a subset of U, the annulus V \U is a sub-annulus of the annulus V \f(U 0), i.e., V \ U ⊆ V \ f(U 0). Thus we have that 1 mod(U \ U 0) ≥ mod(V \ U): 2 ¤ We will use N(X;†) = {x ∈ C | d(x; X) < †} to denote the †-neighborhood of X in the complex plane. Consider a renormalizable quadratic-like map f : U → V whose ∞ Julia set Jf is connected. Let » = {·n}n=0 be the Yoccoz partitions about Jf . Let

0 ∈ · · · ⊆ Cn ⊆ Cn−1 ⊆ · · · ⊆ C2 ⊆ C1 ⊆ C0 be the critical end. Denote ∞ J1 = ∩i=0Ci:

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

◦k1 f1 = f : Ck1+n1 → Cn1 is a degree two branched covering map and such that Cn1 ⊂ N(J1; 1). We can further take domains Ck1+n1 ⊆ U1 ⊂ U and Cn1 ⊆ V1 ⊂ V such that

◦k1 f1 = f : U1 → V1

◦k1 is a quadratic-like map whose Julia set is J1. (Note that f1 = f : U1 → V1 is a simple renormalization of f : U → V .)

◦k1 0 0 Theorem 2. Suppose g = f : U → V is any k1-renormalization of f : U → V . Then its Julia set is always J1. 0 00 Proof. The point 0 is in the intersection U ∩U1. Suppose U is the connected component 0 ◦k1 00 00 0 of U ∩ U1 containing 0. Then h = f : U → V ⊂ V ∩ V1 is also a renormalization of f and its Julia set Jh is connected. It is easy to check that Jh ⊆ Jg ∩ Jf1 . Let flf1 and fif1 be the non-separating and separating fixed points of f1 and let flg and fig be the non-separating and separating fixed points of g. All flf1 , fif1 , flg, and fig are in Jh since each of g, f1 and h has exactly two fixed points in its domain, and every fixed ◦k1 point of h has to be a common fixed point of g and f1. Since f1 = f : U1 → V1 and ◦k1 0 0 g = f : U → V are both degree two branched covering maps and flf1 is in both 0 0 U and U1, all preimages of flf1 under iterates of g and f1 are in both U and U1. But each of Julia sets Jh, J1, and Jg is the closure of the set of all preimages of flf1 under iterates of h, g, and f1. Therefore, Jh = Jg = J1. ¤

Remark 2. From Theorem 2, for any k1-renormalization

◦k1 0 0 f1 = f : U → V 0 of f : U → V , there is a critical piece Ck1+n ⊂ U such that

k1 0 f1 = f : Ck1+n → Cn ⊂ V is a degree two branched covering map. 11 §4.2. Three-dimensional partition. 2 Take an infinitely renormalizable quadratic polynomial qc(z) = z + c. Let U and V be domains bounded by equipotential curves such that f0 = qc : U → V is a quadratic- 0 0 0 0 0 like map. Let fi0 = fi, fl0 = fl, ·n = ·n, » = », Cn = Cn,Λn = Λn, and Λ∞ = Λ∞ be the same as before. Let n1 ≥ 0 and k1 ≥ 2 be two integers such that

◦k1 0 0 ∞ 0 f1 = f0 : Ck1+n1 → Cn1 ⊂ N(J1; 1) where J1 = ∩i=0Ci :

Suppose fl1 and fi1 are the non-separating and separating fixed points of f1, i.e., J1\{fl1} is still connected and J1 \{fi1} is not. The points fl1 and fi1 are also repelling periodic points of qc. There are at least two, but a finite number, external rays of qc landing 1 1 1 0 at fi1. Let Λ0 be the union of external rays landing at fi1. Then Λ0 cuts V0 = Cn1 1 into a finite number of closed domains. Let ·0 be the collection of these domains. Let 1 −n 1 1 1 −n 1 Λn = f1 (Λ0) for any n > 0. Then Λn cuts Vn = f1 (V0 ) into a finite number of 1 1 1 ∞ closed domains. Let ·n be the collection of these domains. The sequence » = {·n}n=0 is a sequence of nested partitions about J1. We call it the first partition. (We also call 0 th 1 ∞ 1 » the 0 partition.) Let Λ∞ = ∪n=0Λn. 1 1 The domain Cn ∈ ·n containing 0 is called the critical piece. It is clear the restriction 1 1 f1 to Cn is degree two branched covering map but to all other domains in ·n are degree one. Let ∞ 1 J2 = ∩n=0Cn:

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

◦k2 1 1 f2 = f1 : Cn2+k2 → Cn2

1 is a degree two branched cover map and such that Cn2 ⊂ N(J2; 1=2). Inductively, for every i ≥ 2, suppose we have already constructed

f = f ◦ki : Ci−1 → Ci−1: i i−1 ni+ki ni

Let fli and fii be the non-separating and separating fixed points of fi; i.e., Ji \{fli} is still connected and Ji \{fii} is not. The points fli and fii are also repelling periodic points of qc. There are at least two, but a finite number, external rays of qc landing at fii. Let i i i i−1 Λ0 be the union of external rays landing at fii. Then Λ0 cuts V0 = Cni into finitely i i −n i many closed domains. Let ·0 be the collection of these domains. Let Λn = fi (Λ0) for i i −n i i any n > 0. Then Λn cuts Vn = fi (V0 ) into finitely many closed domains. Let ·n be i i the collection of these domains. The domain Cn in ·n containing 0 is called the critical i i i piece in ·n. It is clear that fi restricted to all domains but Cn are bijective and fi|Cn is a degree two branched covering map. Let

∞ i Ji+1 = ∩n=0Cn: 12 There are two integers ni+1 ≥ 0, ki+1 ≥ 2 such that

◦ki+1 i i fi+1 = fi : Cni+1+ki+1 → Cni+1

i is a degree two branched cover map and such that Cni+1 ⊂ N(Ji+1; 1=(i + 1)). Let i i ∞ th » = {·n}n=0. It is the sequence of nested partitions about Ji. We call it the i partition.

