THE RENORMALIZATION METHOD AND QUADRATIC-LIKE MAPS

YUNPING JIANG

Abstract. The renormalization of a quadratic-like map is studied. The threedimensional Yoccoz puzzle for an infinitely renormalizable quadratic-like map is discussed. For an unbranched quadratic-like map having the a priori complex bounds, the local connectivity of its Julia set is proved by using the three-dimensional Yoccoz puzzle. The generalized version of Sullivan’s sector theorem is discussed and is used to prove his result that the Feigenbaum quadratic polynomial has the a priori complex bounds and is unbranched. A dense subset on the boundary of the Mandelbrot set is constructed so that for every point of the subset, the corresponding quadratic polynomial is unbranched and has the a priori complex bounds.

Contents

0. Introduction

1. Quadratic Polynomials, Quadratic-Like Maps, and Hyperbolic Geometry

2. Renormalizable Quadratic-Like Maps

3. Two-Dimensional Yoccoz Puzzles and Renormalizability

4. Infinitely Renormalizable Quadratic Julia Sets and Three-Dimensional Yoccoz Puzzles

5. A Generalized Sullivan’s Sector Theorem

6. Feigenbaum-Like Quadratic-Like Maps

7. The Local Connectivity of Certain Infinitely Renormalizable Quadratic Julia Sets

1991 Mathematics Subject Classification. Primary: 58F23, 54D05, Secondary: 30D05. Research is partially supported by an grant from the NSF, by awards from the PSC-CUNY, and,

at MSRI, by NSF grant # DMS-9022140.

1. Introduction

Let P_{c}(z) = z^{2} + c be a quadratic polynomial. A central problem in the study of the dynamics of P_{c} is to understand the topology and geometry of the Julia set J_{c} of P_{c}. The filled-in Julia set K_{c} of P_{c} is, by definition, the set of points not going to infinity under iterations of P_{c}. The Julia set J_{c} of P_{c} is, by definition, the boundary of K_{c}.

In order to have the more penetrating study of the dynamics of a quadratic polynomial P_{c}, Douady and Hubbard [DH3] introduced the concept of a quadratic-like map. In §1, we review the definition of a quadratic-like map and the work of Douady and Hubbard [DH3] which proves that a quadratic-like map with connected Julia set is hybrid equivalent to a unique quadratic polynomial. We also review the result due to Douady and Yoccoz (see [HUB, MI2]) about the landing of external rays at a repelling periodic point of a quadratic polynomial. In the same section, we also review some fundamental results about the Julia set of a quadratic-like map and some basic facts of hyperbolic geometry.

The Mandelbrot set M is the set of complex parameters c such that the Julia set J_{c} of P_{c} is connected. The Julia set J_{c} in the complement of M is a Cantor set. Douady and Hubbard [DH1] proved that M is connected. They further conjectured that M is locally connected. Many important research problems follow from this conjecture. For example, the hyperbolicity conjecture (explained below) would follow from this conjecture [DH2]. A quadratic polynomial is hyperbolic if it has an attractive or super-attractive periodic point in the complex plane. Let HP be the set of parameters c such that P_{c} is hyperbolic. The hyperbolicity conjecture says that HP is open and dense in M. To study the local connectivity of the Mandelbrot set M, it would be helpful to answer the question: for which c in M is the Julia set J_{c} locally connected ? Yoccoz made substantial progress in this direction. He proved that if P_{c} is nonrenormalizable (or finitely renormalizable), then the Julia set J_{c} is locally connected. We discuss his result in §3. (Using this result, Yoccoz further proved that M is locally connected at a finitely renormalizable point c (see [HUB])). Also in §3, we discuss the two-dimensional Yoccoz puzzle of a quadratic-like map and its relation with the renormalizability of this quadratic-like map.

There remain many points in M which are infinitely renormalizable. In §2, we define infinitely renormalizable quadratic-like maps and define infinitely renormalizable folding mappings, and discuss the relation between the two definitions.

In §4, we prove one of our main results: for an unbranched infinitely renormalizable quadratic-like map having the a priori complex bounds, its Julia set is locally connected. We prove this result by using the three-dimensional Yoccoz puzzle of an infinitely renormalizable quadratic-like map. Also in §4, we prove that the filled-in Julia set of any renormalization of a renormalizable quadratic-like map, about the period of the two-dimensional Yoccoz puzzle, does not depend on the choices of renormalization domains. In particular, the renormalized filled-in Julia set is the limiting component in two-dimensional Yoccoz puzzle containing the critical point.

In §5, we discuss Sullivan’s sector theorem. We prove a generalized version. The proof repeatedly applies the sharpest version of Koebe’s distortion theorem (see [BIE]), and uses Sullivan’s idea about using hyperbolic contraction to trap points in a hyperbolic neighborhood.

Using the generalized version of Sullivan’s sector theorem, we prove in §6 his result that the Feigenbaum quadratic polynomial has the a priori complex bounds and is unbranched. By combining this with the result in §4, we complete the proof of the result which was first announced in [JIH] and which says that the Julia set of the Feigenbaum quadratic polynomial is locally connected.

In §7, we construct a subset Υ̃ of the Mandelbrot set which is dense on the boundary ∂M of the Mandelbrot set M such that for every point c in this subset, the corresponding quadratic polynomial P_{c}(z) = z^{2}+c is unbranched and infinitely renormalizable and has the a priori complex bounds. Thus, for c in Υ̃, the Julia set J_{c} of a quadratic polynomial P_{c}(z) = z^{2} + c is locally connected. A similar result concerning about the local connectivity of the Mandelbrot set at infinitely renormalizable points is proved in [JI4].

This article benefited from reading J. Hubbard’s paper [HUB] and J. Milnor’s paper [MI2], from talking with D. Sullivan, and from listening the talk given by J.-C. Yoccoz in Denmark. During the writing of this article, several e-mails from C. McMullen provided great help. Some statements in this article are more precise because of his suggestions. Also from his e-mails, I decided to include the papers [JI1,JIH,JI2,JI3] into this self-contained article. During my study of the work of Sullivan, F. Gardiner and J. Hu provided a lot help. I thank them all. This research was partially carried out [JI1] when the author was in the IMS at Stony Brook. This research is partially supported by an grant from the NSF and by awards from the PSC-CUNY. Research at MSRI is supported in part by NSF grant # DMS-9022140.

2. Quadratic Polynomials, Quadratic-Like Maps, and Hyperbolic Geometry

Let P_{c}(z) = z^{2} + c be a quadratic polynomial. Let C be the complex plane. Let V = {z ∈ C | |z| < r} be a disk in C. For r large enough, U = P_{c}^{−1}(V) is a simply connected domain, its closure is relatively compact in V, and P_{c} : U → V is a holomorphic, proper map of degree two (see Fig. 1). This is a model of an object defined by Douady and Hubbard [DH3].

Definition 1. A quadratic-like map is a triple (U, V, f) where U and V are simply connected domains isomorphic to a disc with \overline{U} ⊂ V, and where f : U → V is a holomorphic, proper map of degree two (see Fig. 1).

Remark 1. A proper map f means that f^{−1}(K) is compact for every compact set K. Douady and Hubbard [DH3] also defined a polynomial-like map.

Suppose (U, V, f) is a quadratic-like map. We use

K_{f} = ∩_{n=0}^{∞}f^{−n}(U)

to denote the set of z in U such that images of z under iterations of f are all in U. The set K_{f} is called the filled-in Julia set. It is a compact subset of U. The Julia set J_{f} of f is the boundary of K_{f}.

A quadratic-like map (U, V, f) has only one branched point b at which the derivative f′(b) of f equals zero. We call b the critical point of f and, without loss of generality, we always assume that b = 0. The following theorem is well-known. Theorem 1. Suppose (U, V, f) is a quadratic-like map. If the critical point 0 is not in K_{f}, then K_{f} = J_{f} is a Cantor set of zero Lebesgue measure in C. And moreover, the set K_{f} (or J_{f}) is connected if and only if the critical point 0 is in K_{f}.

Suppose that Ω is a domain in C and that f is a self-map of Ω. A point z in Ω is called a periodic point of period k ≥ 1 if f^{∘k}(z) = z and f^{∘i}(z) ≠ z for 1 ≤ i < k. The number λ = (f^{∘k})′(z) is called the multiplier (or eigenvalue) of f at z. A periodic point of period 1 is called a fixed point. A periodic point z of f is said to be repelling, attractive, or neutral if |λ| > 1, 0 < |λ| < 1, or |λ| = 1. Moreover, a periodic point is called super-attractive if λ = 0, and is called parabolic if λ = e^{2πip/q} where p and q are integers. For an attractive or super-attractive or parabolic periodic point of period k ≥ 1 of a quadratic-like map f : U → V, let O(p) = {f^{∘i}(p)}_{i=0}^{k−1} be the periodic orbit of f. The set

B(p) = {z ∈ V | f^{∘n}(p) → O(p) as n → ∞}

is called the basin of O(p). Let CB(f^{∘i}(p)) be the connected component containing f^{∘i}(p) of B(p). The set IB(p) = ∪_{i=0}^{k−1}CB(f^{∘i}(p)) is called the immediate basin of O(p). The proofs of the following two theorems can be found in Blanchard’s survey article [BLA].

Theorem 2. Suppose z_{0} is a super-attractive periodic point of period k of a holomorphic function f defined on Ω. Then there is a neighborhood U of z_{0}, a unique holomorphic diffeomorphism h : U → h(U) with h(z_{0}) = 0 and h′(z_{0}) = 1, and a unique integer n > 1 such that

h ∘ f^{∘k} ∘ h^{−1}(z) = z^{n}

on h(U).

Theorem 3. Let (U, V, f) be a quadratic-like map and let J_{f} = ∂K_{f} be its Julia set. Let E_{f} be the set of all repelling periodic points of f. Then

(1) J_{f} is completely invariant, i.e., f(J_{f}) = J_{f} and f^{−1}(J_{f}) = J_{f};

(2) J_{f} is perfect, i.e., J′_{f} = J_{f}, where J′_{f} means the set of limit points of J_{f};

(3) E_{f} is dense in the Julia set J_{f}, i.e., \overline{E_{f}} = J_{f};

(4) for any z in V, the limit set of {f^{−n}(z)}_{n=0}^{∞} is J_{f};

(5) J_{f} has no interior point;

(6) If f has an attractive or super-attractive or parabolic periodic point p in V, then the immediate basin IB(p) contains the critical point 0 and the critical orbit O(0) = {f^{∘n}(0)}_{n=0}^{∞};

(7) If f has neither any attractive, nor any super-attractive, nor any neutral periodic points in V, then K_{f} = J_{f}.

Let (U, V, f) and (U′, V′, g) be two quadratic-like maps. They are topologically conjugate if there is a homeomorphism h from a neighborhood K_{f} ⊂ X ⊂ U to a neighborhood K_{g} ⊂ Y ⊂ U′ such that h ∘ f = g ∘ h on X where K_{f} and K_{g} are filled-in Julia sets. If h is quasiconformal (see [AH1]) (respectively, holomorphic), then they are quasiconformally (respectively, holomorphically) conjugate. If h can be chosen such that h\overline{z} = 0 a.e. on K_{f}, then they are hybrid equivalent. Let

I(f) = {g | g is hybrid equivalent to f}

be the inner class of f. The following theorem is due to Douady and Hubbard.