◦ki+1 0 0 Remark 3. For any ki+1-renormalization fi+1 = fi : U → V of fi : Ui → Vi, we i 0 ◦ki+1 have an integer n > 0 such that Cn ⊂ V ∩ N(Ji+1; 1=(i + 1)) and fi+1 = fi : Ci → Ci is a degree two branched covering map. We will still use »i to mean n+ki+1 n »i ∩ Ci . Therefore, (U ;V ) can be an arbitrary domains such that f = n+ki+1 i+1 i+1 i+1 ◦ki+1 fi : Ui+1 → Vi+1 is a ki+1-renormalization of fi : Ui → Vi. Qi Let mi = j=1 ki, 1 ≤ i < ∞. We have thus constructed a most natural inﬁnite sequence of simple renormalizations,

◦mi ∞ {fi = qc : Ui → Vi}i=1;

i ∞ ∞ and the nested-nested sequence {» }i=0 of partitions about {Ji}i=0 (where J0 = Jc). i ∞ We call Ξ = {» }i=0 the first three-dimensional partition about Jc. Definition 3. We say that the first three-dimensional partition Ξ determines points in X ⊆ Jc dynamically if for any z ∈ X, there are integers n; l; i ≥ 0 such that for ◦l 0 i 0 0 x = qc (z) and any z 6= x ∈ Jc, there is a domains D ∈ ·n such that x ∈ D but z 6∈ D . ◦n ∞ Let Orb(z) = {qc (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 dynamically from all other points in Jc. −1 −1 Proof. Suppose z ∈ Γ. Take n0 = 0, k0 = 1, and Cn0 = U and Cn0 = V . We have j−1 i−1 Cnj for all j ≥ 1. There is the smallest integer i ≥ 0 such that O(z) ∩ Cni 6= ∅ and ◦l i−1 O(z) ∩ Ji+1 = ∅. Take l ≥ 0 the smallest integer such that x = qc (z) ∈ Cni and th i i ∞ i−1 ∞ i consider the i partition » = {·n}n=0 inside Cni . Since Ji+1 = ∩n=0Cn, there is an i integer N ≥ 0 such that O(x) ∩ CN = ∅. Consider

i i i i i ·N = {CN ;BN,1;BN,2;:::;BN,l}

th i i in the i partition » . Assume that BN,1 contains fi(0). (Note that fi(0) is an interior i i i point of BN,1.) Then fi(O(x)) is disjoint with BN,1 since O(x) is disjoint with CN+1 ⊆ 13 i i i i i i CN and fi(CN+1) = BN,1. Suppose x ∈ D is a domain in ·N+1 ∈ » . Suppose D ⊆ BN,j 0 and fi(D) = BN,j0 for 2 ≤ j; j ≤ r. Then fi : D → BN,j0 has the inverse i gjj0 : BN,j0 → D: i i 0 Therefore for each pair (BN,j ;BN,j ), 2 ≤ j; j ≤ r, satisfying gjj0 (BN,j0 ) ⊆ BN,j , we can thicken BN,j0 and BN,j to simply connected domains BÑ,j0,jj0 and BÑ,j,jj0 such that BN,j0 ⊂ BÑ,j,jj0 and BN,j ⊂ BÑ,j,jj0 and such that gjj0 can be extended to a schlicht function from BÑ,j,jj0 into BÑ,j,jj0 . We still use gjj0 to denote this extended function. Now consider BÑ,j,jj0 and BÑ,j,jj0 as hyperbolic Riemann surfaces with hyperbolic distances dH,j,jj0 and dH,j,jj0 . Then gjj0 on BN,j0 strictly contracts these hyperbolic distances; more precisely, there is a constant 0 < ‚jj0 < 1 such that dH,j,jj0 (gjj0 (x); gjj0 (y)) < ‚jj0 dH,j,jj0 (x; y) for x and y in BN,j0 . Since there are only a finite number of such pairs, we have a constant 0 < ‚ < 1 such that dH,j0,jj0 (gjj0 (x); gjj0 (y)) < ‚dH,j,jj0 (x; y) for all such pairs. Let i i i i x ∈ · · · ⊆ Dn(x) ⊆ Dn−1(x) ⊆ · · · ⊆ D1(x) ⊆ D0(x) th i i i ◦m i be any x-end in the i partition » , where Dn(x) ∈ ·n. Then fi (Dn(x)) for n−m > N i is in BN,j for some 2 ≤ j ≤ q. Therefore, there is a constant C > 0 such that for any i Dn(x) and for any n > N, d(Di (x)) = max |y − z| ≤ C‚n−N : n i y,z∈Dn(x)

i 0 Thus d(Dn(x)) tends to zero as n goes to infinity. Thus for any z 6= x ∈ Jc, we have 0 an integer n > 0 such that Dn(x) does not contain z . ¤ 2 Definition 4. An infinitely renormalizable quadratic polynomial qc(z) = z + c is said to have complex bounds if there are a constant ‚ > 0 and an infinite sequence of simple ◦mis ∞ renormalizations {fis = qc : Uis → Vis }s=1 such that the modulus mod(Vis \ U is ) is greater than ‚ for every s ≥ 1. 2 Theorem 4. Suppose qc(z) = z +c is an infinitely renormalizable quadratic polynomial having complex bounds. Then the first three-dimensional partition Ξ determinants 0 dynamically from other points. Proof. To make notations simple, we assume

◦mi ∞ {fi = qc : Ui → Vi}i=1 is the infinite sequence of simple renormalizations in Definition 4. Let ‚ > 0 be the ∞ constant in Definition 4. Then {Ui}i=1 is a sequence of nested domains containing 0. From Theorem 1 (the modulus inequality), we have 1 ‚ mod(U \ U ) ≥ mod(V \ U ) > : i i+1 2 i i 2 14 ∞ ∞ Let Ai = Ui \ U i+1 for i ≥ 1 and X = ∩i=1Ui. Since U1 \ X = ∪i=1Ai, X∞ mod(U1 \ X) ≥ mod(Ai) = ∞: i=1 Thus, X = {0} (following the Grötzsch 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. th i i ∞ i Consider the i partition » = {·n}n=0 for every i ≥ 0. Remember that Cn is the i member in ·n containing 0. Consider the corresponding critical end i i i i 0 ∈ · · · ⊆ Cn ⊆ Cn−1 ⊆ · · · ⊆ C1 ⊆ C0

i ∞ i i in » . Since Ji+1 = ∩j=0Cj is the Julia set of fi+1 : Ui+1 → Vi+1, there is a Cl(i) i i contained in Ui+1. The diameter d(Cl(i)) of Cl(i) tends to zero as i goes to infinity. i Thus for any z 6= 0 ∈ Jc, we can find Cl(i) does not containing z. Since 0 is an i i interior point of Cl(i), we have an integer n > l(i) and a domain z ∈ D ∈ ·n such that i D ∩ Cn = ∅. ¤ ∞ i Following Theorem 4, if qc has complex bounds, then ∩i=1Ji = {0} since Ji ⊆ Cl(i). Now we construct our second three-dimensional partition Υ about Jc. Denote by •1 0 0 ∞ the first partition which will be constructed as follows: Consider » = {·n}n=0 in Ξ. 0 0 0 0 Take Ck(0) ∈ ·k(0) ∈ » where k(0) = n1 + k1. Put all domains in ·k(0)+1 which are 0 0c the preimages of Ck(0) under qc into •1 and let ·k(0)+1 be the rest of the domains. 0 0c 0 Consider ·k(0)+2 ∩ ·k(0)+1 consisting of all domains in ·k(0)+2 which are subdomains of 0c 0 0c the domains in ·k(0)+1. Put all domains in ·k(0)+2 ∩ ·k(0)+1 which are the preimages of 0 ◦2 0c Ck(0) under qc into •1 and let ·k(0)+2 be the rest of the domains. Suppose we already have ·0c for s ≥ 2. Consider ·0 ∩ ·0c consisting of all domains in ·0 k(0)+s k(0)+s+1 k0+s k(0)+s+1 0c 0 0c which are subdomains of the domains in ·k(0)+s. Put all domains in ·k(0)+s+1 ∩ ·k(0)+s 0 ◦(s+1) 0c which are the preimages of Ck(0) under qc into •1 and let ·k(0)+s+1 be the rest of the domains. Thus we can construct the partition •1 inductively. This partition 0 covers points in Jc minus all points not entering the interior of Ck(0) under all forward iterations of qc. 1 1 ∞ 1 1 1 Next consider the » = {·n}n=0 in Ξ. Take Ck(1) ∈ ·k(1) ∈ » where k(1) = n2 + k2. We can use similar arguments to those in the previous paragraph by considering f1 : C0 → C0 (to replacing q : U → V ) to get a partition • in C0 . Then we k(0) k(0)−k1 c 1,1 k(0) use all iterations of qc to pull back this partition following •1 to get a partition •2. It is a sub-partition of •1 and covers points in Jc minus all points not entering the interior 1 of Ck(1) under iterations of qc. th Suppose we have already constructed the (j − 1) partition •j−1 for j ≥ 2. Consider j j ∞ j j j the partition » = {·n}n=0 in Ξ. Take Ck(j) ∈ ·k(j) ∈ » where k(j) = nj+1 + kj+1. 15 j−1 j−1 j−1 Similarly, by considering fj : C → C , we get a partition •j,1 in C . k(j−1) k(j−1)−kj k(j−1) Then we use all backward iterations of fj−1 to pull back this partition following •j−1 j−2 to get a partition •j,2 in Ck(j−2) and all backward iterations of fj−2 to pull back this j−3 partition following •j−1 to get a partition •j,3 in Ck(j−3), and so on to obtain a partition •j = •j,j in V . It is a sub-partition of •j−1 and covers points in the Julia set minus all j points not entering the interior of Ck(j) under forward iterations of qc. By induction, we have a sequence of nested partitions

∞ Υ = {•j}j=1

∞ which covers points in Jc \ Γ. We call Υ = {•j}j=1 the second three-dimensional partition about Jc. Definition 5. We say that the second three-dimensional partition Υ determines points 0 in X ⊆ Jc dynamically if for any x ∈ X and x 6= x ∈ Jc, there is an integer i ≥ 1 and 0 a domain x ∈ D ∈ •i ∈ Υ such that x 6∈ D. 2 Definition 6. We say an infinitely renormalizable quadratic polynomial qc(z) = z + c ∞ is unbranched if there are {Jil }l=1, neighborhoods Wl of Jil , and a constant „ > 0 such that the modulus mod(Wl \ Jil ) is greater than „ and Wl \ Jil contains no point in the critical orbit CO of qc. 2 Theorem 5. Suppose qc(z) = z + c is an infinitely renormalizable quadratic poly- ∞ nomial. If ∩i=1Ji = {0} and if qc is unbranched, then the second three-dimensional partition Υ determines points in Jc \ Γ dynamically. th i i ∞ Proof. Consider the i partition » = {·n}n=0 for every i ≥ 0 and the corresponding critical end i i i i 0 ∈ · · · ⊆ Cn ⊆ Cn−1 ⊆ · · · ⊆ C1 ⊆ C0: Consider f = f ki+1 : Ci → Ci . Let k(i) = n +k . Since ∩∞ J = {0} i+1 i ni+1+ki+1 ni+1 i+1 i+1 i=1 i i and Ck(i) ⊂ N(Ji+1; 1=(i + 1)),

∞ i ∩i=0Ck(i) = {0}:

∞ Let „ > 0 and {Wl}l=1 be a constant and domains satisfying Deﬁnition 6. Without loss of generality, we assume in Deﬁnition 6 that l = i and il = i + 1. Since

∞ i mod(Wi \ Ji+1) ≥ „ and Ji+1 = ∩n=0Cn; by choosing k(i) large enough, we can assume „ mod(W \ Ci ) ≥ : i k(i) 2 16 for all i ≥ 1. Also, by modifying Wi, we can assume the diameter diam(Wi) tends to zero as i goes to ∞. ◦n ∞ i Suppose x 6= 0 in Jc \ Γ. Then the orbit Orb(x) = {qc (x)}n=0 enters every Ck(i) ∞ inﬁnitely many times. Consider the second three-dimensional partition Υ = {•j}j=1 and the x-end in this partition,

x ∈ · · · ⊆ Dj(x) ⊆ Dj−1(x) ⊆ · · · ⊆ D1(x); where Dj(x) ∈ •j. Let ij(x) ≥ 0 be the unique integer such that

◦ij (x) j qc : Dj+1(x) → Ck(j) is a proper holomorphic diﬀeomorphism (see Fig. 3). Let

j gj,x : Ck(j) → Dj+1(x)

◦i ∞ j be its inverse. Since there is no critical values {ci = q (0)}i=1 in Wj \ Ck(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 i ◦ii(x) i an integer j = j(i) > i such that Wj ⊂ Ck(i). Let xi = qc (x) ∈ Ck(i) and xj = ◦ij (x) ◦ij (xi) j qc (x) = qc (xi) ∈ Ck(j): Consider the xi-end in the second three-dimensional partition Υ,

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

◦i (x ) j+1−ii(x) i j where Dl(xi) ∈ •l. Then qc : Dj+1−ii(x)(xi) → Ck(j) is a proper holomorphic 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. Since i Ck(i) is bounded by external rays landing at some pre-images of ﬁi under iterations of i qc and by equipotential curves of qc, Wij = gij(Wj) is contained in Ck(i). Thus

„ mod(W \ W ) ≥ : i ij 2

Consider Xi = gi,x(Wi) and Xj = gj,x(Wj) = gi,x(Wij). Then „ mod(X \ X ) ≥ i j 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 that „ mod(X \ X ) ≥ it it+1 2 ∞ ∞ for t ≥ 1. Let X = ∩t=1Xit . Then Xi1 = ∪t=1(Xit \ Xit+1 ) and

X∞

mod(Xi1 \ X) ≥ mod(Xit \ Xit+1 ) = ∞: t=1

Thus, X = {x} (following the Gr¨otzsch 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. Since it Di +1(x) = gi,x(C ), we have that t k(it)

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 \ Γ dynamically from other points in Jc. ¤ Our three-dimensional partition Σ is constructed from Ξ and Υ as follows. First ∞ consider the second three-dimensional partition Υ = {•j}j=1. For each domain D ∈ •j, 18 ◦s j there is a unique integer s ≥ 0 such that qc : D → Ck(j) is diﬀeomorphic. Now consider »j+1 ∈ Ξ. Then j j »j+1(Ck(j)) = »j+1 ∩ Ck(j)

j j ◦s j is a partition insider Ck(j). Pull back »j+1(Ck(j)) by qc : D → Ck(j) to D, we get a partition inside D, which we denote as »j+1(D). Let

j = tD∈κj »j+1(D):

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

∞ Σ = {j}j=1 the three-dimensional partition about Jc.