Theorem 4 [DH3]. If (U, V, f) is a quadratic-like map such that K_{f} is connected, then there is a unique quadratic polynomial P (z) = z^{2} + c_{f} in I(f).

Let \overline{C} = C ∪ \{∞\} be the extended complex plane. Then ∞ is a super-attractive fixed point of any quadratic polynomial P_{c}(z) = z^{2} + c. The filled-in Julia set K_{c} of P_{c} is the set of all points not going to infinity under iterations of P_{c}. Let D_{r} = {z ∈ C | |z| < r}. As we mentioned in the beginning of this section, for r > 1 large enough, U = P_{c}^{−1}(D_{r}) is a simply connected domain and \overline{U} ⊂ D_{r}.

Thus (U, V, P_{c}) for V = D_{r} is a quadratic-like map. In particular, (D_{r}, D_{r^{2}}, P_{0}) is a quadratic-like map for every r > 1; the filled-in Julia set K_{0} of P_{0} is the closed unit disk \overline{D}_{1}. Applying Theorem 2, there is a holomorphic diffeomorphism h_{1} defined on a neighborhood \overline{C} \ \overline{D}_{r} (for r > 1 large) about ∞ such that h_{1}(∞) = ∞, h′_{1}(∞) = 1, and such that

h_{1}^{−1} ∘ P_{c} ∘ h_{1}(z) = z^{2}

on \overline{C}\\overline{D}_{r}. Let B_{1}(∞) = h_{1}(\overline{C}\\overline{D}_{r}) and let B_{n}(∞) = P_{c}^{−(n−1)}(B_{1}(∞)). If the filled-in Julia set K_{c} of P_{c} is connected, then all

P_{c} : B_{n}(∞) ∩ C → B_{n−1}(∞) ∩ C

are unramified covering maps of degree two. We can inductively define holomorphic diffeomorphisms h_{n} on \overline{C} \ \overline{D}_{r^{\frac{1}{2^{n}}}} such that

h_{n}^{−1} ∘ P_{c} ∘ h_{n}(z) = z^{2}

for z in \overline{C} \ \overline{D}_{r^{\frac{1}{2^{n}}}} and for n > 1. As n tends to infinity, we get a holomorphic diffeomorphism h_{∞} defined on \overline{C} \ \overline{D}_{1} such that

h_{∞}^{−1} ∘ P_{c} ∘ h_{∞}(z) = z^{2}  (1)

for all z in \overline{C} \ \overline{D}_{1}. Therefore, B_{c}(∞) = h_{∞}(\overline{C} \ \overline{D}_{1}) is the basin of ∞ for P_{c} and K_{c} = \overline{C}\B_{c}(∞). Furthermore, for every r > 1 and for U_{r} = h_{∞}(D_{r}), (U_{r}, U_{r^{2}}, P_{c}) is a quadratic-like map and its filled-in Julia set is always K_{c}. Let S^{R} = {z ∈ C | |z| = R} and let s_{R} = h_{∞}(S^{R}) for R > 1. Then

P_{c}(s_{R}) = s_{R^{2}}.  (2)

The topological circle s_{R} for every R > 1 is called an equipotential curve of P_{c}. A curve

e_{θ} = h_{∞}(\{z ∈ \overline{C} | |z| > 1, arg(z) = θ\})

for 0 ≤ θ < 2π is called an external ray of P_{c}. Then

P_{c}(e_{θ}) = e_{2θ}.  (3)

Remark 2. Let

G(z) = max\{0, lim_{n→∞} \frac{1}{2^{n}} log |P_{c}^{∘n}(z)|\}

be the Green’s function of K_{c} in \overline{C}. Then G(P_{c}(z)) = 2G(z). For any R > 1, the equipotential curve s_{R} = G^{−1}(log R) is a level curve of G.

If the mapping h_{∞} in Equation (1) can be extended continuously to the unit circle S^{1}, then we have a unique continuous map H : \overline{C} \ D_{1} → \overline{C} \ K̊_{f} such that H|(\overline{C}\\overline{D}_{1}) = h_{∞}. Using H, we can define an equivalence relation on S^{1}: z_{1} ∼ z_{2} if and only if H(z_{1}) = H(z_{2}). Let [z] be the equivalent class of z. Then P̃_{0}([z]) = [P_{0}(z)] defines a map of the quotient space X = S^{1}/ ∼, since z_{1}^{2} ∼ z_{2}^{2} if z_{1} ∼ z_{2}. The dynamical system (P̃_{0}, X) is topologically conjugate to (P_{c}, J_{c}) by H̃([z]) = H(z). The question arises:

Question 1. For which c can h_{∞} be extended continuously to the unit circle S^{1} ?

A connected set X in C is locally connected if for any point p in X and any neighborhood V about p, there is another neighborhood U ⊂ V about p such that U ∩ X is connected. The following classical theorem proved by Carathéodory in one complex variable gives a sufficient and necessary condition to extend h_{∞} continuously to S^{1}. The proof of this theorem can be found in [MI1].

Theorem 5 [CAR]. Let h be a Riemann mapping from D_{1} onto a simply connected open domain Ω. Then h can be extended continuously to the unit circle S^{1} if and only if the boundary ∂Ω (as well as Ω) is locally connected.

Remark 3. If ∂Ω is a Jordan curve, then h can be extended to a homeomorphism from \overline{D}_{1} onto \overline{Ω}. Moreover, if ∂Ω is made of finite number of analytic curves, then the extension restricted to the unit circle S^{1} has non-zero derivative at every point other than a corner (see [BIE]).

We have an equivalent question by just concerning the topology of a Julia set now:

Question 2. For which c is J_{c} locally connected ?