Deﬁnition 7. We say that the three-dimensional partition Σ determines points in Jc 0 0 dynamically if for any z 6= z ∈ Jc, there is an integer j ≥ 1 and domains z ∈ D; z ∈ 0 0 D ∈ j ∈ Σ such that D ∩ D = ∅. Combining Theorems 3 and 5, we have that 2 Theorem 6. Suppose qc(z) = z +c is a unbranched inﬁnitely renormalizable quadratic polynomial having complex bounds. Then the three-dimensional partition Σ determines points in Jc dynamically. 0 Proof. If one of z and z is in Jc \ Γ, it follows directly from the proofs of Theorems 4 and 5. If z ∈ Γ, from the proof of Theorem 3, there are the smallest integer l ≥ 0 and ◦l the x = qc (z)-end

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

th i+1 i+1 in the (i + 1) partition » ∈ Ξ such that the diameter d(Dn (x)) tends to zero as n goes to inﬁnity. Take z ∈ D ∈ •i. There is a smallest integer s ≥ 0 such that ◦s i q : D → Ck(i) is diﬀeomorphic. Therefore, there is a z-end in »i+1(D) ∈ Σ,

i+1 i+1 i+1 i+1 z ∈ · · · ⊆ Dn (z) ⊆ Dn−1(z) ⊆ · · · ⊆ D1 (z) ⊆ D0 (z);

i+1 such that the diameter d(Dn (z)) tends to zero as n goes to inﬁnity. This implies that if one of z and z0 in Γ, then the theorem still holds. ¤ §4.3 Local connectivity. The three-dimensional partition Σ for an inﬁnitely renormalizable quadratic poly- 2 nomial qc(z) = z + c provides a nice dynamical description of points in the Julia set Jc. One can expect that it is useful in the study of topological, geometrical, and measure-theoretical properties of qc. In this subsection, we show one application. 19 Lemma 1. For any domain D in Σ, D ∩ Jc is connected.

m Proof. Since the domain D is bounded by ﬁnitely many external rays Π = {eθj }j=1 and some equipotential curve for qc, then @D ∩ Jc consists of a ﬁnite number of points m0 {pi}j=1. Every pi is a landing point of two external rays in Π. Suppose D ∩ Jc is not connected for some D. Then there are two disjoint open sets X and Y such that D∩Jc = (D∩Jc ∩X)∪(D∩Jc ∩Y ). Suppose that p1, ::: , pm00 are in X and that pm00+1, ::: , pm0 are in Y . The two external rays in Π landing at pj cut C into two domains. 0 m00 One of them, denoted as Zj, is disjoint with D, i.e., D ∩ Zj = ∅. Then U = U ∪ ∪j=1Zj 0 m0 0 0 and V = V ∪ ∪j=m00+1Zj are two disjoint open sets, and Jc = (U ∩ Jc) ∪ (V ∩ Jc). This contradicts the fact that Jc is connected. ¤

2 Corollary B (Jiang [Ji2]). If qc(z) = z + c is a unbranched inﬁnitely renormalizable quadratic 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 domains Dn(z) from the three-dimensional partition Σ such that z ∈ Dn(z) ⊆ Dn−1(z) and the diameter d(Dn(z)) tends to 0 as n goes to inﬁnity. This fact with Lemma 1 implies that ∞ {Dn(z)}n=0 is a basis of connected neighborhoods about z. So Jc is locally connected at z. ¤

Corollary B indicates that unbranched and complex bounds conditions are important in the study of local connectivity of the Julia set of an infinitely renormalizable quadratic polynomial. A real infinitely renormalizable quadratic polynomial is always unbranched if it has complex bounds. Many people have worked out some results about complex bounds for real infinitely renormalizable quadratic polynomials. The Feigenbaum poly- 2 nomial is a real infinitely renormalizable quadratic polynomial q∞(z) = z + t∞, t∞ real, such that ◦2i fi(z) = q∞ (z): Ui → Vi is a sequence of simple renormalizations. Sullivan [Su] (see also [MS,Ji1]) proved that q∞ 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 locally connected.

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

Corollary D (Levins-van Strien [LS] and Lyubich-Yampolsky [LY). A real infinitely renormalizable quadratic polynomial has complex bounds. Therefore, its Julia set 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 −1 2 2 linear map h(z) = ez + s, we get h ◦ f ◦ h (z) = z + c. We call qc(z) = z + c the Douady-Hubbard family of quadratic polynomials. The Mandelbrot set M is the compact set of all parameters c in C such that the Julia set Jc of qc is connected. From Theorem B, M consists of all parameters c such that 0 has bounded orbit. We will often identity c with the corresponding polynomial qc.A point c ∈ M (really means qc) is called hyperbolic if and only if it has a unique attracting periodic orbit. Let {z0; ··· ; zp−1} be the attracting periodic orbit for a hyperbolic c and ◦p let ‚p(c) = qc (z0). The hyperbolic maps form an open subset in C. When c changes in one connected component H of this open set, the period p and the combinatorial type of the attracting periodic orbit are fixed. Here p is called the period of H. Moreover, ‚H(c) maps H to D1 = {z ∈ C | |z| < 1} conformally and can be extends uniquely to a homeomorphism ‚H(c): H → D1: −1 −1 Note that H has a unique center cH = ‚H (0). The point rH = ‚H (1) is called the root of H. For example, the cardioid H0 is the hyperbolic√ component of c such that qc has an attracting fixed point. Then ‚H0 (c) = 1 − 1 − 4c. The center is 0, the root is 1=4, and ‚−1 (e2πiθ) = e2πiθ(2 − e2πiθ)=4. H0 Consider a hyperbolic component H of period p ≥ 1 and the corresponding ‚H(c). m 2πi 0 0 For each r ∈ @H such that ‚H(r) = e p ,(m; p ) = 1, it is the root of a hyperbolic component H0 of period pp0. (Note that in the case m = 0 and p0 = 1, H0 = H, and in all other cases, H0 6= H.) Here H0 is called the satellite of H with internal angle m=p0 and denoted as H0 = H ∗ H(m=p0). For each c ∈ C, let

hc : Uc = {z ∈ C | Gc(z) > Gc(0)} → {z ∈ C | |z| > Gc(0)} be the B¨ottcher change of coordinate from §2, where Gc is the Green function for Kc. Then −1 2 hc ◦ qc ◦ hc (z) = z : 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

−1 Sr = ΦM ({c ∈ C | |c| = r}) and the external ray of angle 0 ≤ < 1 of M is

−1 Eθ = ΦM ({c ∈ C | |c| > 1 and arg(c) = 2…}):

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

(1) every external ray Eθ of rational angle = m=p lands at a point in M. 0 (2) If = 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 Misiurewicz point). Conversely, every Misiurewicz point is a landing point of an external ray 0 Eθ of angle = m=p with (m; p) = 1 and p = 2p . (3) For = m=p with (m; p) = 1 and p = 2p0, let c be the landing (Misiurewicz) point of Eθ, the external ray eθ in the z-plane for qc lands at its critical value of c. (Tan Lei [Ta] showed some similarity between M around a Misiurewicz c and corresponding Jc around c.) (4) Let H be a hyperbolic component of period p > 1, there are exactly two external − − p + rays Eθ− and Eθ+ land at the root rH where = m =(2 − 1) and = 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). Then + the closure of Eθ− ∪ Eθ cuts C into two connected components. Let Wm/p be the one containing H(m=p).

Theorem H (Goldberg-Milnor [GM]). For any c ∈ Wm/p, consider qc. There are exactly p external rays in the z-plane that land at the separating fixed point fic of qc. These p external rays have angles 2i+, i = 0; ··· ; p − 1. Moreover, c is in the domain bounded by two external rays of angle − and +. The main conjecture in this direction is that the Mandelbrot set M is locally connected. From the Carathéodory theorem, if M is locally connected, then ΦM can be extended to a homeomorphism from C \ M˚ to C \ D1, thus every external ray lands at a unique point in M. Yoccoz proved that M is locally connected at those points c such that qc are not infinitely renormalizable. Therefore, the study of the above conjecture is concentrated 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 let ‚ = P 0(p). Suppose there are totally m0 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 + k0)th( mod m0) external ray. Let r = gcd(m0; k0). Then there are r cycles of external rays landing at p. Let m0 = rm and k0 = rk,(m; k) = 1. The Yoccoz inequality says that there is a branch ¿ of log ‚ satisfying <¿ rm ≥ : k 2 2 log d |¿ − 2…i m | In other words, ¿ belongs to the closed disc of radius (log d)=(rm) tangent to the imagi- nary axis at 2…ik=m. Using this inequality, Yoccoz showed that M is locally connected at 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 generic subset of infinitely renormalizable quadratic polynomials whose Julia set is non-locally connected but M is locally connected at every point in this subset. They used a method called 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) and Eθ+(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 the Mandelbrot 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 ◦m(H) qc : U → V: Let H0 be a satellite of H of period p0. We can similarly construct W(H0) and M(H0) for H0 but we will denote them as W(H ∗ H0) and M(H ∗ H0). (Without having confusion, we also denote them as W(p ∗ p0) and M(p ∗ p0)). For every c ∈ W(p ∗ p0), ◦m(p) qc : U → V is renormalizable and has a simple renormalization ◦m(p)p0 0 0 qc : U ⊂ U → V ⊂ V: This processing is called tuning. Starting from the cardioid and a sequence of positive integers, 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 is locally 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 is dense on the boundary of M and consists of unbranched infinitely renormalizable points c having complex bounds (therefore, the Julia set Jc of qc is locally connected according to 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 such that (1) c is a limit point of A(c), 0 2 0 (2) for every c ∈ A(c), qc0 (z) = z + c is an unbranched infinitely renormalizable quadratic polynomial having complex bounds, (thus, the corresponding Julia set is locally connected according to Corollary B), and (3) the Mandelbrot set M is locally connected at every point c0 ∈ 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 (Rouch´eTheorem). Suppose D is a domain in the complex plane C. Let ⊂ D be a closed path homologous to 0 and assume that has an interior. Let f and g be analytic on D, and

|f(z) − g(z)| < |f(z)|; z ∈ :

Then f and g have the same number of zeros in the interior of . 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) and A(−2) consists of unbranched inﬁnitely renormalizable points having complex bounds at which M is locally connected. (Therefore, the corresponding Julia sets are all locally connected according to Corollary B).

2 Basic construction. Let q−2(z) = z − 2. Its Julia set J2 is [−2; 2]. It has the non- separate fixed point fl(−2) = 2 and the separate fixed point fi(−2), i.e., J−2 \{fl(−2)} is still connected and J−2 \{fi(2)} is disconnected. Let G0,l(−2) be the closure of the union of two external rays landing at fi(−2). Let h−2 be the Bötteche coordinate for q−2, i.e., −1 2 h−2 ◦ q−2 ◦ h−2(z) = z :

Then h−2 maps G0,l(−2) to two straight rays in C \ D1 (which have angles 1=3 and 2=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 > 0} 24 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:

n+1 n+1 Let Λn,r = gr,0 (Λ0,l) and Λn+1,l = gl,0 (Λn,r) for n = 0; 1; ··· . It is easy to see that Λn,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; ∞)):

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

∞ n n A(−2) = [∪n=0(gr,−2(ﬁ(−2)) ∪ gl,−2(ﬁ(−2)))] ∪ {−2; 2}:

−1 −1 We also denote Gn,r(−2) = h−2(Λn,r) and Gn,l(−2) = h−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 fi(c). Let fi(c) be another preimage of fi(c). Let G0,r(c) be the closure of union of two external rays landing at fi(c). Then G0,l(c) ∪ C0,r(c) cuts C into three domains. One we call Er(c) contains the non-separating fixed point fl(c) of qc. The other we call El(c) contains the other preimage fl(c) of fl(c) under qc. The restriction qc|(El,c(c) ∪ Er,c)(c) has two inverse branches

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

2 Let hc be the B¨ottcher changes of coordinate for qc(z) = z + c, i.e.,

−1 2 hc ◦ qc ◦ hc = z :

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

∞ n n A(c) = [∪n=0(gr,c(fi(c)) ∪ gl,c(fi(c)))] ∪ {fl(c); fl(c)}:

−1 −1 We also denote Gn,r(c) = hc (Λn,r) and Gn,l(c) = hc (Λn,l). Let −1 `c = hc ◦ h−2:

When restricted to G(−2), `c conjugates q−2|G(−2) to qc|G(c). (The other way to construct the graph G(c) from G(−2) for c close to −2 is by using the structure stability theorem. Since the graph G−2 does not contain 0, and since fi(−2), fl(2) and ∞ are only periodic points of q−2 on G(−2), for a small neighborhood Ω of G(−2), q−2|Ω is a hyperbolic endomorphism. From the structural stability theorem (see [Sh,Pr]), 1 any C -perturbation f of q−2|Ω is topologically conjugate to q−2|Ω, i.e., there is a homeomorphism ` :Ω → `(Ω) (close to the identity) such that f ◦ ` = ` ◦ q−2 on −1 q−2(Ω). Therefore, the graph G(−2) is preserved for qc as long as c is close to −2. Let 25 `c be the corresponding homeomorphism. Then G(c) = `c(G(−2)) is the corresponding graph.) Let U(−2) be a fixed domain bounded by an equipotential curve s(−2) for q−2. Then q−2 : U(−2) → V (−2) is a quadratic-like map where V (−2) = q−2(U(−2)). The set G0,l(−2) ∪ G0,r(−2) cuts U(−2) into three disjoint domains. Denote D0(−2) be the closure of the one containing 0 (see Fig. 4).