An external ray e_{θ} lands at J_{c} if e_{θ} has only one limit point at J_{c}. An external ray is periodic with period m if e_{θ} ∩ P_{c}^{∘i}(e_{θ}) = {∞} for 1 ≤ i < m and if P_{c}^{∘m}(e_{θ}) = e_{θ}. The following theorem is proved by Douady and Yoccoz (refer to [MI1,MI2,HUB]).

Theorem 6. Let P_{c}(z) = z^{2} + c be a quadratic polynomial with connected Julia set J_{c}. Then every repelling periodic point of P_{c} is a landing point of finitely many periodic external rays with the same period.

Let S be a Riemann surface and let (S̃, π) be the universal cover of S, where S̃ is a simple connected Riemann surface and π : S̃ → S is the universal covering map. From the Uniformization Theorem (see [AH2]), we can identify S̃ with one of the extended complex plane \overline{C}, the complex plane C, or the open unit disk D_{1}. A Riemann surface S is hyperbolic if S̃ = D_{1}.

Let D be the hyperbolic disk which is the open unit disk with the hyperbolic metric

d_{H}s = \frac{|dz|}{1 − |z|^{2}} .

Let d_{H} be the hyperbolic distance. Every holomorphic diffeomorphism h : D_{1} → D_{1}, which is not a linear fractional transformation, strictly decreases the hyperbolic distance d_{H}, i.e,

d_{H}(h(z_{1}), h(z_{2})) < d_{H}(z_{1}, z_{2})

for all z_{1} and z_{2} in D. For a hyperbolic Riemann surface S, one can define the hyperbolic distance d_{H,S} from d_{H} and π. Any holomorphic map h : S → S, which is not an isometry with respect to d_{H,S}, strictly decreases this hyperbolic distance d_{H,S}, i.e., for any z_{1} and z_{2} in S,

d_{H,S}(h(z_{1}), h(z_{2})) < d_{H,S}(z_{1}, z_{2}).

A bounded domain Ω in C is a hyperbolic Riemann surface. An important family of hyperbolic Riemann surfaces is the family of bounded doubly connected domains in C. A bounded doubly connected domain Ω is called an annulus. From complex analysis (see [BIE]), any annulus is holomorphically diffeomorphic to a unique round annulus A_{r} = {z ∈ C | r < |z| < 1} for 0 < r < 1. The number

mod(Ω) = − log r

is called the modulus of Ω. It is a conformal invariant, i.e., mod(h(Ω)) = mod(Ω) whenever h is a conformal homeomorphism from Ω onto h(Ω).

Let Ω be a bounded doubly connected domain in C. Then the complement C \ Ω of Ω in C has two components. One is a connected and simply connected compact set E and the other is an unbounded set F. From the Grötzsch argument (see [AH1]), the bounded component E is a single point if and only if mod(Ω) = ∞ (see [BRH]).

Let Ω be an annulus. If Ω_{1} ⊆ Ω is a subannulus, then

mod(Ω_{1}) ≤ mod(Ω).  (4)

If Ω_{1}, Ω_{2} ⊆ Ω are two disjoint subannuli, then

mod(Ω_{1}) + mod(Ω_{2}) ≤ mod(Ω).  (5)

The proofs of these two inequalities can be found in the book of Ahlfors [AH1].

Let E be a connected and simply connected compact subset of the open unit disk D_{1}. Let

mod(D_{1}, E) = sup_{Ω}{mod(Ω), where Ω ⊆ D_{1} \ E is a round subannulus }.

Note that mod(D_{1}, E) = ∞ if E is a single point.

Let diam_{H}(E) = sup_{z_{1},z_{2}∈E} d_{H}(z_{1}, z_{2}) be the hyperbolic diameter of E in D_{1}. The following theorem can be found in the book of McMullen [MC1].

Theorem 7. The hyperbolic diameter d_{H}(E) and the mod(D_{1}, E) are inversely related:

diam_{H}(E) → 0 ⟺ mod(D_{1}, E) → ∞

and

diam_{H}(E) → ∞ ⟺ mod(D_{1}, E) → 0.

More precisely, there is a constant C > 0 such that

C^{−1}diam_{H}(E) ≤ exp(−mod(D_{1}, E)) ≤ Cdiam_{H}(E)

when diam_{H}(E) is small, while

\frac{C}{diam_{H}(E)} ≥ mod(D_{1}, E) ≥ C^{−1} exp(−diam_{H}(E))

when dima_{H}(E) is large.

3. Renormalizable Quadratic-Like Maps

Let (U_{0}, V_{0}, f_{0}) be a quadratic-like map and suppose its filled-in Julia set K_{f_{0}} is connected. The map f_{0} : U_{0} → V_{0} is renormalizable if there are an integer n ≥ 2 and an open subdomain U_{1} of U_{0} such that 0 ∈ U_{1} and such that (U_{1}, V_{1}, f_{0}^{∘n}) is a quadratic-like map with connected filled-in Julia set, where V_{1} = f^{∘n}(U_{1}). Let f_{1} = f_{0}^{∘n}|U_{1}. The filled-in Julia set K_{f_{1}} = K_{f_{1}}(n, U_{1}) is a priorily dependent on the choice of U_{1} (refer to Theorem 13). The domain U_{1} is called a renormalization and (U_{0}, V_{0}, f_{0}) is called renormalizable about n. Otherwise, (U_{0}, V_{0}, f_{0}) is called non-renormalizable.

The quadratic-like map (U_{0}, V_{0}, f_{0}) is infinitely renormalizable if there is a strictly increasing sequence of integers {n_{k}}_{k=1}^{∞} such that (U_{0}, V_{0}, f_{0}) is renormalizable about n_{k} for k = 1, 2, . . .. Otherwise, (U_{0}, V_{0}, f_{0}) is called finitely renormalizable.