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

◦n ◦n Let Dn(−2) = gr,−2(D0(−2)) and let Bn(−2) = gl,−2(Dn−1(−2)) for n ≥ 1 (see Fig. 4). Since 2 is an expanding fixed point of q−2 and q(−2) = 2, the diameter diam(Bn) of Bn tends to zero exponentially as n goes to infinity. Let 0 6∈ U0 be a small neighborhood about −2 in the parameter space such that diam(U0) ≤ 1 and such that the corresponding graph G(c) for qc exists for c in U0. For c in U0, let `c : G(−2) → G(c) be the conjugacy from q−2 to qc. Let sc = `c(s) be the corresponding equipotential curve for 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 containing ◦n 0. Let Dn(c) = gr,c(D0(c)) and let Bn(c) = gl,c(Dn−1(c)) for n ≥ 1. Note that fl(c) and fi(c) are the non-separate and separate fixed points of qc and fl(c) is the other inverse image of fl(c) under qc. Then Bn(c) and Dn(c) tend to fl(c) and fl(c), respectively, as n goes to infinity. Since fl(c) is an expanding fixed point of qc and since there is a 0 constant „ > 1 such that |qc(fl(c))| ≥ „ for all c in U0, the diameter diam(Bn(c)) tends to 0 uniformly on U0 and the set Bn(c) approaches to fl(c) uniformly on U0 as n goes to 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) − ﬂ(c) and g(c) = qc(0) − x for x ∈ Bn(c). Suppose = @U0 is a closed path homologous to 0 in U0 such that

m = min = min |qc(0) − fl(c)| > 0: c∈∂U0 c∈∂U0 26 Because Bn(c) approaches fl(c) uniformly on U0 as n goes to infinity, there is an integer N0 > 0 such that for n ≥ N0,

|f(c) − g(c)| = |ﬂ(c) − x| < m ≤ |f(c)|; c ∈ @U0:

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

Fig. 5

Lemma 2. The intersection M˜ n = 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 pieces −n ◦(n+1) of the equipotential curve sn(c) = qc (s(c)). Note that qc (Gn,l(c)) = G0,l(c) and ◦n qc (Gn−1,r(c)) = G0,l(c). The domain Wn is bounded by the closure of two external rays 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 and an−1. The closure of Gn (or Gn−1) in the extended complex plane is a topological curve isomorphic to a circle and it intersects M only at an (or an−1). Let us still denote these extended curves as Gn and Gn−1. Then Gn and Gn−1 cut C into three domains. The one on an side is called Xn and the one on an−1 side is called Xn−1. 27 If M˜ n is disconnected. Let U and V be two non-empty domains such that U ∩ V = ∅ and such that (U ∩ M˜ n) ∪ (V ∩ M˜ n) = M˜ n:

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˜ = ∅ and such that (U˜ ∩ M) ∪ (V˜ ∩ M) = M:

This would say that M is disconnected. This contradiction implies that M˜ n must be connected. ¤ −1 Proof of Theorem 8. For each Wn where n ≥ N0, since c ∈ Bn(c), Cn(c) = qc (Bn(c)) is the closure of a connected and simply connected domain which contains 0 and which is a sub-domain of D0(c). Let

◦(n+1) ˚ ˚ fn,c = qc : Cn(c) → D0(c):

Then it is a quadratic-like map (see Fig. 6). Furthermore, since diam(Bn(c)) tends to zero as n goes to inﬁnity uniformly on U0, for N0 large enough we can have that the modulus mod(An(c)) ≥ 1 where An(c) = D˚0(c) \ Cn(c). Moreover, the family

{fn,c : C˚n(c) → D˚0(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 ∈ A = ∪∞ M ; q is once renormalizable. 1 n≥N0 n c

Fig. 6

For a fixed integer i0 ≥ N0, consider Wi0 and Mi0 ; there is a parameter ci0 ∈ Mi0 such that f = f : C˚ = C˚ (c ) → D˚ = D˚ (c ) is hybrid equivalent to i0 i0,ci0 i0 i0 i0 i0 0 i0 2 ˚ ˚ q−2(z) = z − 2. The quadratic-like map fi0 : Ci0 → Di0 has the non-separate fixed 28 point and the separate fixed point, which we simple denote as fl and fi. Let fl be another pre-image of fl under fi0 . Let G be the closure of the union of two external rays for q landing at fi. Then ci0 f (G) = G and G ∩ J = {fi}. Let G˜ = f −1(G). Then G˜ cuts C into three domains. i0 fi0 i0 i0

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

◦n Di0n = gi01(Di00) and Bi0n = gi00(Di0(n−1)) for n ≥ 1. Again by the Böttcher changes of coordinate h and h (or the structure ci0 c ˜ stability theorem), we can similarly prove that fl, fl, fi, and G, G, and @Ci0 are preserved for c close to ci0 . Similar to the argument in the above, we can find a small neighborhood

Ui0 about ci0 with diam(Ui0 ) ≤ 1=2 such that the corresponding domains Bi0 (c) and

Di0 (c) can be constructed for qc for c ∈ Ui0 . Let

Wi0n = {c ∈ C | fi0,c(0) ∈ Bi0n(c)}:

Then {Wi0n}i0≥1,n≥1 give a sub-partition of {Wn}n≥1.

The diameter diam(Bi0n(c)) tends to zero uniformly on Ui0 and the set Bi0n(c) approaches to ﬂ(c) uniformly on Ui0 as n goes to inﬁnity. Suppose Ui0 is simply connected and the boundary curve = @Ui0 is a closed path homologous to 0 in Ui0 .