A quadratic-like map (U_{0}, V_{0}, f_{0}) is called real if f_{0}(U_{0} ∩ R^{1}) ⊂ V_{0} ∩ R^{1} and g = f_{0}|U_{0} ∩ R^{1} is a real folding map. Let (U_{0}, V_{0}, f_{0}) be a real quadratic-like map and suppose its filled-in Julia set K_{f_{0}} is connected. Suppose g = f_{0}|U_{0} ∩ R^{1} has a fixed point p ∈ R^{1} with positive multiplier f′_{0}(p). Let p′ ≠ p be another inverse image of p under g, that is, g(p′) = p. Conjugating by a linear fractional transformation, we may assume that p = −1 and p′ = 1 and that

[−1, 1] = ∩_{n=0}^{∞}g^{−n}(V_{0} ∩ R^{1}) = K_{f} ∩ R^{1}.

Hence g is a folding map with unique quadratic critical point 0.

Let g be a folding map of [−1, 1] such that g(−1) = g(1) = −1 and such that 0 is a unique quadratic critical point. We say g is renormalizable about n > 1 if there is a subinterval I of [−1, 1] such that 0 ∈ I̊, such that g^{∘i}(I) ∩ I̊ = ∅ for all 0 < i < n, and such that g^{∘n}(I) ⊆ I. Otherwise, g is non-renormalizable. We say g is infinitely renormalizable if there is a strictly increasing sequence {n_{k}}_{k=1}^{∞} such that g is renormalizable about n_{k} for all k > 0. Otherwise, g is called finitely renormalizable. The next theorem shows that for a real quadratic-like map, both definitions of renormalization are essentially equivalent.

Theorem 8. Let (U_{0}, V_{0}, f_{0}) be a real quadratic-like map and suppose its filled-in Julia set K_{f_{0}} is connected. Suppose f_{0} has neither neutral, nor attractive, nor superattractive periodic points. Then f_{0} is renormalizable if and only if the folding map g = f_{0}|[−1, 1] is renormalizable.

Proof. Suppose the folding map g = f_{0}|[−1, 1] is renormalizable. This means that there is a maximal closed subinterval I of [−1, 1] and an integer n > 1 such that 0 is in I̊, such that g^{∘i}(I) ∩ I̊= ∅ for 0 < i < n, and such that g^{∘n}(I) ⊆ I. One of the endpoints of I is fixed by g^{∘n}. It is a repelling fixed point. Take a neighborhood T′ of I such that f^{∘n}|(L∪R) is expanding, where L∪R = T′\I. Let I_{i} = g^{∘i}(I) for 0 ≤ i ≤ n. The inverse h_{n} of g^{∘(n−1)} : I_{1} → I_{n} is a diffeomorphism and can be extended to a diffeomorphism on an open interval T ⊃ I. Take T ⊂ T′. Let M′ = h_{n}(T) and M = g^{−1}(M′). Then \overline{M} is a subset of T. Because the critical orbit CO = \{f_{0}^{∘k}(0)\}_{k=0}^{∞} of f_{0} is in the real line R^{1}, h_{n} can be extended to V_{1} = (V_{0} \ R^{1}) ∪ T analytically. The image V′_{1} of V_{1} under this extension is contained in U′ = (U_{0} \ R^{1}) ∪ M′. Let U_{1} = f_{0}^{−1}(V′_{1}). Then \overline{U}_{1} ⊂ V_{1} and (U_{1}, V_{1}, f_{1}) for f_{1} = f_{0}^{∘n}|U_{1} is a quadratic-like map. Since g^{∘n}(I) ⊆ I, the filled-in Julia set K_{f_{1}} is connected. Therefore, (U_{0}, V_{0}, f_{0}) is renormalizable about n and U_{1} is a renormalization.

Suppose (U_{0}, V_{0}, f_{0}) is renormalizable about n > 1. Let U_{1} be a renormalization and set V_{1} = f^{∘n}(U_{1}). Then (U_{1}, V_{1}, f_{1}) for f_{1} = f_{0}^{∘n}|U_{1} is a quadratic-like map with the connected filled-in Julia set K_{f_{1}}. Let I = K_{f_{1}} ∩ R^{1}. Then I = ∩_{i=0}^{∞}f_{1}^{−i}(V_{1} ∩ R^{1}). For every 1 ≤ i < n, f^{∘i}(I) ∩ I̊ = ∅ else f^{∘n} would have at least three fixed points in I ∪ f^{∘i}(I) with one of them either attractive or parabolic. Since f_{1}(0) is in I, f_{0}^{∘n}(I) ⊆ I. Therefore, g = f_{0}|[−1, 1] is a renormalizable folding map.

4. Two-Dimensional Yoccoz Puzzles and Renormalizability

In this section, we discuss a technique in the study of non-renormalizable quadratic polynomials, due to Yoccoz, and some of its applications to renormalization.

Let P_{c}(z) = z^{2} + c be a quadratic polynomial with connected filled-in Julia set K_{c}. The external ray e_{0} of P_{c} is the only one fixed by P_{c} (see Fig. 2). It lands either at a repelling or at a parabolic fixed point β of P_{c} (see [MI1]). Suppose β is repelling. Applying Theorem 6, we see that e_{0} is the only external ray landing at β. Thus K_{c} \{β} is connected. We call β the non-separate fixed point of P_{c}. Let α ≠ β be the other fixed point of P_{c}. If α is either an attractive or a super-attractive fixed point, then J_{c} = K_{c} \ (∪_{n=0}^{∞}P_{c}^{−n}(D(α)) for a small disk centered at α. The Julia set J_{c} is a Jordon curve; every external ray lands at a unique point in J_{c} (see Remark 3). If α is a repelling fixed point, there are at least two periodic external rays landing at α. We use R_{0}(α) to denote the union of a cycle of periodic external rays of period q landing at α (see Fig. 2). The set R_{0}(α) cuts C into finitely many simply connected domains Ω_{0}, Ω_{1}, …, Ω_{q−1}. Each domain contains points in the Julia set J_{c}. Thus K_{c} \ {α} is disconnected. We call α the separate fixed point of P_{c}.

Henceforth, we assume that the fixed points β and α are both repelling. Let s_{r} be a fixed equipotential curve of P_{c} and let U_{r} be the open domain bounded by s_{r}. Then (U\sqrt{r}, U_{r}, P_{c}) is a quadratic-like map. The set R_{0}(α) cuts U_{r} into finitely many simply connected domains. Let C_{0} be the closure of the domain containing 0, and let B_{0,i} be the closure of the domain containing P_{c}^{∘i}(0) for 1 ≤ i < q. Since R_{0}(α) is forward invariant under P_{c}, the image under P_{c} of C_{0} ∩ K_{c} or B_{0,i} ∩ K_{c}, for every 1 ≤ i < q, is the union of some of C_{0} ∩ K_{c}, B_{0,1} ∩ K_{c}, …, B_{0,q−1} ∩ K_{c}. The set η_{0} = {C_{0}, B_{0,1}, …, B_{0,q−1}} is called the original partition. We note that it is not a Markov partition because P_{c}|C_{0} is a proper, holomorphic map of degree two. (But P_{c}|B_{0,i} is a holomorphic diffeomorphism for every 1 ≤ i < q.)

Let Γ_{n} = P_{c}^{−n}(α) and let R_{n}(α) = P_{c}^{−n}(R_{0}(α)) for n ≥ 0. The set R_{n}(α) is the union of some external rays landing at points in Γ_{n}; it cuts the domain U_{r^{\frac{1}{2^{n}}}} into a finite number of simply connected domains. Let C_{n} be the closure of the domain containing 0 and let B_{n,1}, …, B_{n,k_{n}} be the closures of others. Since P_{c}(Γ_{n}) = Γ_{n−1}, the image of C_{n} or B_{n,i} under P_{c} is one of C_{n−1}, B_{n−1,1}, …, B_{n−1,k_{n−1}}, for 1 ≤ i ≤ k_{n} and n ≥ 1. Then P_{c}|C_{n} is holomorphic, proper branch covering map of degree two; all P_{c}|B_{n,i} are holomorphic diffeomorphisms. The set η_{n} = {C_{n}, B_{n,1}, …, B_{n,k_{n}}} is called the n^{th}-partition. The sequence η = {η_{n}}_{n=0}^{∞} is called the two-dimensional Yoccoz puzzle for P_{c}. A similar puzzle for certain cubic polynomial is constructed by Branner and Hubbard [BRH]. Yoccoz used this puzzle while studying the local connectivity of a non-renormalizable quadratic polynomial as follows.

Let Γ_{∞} = ∪_{n=0}^{∞}Γ_{n}. For any x in K_{c} \ Γ_{∞}, there is one and only one sequence {D_{n}(x)}_{n=0}^{∞} such that x ∈ D_{n}(x) ∈ η_{n}. For any x in Γ_{∞}, there are q such sequences. We call such a sequence

x ∈ ⋯ ⊆ D_{n}(x) ⊆ D_{n−1}(x) ⊆ ⋯ ⊆ D_{1}(x) ⊆ D_{0}(x)

an x-end. In particular,

0 ∈ ⋯ ⊆ C_{n} ⊆ C_{n−1} ⊆ ⋯ ⊆ C_{2} ⊆ C_{1} ⊆ C_{0}

is called the critical end.

Suppose D_{n+1} ⊆ D_{n} are domains in η_{n+1} and η_{n} for n ≥ 0. Define A_{n} = D_{n} \ D̊_{n+1}. If D_{n+1} ⊂ D̊_{n}, then Å_{n} is a non-degenerate annulus and its modulus (denoted mod(A_{n})) is greater than 0. Otherwise, A_{n} is a degenerate annulus and its modulus mod(A_{n}) is zero. The domain D_{n} is critical if D_{n} = C_{n} and D_{n+1} = C_{n+1}; it is semi-critical if D_{n} = C_{n} but D_{n+1} ≠ C_{n+1}; it is non-critical if D_{n} ≠ C_{n}. If D_{n} is non-critical, then

mod(A_{n}) = mod(P_{c}(A_{n}))

since P_{c} : A_{n} → P_{c}(A_{n}) is a conformal homeomorphism. If D_{n} is critical, then

mod(A_{n}) = \frac{mod(P_{c}(A_{n}))}{2}

since P_{c} : A_{n} → P_{c}(A_{n}) is a proper, holomorphic, unramified covering map of degree two. If D_{n} is semi-critical, then

\frac{mod(P_{c}(A_{n}))}{2} ≤ mod(A_{n}).

For a point x in K_{c} and an x-end

x ∈ ⋯ ⊆ D_{n}(x) ⊆ D_{n−1}(x) ⊆ ⋯ ⊆ D_{1}(x) ⊆ D_{0}(x),

we define

D_{nm}(x) = P_{c}^{∘m}(D_{n+m}(x))(= D_{n}(P_{c}^{∘m}(x)))

and

A_{nm}(x) = D_{nm}(x) \ D̊_{(n+1)m}(x)

for n, m ≥ 0. The tableau T (x) = (a_{nm})_{n≥0,m≥0} is an ∞ × ∞-matrix defined as follows: a_{nm} = 1 if D_{nm}(x) = C_{n}, and a_{nm} = 0 if D_{nm}(x) ≠ C_{n}. The tableau

T (0) = (a_{nm}^{0})_{n≥0,m≥0} is called the critical tableau. Note that

P_{c}^{∘m}(x) ∈ ⋯ ⊆ D_{nm}(x) ⊆ D_{(n−1)m}(x) ⊆ ⋯ ⊆ D_{1m}(x) ⊆ D_{0m}(x)

is a P_{c}^{∘m}(x)-end.