Since the equation fi0,c(0) − fli0 (c) = 0 has a unique solution ci0 (following Thurston’s theorem for critically finite rational maps (see [DH4]) and also refer to [DH3]), m = min |f (0) − fl (c)| > 0. Let f(c) = f (0) − fl (c) and g(c) = f (0) − x for c∈∂Ui0 i0,c i0 i0,c i0 i0,c x ∈ Bi0n(c). There is an integer Ni0 > 0 such that for n ≥ Ni0 ,

|f(c) − g(c)| = |ﬂi0 (c) − x| < m ≤ |f(c)|; c ∈ @Ui0 :

Theorem I (Rouch´eTheorem) now implies that g(c) = fi0,c(0) − x = 0 has a unique solution in Ui0 for any x in Bi0n(c) for n ≥ Ni0 . Therefore Wi0n ⊆ Ui0 for n ≥ Ni0 . So diam(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 W , n ≥ N , let C (c) = f −1 (B (c)). Then i0n i0 i0n i0,c i0n

f = f ◦(n+1) : C˚ (c) → D˚ (c) i0n,c i0,c i0n i00

is a quadratic-like map. Furthermore, since diam(Bi0n(c)) tends to zero as n goes to inﬁnity 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 ∈ Wi0n} 29 is a full family. Thus Wi0n contains a copy Mi0n of the Mandelbrot set M (refer to [Mi2, pp 102-106]). For c ∈ A = ∪ ∪ M , the Julia set of f : C˚ (c) → 2 i0≥N0 i1≥Ni0 i0i1 i0n,c i0n ˚ 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 that ˚ ˚ ˚ ˚ 2 fw = fw,cw : Cw = Cw(cw) → Dw = Dv0(cw) is hybrid equivalent to q−2(z) = z − 2. The quadratic-like map fw : C˚w → D˚w has the non-separate fixed point flw and the separate fixed point fiw. Let Gw be the closure of the union of two external rays for ˜ −1 ˜ qcw which land at fiw. Let Gw = fw (Gw). Then Gw cuts Cw into three domains. Let Dw0 be the one containing 0. Denote flw be another preimage of flw under fw. Let flw ∈ Ew0 and flw ∈ Ew1 be the components of the closure of Cw \ Dw0. Let gw0 and gw1 be the inverses of fw|Ew0 and fw|Ew1. Let

◦n Dwn = gw1(Dw0) and Bwn = gw0(Dw(n−1))

for n ≥ 1. By the Böttcher changes of coordinate hcw and hc (or the structure stability ˜ theorem), flw, flw, fiw, and Gw, Gw, and @Cw are preserved for c close to cw. We can find k a small neighborhood Uw about cw with diam(Uw) ≤ 1=2 such that the corresponding domains 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) approaches to flw(c) uniformly on Uw as n goes to infinity. Suppose Uw is simply connected and the boundary curve @Uw is a closed path homologous to 0. Since the equation fw,c(0) − flw(c) = 0 has a unique solution cw (following from Thurston’s theorem for critically finite rational maps (see [DH4]) and also refer to [DH3]),

m = min |fw,c(0) − ﬂw(c)| > 0: c∈γ

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

|f(c) − g(c)| = |ﬂw(c) − x| < m ≤ |f(c)|; c ∈ @Uw:

Now Theorem I (the Rouch´eTheorem) implies that g(c) = fw,c(0) − x = 0 has a unique solution in Uw for any x in Bwn(c) and n ≥ Nw. Therefore Wwn ⊆ Uw for n ≥ Nw. So k diam(Wwn) ≤ 1=2 for n ≥ Nw. 30 Similar to the proof of Lemma 2, we have M˜ wn = Wwn ∩M for n ≥ Nw is connected. −1 For each c in Wwn where n ≥ Nw, let Cwn(c) = fw,c(Bwn(c)). Then

◦(n+1) ˚ ˚ fwn,c = fw,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) = D˚w0(c) \ Cwn(c). Moreover,

{fwn,c : C˚wn(c) → D˚w0(c) | c ∈ Wwn} is a full family. Therefore, Wwn contains a copy Mwn ⊆ M˜ wn of M (refer to [Mi2, pp,

102-106]). For c ∈ Ak+1 = ∪w ∪ik≥Nw Mwik , qc is (k + 1)-times renormalizable where w = i0i1 : : : ik−1 runs over all sequences of integers of length k. We thus construct a three-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 has complex 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. ◦m Then there is the smallest integer m ≥ 1 such that p = qc0 (0) is a repelling periodic point of qc0 of period k ≥ 1. We start the construction of our subset A(c0) and a three-dimensional partition {Ww | w = i0i1 ··· in; n = 0; 1; ···} around it as follows.

Let fi be the separating fixed point of qc0 . Let G be the closure of the union of ∞ external rays landing at fi. Let » = {·n}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) ∈ ·n. 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) ∈ ·n. We have ◦(m−1) qc0 (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 such ◦k 0 ◦(m−1) that |(qc0 ) (x)| ≥ ‚ > 1 for all x ∈ Dl(p) and such that qc0 : El+m−1(c0) → Dl(p) ◦t is a homeomorphism. Let t ≥ 0 be the integer such that f = qc0 : Dl(p) → Cr0 is 31 a homeomorphism, where Cr0 is the domain containing 0 in ·r0 , r0 ≥ 0. There is an −1 integer r > r0 such that r + t > l and such that B0 = f (Cr ∩ Dl(p)) does not contain ◦t p. Thus qc0 : B0 → Cr is a homeomorphism. Deﬁne

³ ´−1 ◦nk Bn = qc0 |Dl+nk(p) (B0) ⊆ Dl+nk(p)

◦(q+nk) for n ≥ 1. Note that Bn is in ·r+q+nk. Then qc0 : Bn → Cr is a homeomorphism.

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

m Wn = Wn(c0) = {c ∈ C | qc (0) ∈ Bn(c)}; n ≥ 1:

They give a partition in the parameter space. Suppose U0 is simply connected and the boundary is a closed path homologous to m 0 in U0. Since the equation qc (0) − p(c) = 0 has a unique solution c0 in U0 (following Thurston’s Theorem for critically ﬁnite rational maps (see [DH4]) and also ◦m ◦m refer to [DH3]), m = minc∈γ |qc (0) − p(c)| > 0. Let f(c) = qc (0) − p(c) and ◦m g(c) = qc (0) − x for x ∈ Bn(c). Since Bn(c) approaches p(c) uniformly on U0 as n 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:

◦m Theorem I (the Rouch´eTheorem) now implies that g(c) = qc (0) − x = 0 has a unique solution, 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 implies that, for n ≥ N0, M˜ n = 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

◦(m−1) qc0 : El+m−1(c) → Dl(p; c)

−1 and let Cm+r+q+nk(c) = qc (Rn(c)). Then Cm+r+q+nk(c) is the closure of a domain which contains 0 and which is in ·m+r+q+nk. Hence

◦(q+nk+m) ˚ ˚ fn,c = qc : Cm+r+q+nk(c) → Cr(c) 32 is a quadratic-like map. Furthermore, since diam(Bn(c)) tends to zero as n goes to inﬁnity uniformly on U0, for N0 large enough we can have the modulus mod(An(c)) ≥ 1 where An(c) = C˚r(c) \ Cm+r+q+nk(c). Moreover,