Lemma 1. The tableau T (x) satisfies the following rules:

(T1) if a_{nm} = 1 for n, m ≥ 0, then a_{im} = 1 for all 0 ≤ i ≤ n,

(T2) if a_{nm} = 1 for n, m ≥ 0, then a_{(n−i)(m+j)} = a_{(n−i)j}^{0} for all 0 ≤ i + j ≤ n,

(T3) if i) a_{nm} = 1 and a_{(n+1)m} = 0 for n, m ≥ 0 and if ii) a_{(n−i)(m+i)} = 1 for 1 ≤ i ≤ n and a_{(n−j)(m+j)} = 0 for 0 < j < i,

then a_{(n−i+1)i}^{0} = 1 implies a_{(n−i+1)(m+i)} = 0.

Proof. Rule (T1) is valid because

0 ∈ ⋯ ⊆ C_{n} ⊆ C_{n−1} ⊆ ⋯ ⊆ C_{1} ⊆ C_{0}.

Rule (T2) is valid because if a_{nm} = 1, then D_{im}(x) = C_{i} for all 0 ≤ i ≤ n. Thus D_{(n−i)(m+j)}(x) = f^{∘j}(C_{n−i}) for all 0 ≤ i + j ≤ n.

We prove Rule (T3). Conditions i) and ii) imply that P_{c}^{∘i} : C_{n} → C_{n−i} is a degree two holomorphic proper branch covering map. Condition i) and Rule (T2) imply a_{k(m+j)} = a_{kj}^{0} for 0 ≤ k + j ≤ n. Now a_{(n−i+1)i}^{0} = 1 and a_{j0}^{0} = 1 for all 0 ≤ j < ∞ imply that P_{c}^{∘i} : C_{n+1} → C_{n−i+1} is a degree two holomorphic proper branch covering map. Thus C_{n+1} = P_{c}^{−i}(C_{n−i+1}) ∩ C_{n}. Assume a_{(n−i+1)(m+i)} = 1. Then D_{(n+1)m}(x) = P_{c}^{−i}(C_{n−i+1}) ∩ C_{n} = C_{n+1}. This contradicts a_{(n+1)m} = 0.

Lemma 2. For any domain D in η_{n}, for n ≥ 0, D ∩ K_{c} is connected.

Proof. Since the domain D is bounded by finitely many external rays Π = {e_{θ_{i}}}_{i=1}^{m} and by some equipotential curve, then ∂D ∩ K_{c} consists of a finite number of points {p_{i}}_{i=1}^{n}. Every p_{i} is a landing point of two external rays in Π. Suppose D ∩ K_{c} is not connected for some D in η_{n}. Then there are two disjoint open sets U and V such that D ∩ K_{c} = (D ∩ K_{c} ∩ U) ∪ (D ∩ K_{c} ∩ V). Suppose that p_{1}, …, p_{k} are in U and that p_{k+1}, …, p_{n} are in V. The two external rays in Π landing at p_{i} cut C into two open domains. Let W_{i} be the one which is disjoint from D. Then U′ = U ∪ ∪_{i=1}^{k}W_{i} and V′ = V ∪ ∪_{i=k+1}^{n}W_{i} are two disjoint open sets and K_{c} = (U′ ∩ K_{c}) ∪ (V′ ∩ K_{c}). This contradicts the fact that K_{c} is connected.

Lemma 2 implies that for any x-end

x ∈ ⋯ ⊆ D_{n}(x) ⊆ D_{n−1}(x) ⊆ ⋯ ⊆ D_{1}(x) ⊆ D_{0}(x),

the intersection L_{x} = ∩_{n=0}^{∞}D_{n}(x) is a compact connected non-empty set containing x. Let T (x) = (a_{nm})_{n≥0,m≥0} be the tableau of the x-end. It is non-recurrent if there is an integer N ≥ 0 such that a_{nm} = 0 for all n ≥ N and all m ≥ 1. Otherwise, T (x) is recurrent.

Lemma 3. If T (x) is non-recurrent, then L_{x} = {x}.

Proof. Suppose N ≥ 0 is an integer such that a_{nm} = 0 for all n ≥ N and all m ≥ 1. Then, for n > N, every

P_{c}^{∘(n−N−1)} : D_{n1}(x) → D_{N(n−N)}(x)

is a holomorphic diffeomorphism. Thus mod(A_{n0}(x)) is greater than or equal to mod(A_{N(n−N)}(x))/2 for every n > N. There are only finitely many different annuli in {A_{Nm}(x)}_{m=0}^{∞} because η_{N} has only finitely many domains. If there were infinitely many non-degenerate annuli in {A_{Nm}(x)}_{m=0}^{∞}, then there would be infinitely many non-degenerate annuli in {A_{n0}(x)}_{n=0}^{∞} whose moduli are the same. This would imply that

mod(D_{0}(x) \ L_{x}) ≥ ∑_{n=0}^{∞} mod(A_{n0}(x)) = ∞.

Therefore L_{x} = {x}.