{fn,c : C˚m+r+q+nk(c) → C˚r(c) | c ∈ Wn} is a full family. Thus Wn contains a copy Mn = Mn(c0) of the Mandelbrot set M ˜ where Mn ⊆ Mn (refer to [Mi2, pp 102-106]). For any c ∈ A1(c0) = ∪n≥N0 Mn; qc is once 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 construct 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 infinitely renormalizable at which M is locally connected. Furthermore, qc is unbranched and has complex bounds from our construction. Therefore the Julia set Jc is locally connected following Corollary B. It completes the proof. ¤ Remark 4. Eckmann and Epstein [EE] and Douady-Hubbard [DH3] have estimated the size of Mn. Since M˜ n \Mn contains hairs (see Fig. 7) which may destroy the local connectivity of M, we must estimate the size of M˜ n. This is the key point in the proof of Theorems 7 and 8. Lyubich [Ly2] announced another subset consisting of infinitely renormalizable points such that M is locally connected at every point in this subset. A point in his subset must satisfies several complicate conditions. His idea is more close to Douady and Hubbard’s tuning idea (see §5.1). Our idea is different and simple. A point in our subset may not satisfy those conditions. In our proof, the use of the Rouché 33 Theorem is interesting. Actually, the three-dimensional partition in the parameter space can be constructed more generally following the construction of the three-dimensional partition in the phase space for every infinitely renormalization quadratic polynomial. However, finding a proper tool to replace the RouchéTheorem in the proof of Theorems 7 and 8 is an interesting problem.

References

[Ah] L. V. Ahlfors, Lectures on Quasiconformal Mappings, vol. 10, Van Nostrand Mathematical studies, D. Van Nostrand Co. Inc., Toronto-New York-London, 1966. [Bo1] R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970), 725–747. [Bo2] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin-New York, 1975. [Bo3] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin-New York, 1975. [BH1] B. Branner and J. Hubbard,, The iteration of cubic polynomials. I. The global topology of parameter space, Acta Math 160, no. 3-4 (1988), 143–206. [BH2] B. Branner and J. Hubbard,, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math 169, no. 3-4 (1992), 229–325. [CG] L. Carleson and T. Gamelin, Complex Dynamics. Universitext:Tracts in Mathematics, Springer- Verlag, New York, 1993. [DH1] A. Douady and J. H. Hubbard, On the dynamics of polynomial-like maps, Ann. Sci. Ecole.´ Norm. Sup (4) 18, no. 2 (1985), 287–344. [DH2] A. Douady and J. H. Hubbard, Itération des polynômesquadratiques complexes, C. R. Acad. Sci. Paris 294 (1982), 123-126. [DH3] A. Douady and J. H. Hubbard, Etude´ dynamique des polynômescomplexes I& II, Publ. Math. Orsay (1984-85). [DH4] A. Douady and J. H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), 263-297. [EE] J. Eckmann and H. Epstein, Scaling of Mandelbrot sets by critical point preperiodicity, Comm. Math. Phys. 101 (1985). [MG] L. Goldberg and J. Milnor, Fixed points of polynomial maps, Part II: fixed point portaits, Ann. Sci. Ec.¨ Norm. Sup. 26 (1993), 51-98. [HJ] J. Hu and Y. Jiang, The Julia set of Feigenbaum quadratic polynomial. Proceedings of the International Conference in Honor of Professor Shantao Liao (edited by Jiang and Wen) (1999), World Scientific Publishing Co. Inc., River Edge, NJ, 99-124. [Hu] J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J. -C. Yoccoz. Topological Methods in Modern Mathematics (Stony Brook, NY, 1991) (1993), Publish or Perish, Houston, TX, 467–511. [Ji1] Y. Jiang, Renormalization and Geometry in One-Dimensional and Complex Dynamics, vol. 10, Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. [Ji2] Y. Jiang, Infinitely renormalizable quadratic polynomials, Trans. Amer. Math. Soc., 352, no 11 (2000), 5077-5091. [Ji3] Y. Jiang, Markov partitions and Feigenbaum-like mappings, Comm. Math. Phys. 171 (1995), 351-363. [La] S. Lang, Complex Analysis, 4th Edition, Springer-Verlag, New York, 1999. 34 [LS] G. Levin and S. van Strien, Local connectivity of the Julia sets of real polynomials, Ann. of Math (2). 147, no. 3 (1998), 471–541. [Ly1] M. Lyubich, On the Lebesgue measure of the Julia set of a quadratic polynomial. Stony Brook IMS Preprint 1991/10. [Ly2] M. Lyubich, Geometry of quadratic polynomials: Moduli, rigidity, and local connectivity. Stony Brook IMS Preprint 1993/9. [LY] M. Lyubich and M. Yampolsky, Dynamics of quadratic polynomials: complex bounds for real maps, Ann. Inst. Fourier (Grenoble) 47, no. 4 (1997), 1219–1255. [Mc1] C. McMullen, Complex Dynamics and Renormalization, vol. 135, Annals of Mathematics Stud- ies, Princeton Univ. Press, Princeton, NJ, 1994. [Mc2] C. McMullen, Renormalization And 3-Manifolds Which Fiber over the Circle, vol. 142, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1996. [MS] W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer-Verlag, Berlin, 1993. [Mi1] J. Milnor, Dynamics in One Complex Variable, 2nd Edition, Vieweg & Sohn, 2000. [Mi2] J. Milnor, Local connectivity of Julia sets: expository lectures in The Mandelbrot Set, Theme and Variations (edited by Tan Lei) (2000), Cambridge University Press, Cambrigde, 67-116.. [Pr] F. Przytycki, On U-stability and structural stability of endomorphisms satisfying Axiom A, Studia Math 60 (1977), 61–77. [Ro] P. Rœsch, Holomorphic motions and puzzles (following M. Shishikura) in The Mandelbrot Set, Theme and Variations (edited by Tan Lei) (2000), Cambridge University Press, Cambrigde, 117-131. [Sc] D. Schleicher, Rational parameter rays of the Mandelbrot set, Stony Brook I.M.S Preprint 1997/#13. [Sk] M. Shishikura, Private conversation in 1991. [Sh] M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175– 199. [Si1] Ya. G. Sinai, Markov partitions and C-diffeomorphisms (Russian), Funkcional Anal. i Prilov zen 2, no. 1 (1968), 64–89. [Si2] Ya. G. Sinai, Construction of Markov partitions (Russian), Funkcional Anal. i Prilov zen 2, no. 3 (1968), 70–80 (loose errata). [Sm] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. [Su] D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathe- matical Society Centennial Publications 2 (1988), Mathematics into the Twenty-First Century, AMS, Providence, RI, 417–466. [Ta] L. Tan, Similarity between the Mandelbrot set and Julia sets, Comm. Math. Phys. 134 (1990), 587–617.

Department of Mathematics, Queens College of CUNY, Flushing, NY 11367 and De- partment of Mathematics, CUNY Graduate Center, 365 Fifth Avenue New York, NY 10016 E-mail address: [email protected]