If there are only finitely many non-degenerate annuli in {A_{Nm}(x)}_{m=0}^{∞}, The proof uses results from hyperbolic geometry. Let B_{N1} be the domain in η_{N} containing the critical value P_{c}(0). Since P_{c}^{∘i}(x) do not enter C_{N+1} for all 0 < i < ∞, then P_{c}^{∘i}(x) does not enter B_{N1} for all 2 ≤ i < ∞. Let us thicken B_{Ni} to an open simply connected domain B̃_{Ni} such that B_{Ni} ⊂ B̃_{Ni} and such that P_{c}(0) is not in B̃_{Ni} for 1 < i ≤ k_{N}. The map P_{c} has two inverse branches g_{i1} and g_{i2} defined on B̃_{Ni} for every 1 < i ≤ k_{N}. we consider B̃_{Ni} to be a hyperbolic Riemann surface with hyperbolic distance d_{H,i} for every 1 < i ≤ k_{N}, where k_{N} is the number of elements in η_{N}. Then if g_{ik}, for 1 < i ≤ k_{N} and k = 1 or 2, sends B_{Ni} into B_{Nj} for some 1 < j ≤ k_{N}, then it strictly contracts these hyperbolic distances; more precisely, there is a constant 0 < λ < 1 such that d_{H,j}(g_{ik}(x), g_{ik}(y)) < λd_{H,i}(x, y) for x and y in B_{Ni} and for k = 0 and 1. Therefore, there is a constant C > 0 such that for any D_{n1}(x) and for any n > N,

d(D_{n1}(x)) = max_{y,z∈D_{n0}(x)} |y − z| ≤ Cλ^{n−N−1}

since D_{(n−i)i}(x) is in one of B_{N2}, …, B_{Nk_{N}} for every 2 ≤ i ≤ n − N. Thus d(D_{n0}(x)) tends to zero as n goes to infinity and L_{x} = {x}.

The critical end

0 ∈ ⋯ ⊆ C_{n} ⊆ C_{n−1} ⊆ ⋯ ⊆ C_{2} ⊆ C_{1} ⊆ C_{0}

is important. Let A_{n0}(0) = C_{n} \ C̊_{n+1}.

Lemma 4. If ∑_{n=0}^{∞} mod(A_{n0}(0)) = ∞, then for any x in K_{c} and any x-end

x ∈ ⋯ ⊆ D_{n}(x) ⊆ D_{n−1}(x) ⊆ ⋯ ⊆ D_{1}(x) ⊆ D_{0}(x),

L_{x} = {x}.

Proof. Consider the tableau T (x) = (a_{nm})_{n≥0,m≥0}. If T (x) is non-recurrent, the lemma follows from Lemma 3.

Suppose T (x) is recurrent. If there is a column which is entirely 1’s, then there are integers M ≥ 0 and N ≥ 0 such that a_{iM} = 1 for all i ≥ 0 and a_{nm} = 0 for all n ≥ N and 0 ≤ m < M. Thus P_{c}^{∘M} : D_{n0}(x) → D_{(n−M)M} = C_{n−M} is a holomorphic diffeomorphism for every n ≥ N. This implies

m(D_{0}(x) \ L_{x}) ≥ ∑_{n=0}^{∞} mod(A_{n0}(x)) ≥ ∑_{n=N}^{∞} mod(A_{n0}(x)) = ∑_{n=N−M}^{∞} mod(A_{n0}(0)) = ∞.

So L_{x} = {x}.

Suppose that there is no column which is entirely 1’s. Let N > 0 be an integer such that a_{n0} = 0 for n ≥ N. For any n ≥ N, let m_{n} > 0 be the integer such that a_{nm_{n}} = 1 and a_{ni} = 0 for 0 ≤ i < m_{n}. Then P_{c}^{∘m_{n}} : D_{(n+m_{n}−1)0}(x) → D_{(n−1)m_{n}}(x) = C_{n−1} is a holomorphic diffeomorphism. Remember that A_{(n−1)0}(x) = D_{(n−1)0}(x) \ D̊_{n0}(x) and A_{(n−1)0}(0) = C_{n−1} \ C̊_{n}. We have

mod(A_{(n+m_{n}−1)0}(x)) = mod(A_{(n−1)0}(0)).

Let q_{n} = n + m_{n} − 1. Then q_{N} < q_{N+1} < ⋯ < q_{n} < q_{n+1} < · · ·. Thus

mod(D_{0}(x) \ L_{x}) ≥ ∑_{n=N}^{∞} mod(A_{n0}(x)) ≥ ∑_{n=N}^{∞} mod(A_{q_{n}0}(x))

= ∑_{n=N}^{∞} mod(A_{(n−1)0}(0)) = ∞.

This implies that L_{x} = {x}.

The first column of the critical tableau T (0) = (a_{nm}^{0})_{n≥0,m≥0} is entirely 1’s. If T (0) has another column which is entirely 1’s, that is, if there is an integer m > 0 such that a_{im} = 1 for all i ≥ 0, then we call T (0) a periodic critical tableau.

Theorem 9 [YOC]. The critical tableau T (0) is periodic if and only if P_{c} is renormalizable.

Proof. Suppose T (0) is periodic. Let n_{1} > 0 be the smallest integer such that a_{in_{1}} = 1 for all i ≥ 0. Let N ≥ 0 be the smallest integer such that a_{ij} = 0 for all i ≥ N and 0 < j < n_{1}. For any n ≥ n_{1} + N, P_{c}^{∘n_{1}} : C_{n} → C_{n−n_{1}} is a degree two proper holomorphic branch covering map. Thus {P^{∘kn_{1}}(0)}_{k=0}^{∞} is contained in C_{n_{1}+N}. If C_{n_{1}+N} ⊂ C̊_{N}, then P_{c}^{∘n_{1}} : C̊_{n_{1}+N} → C̊_{N} is a quadratic-like map with connected filled-in Julia set and C̊_{n_{1}+N} is a renormalization about n_{1}. Thus P_{c} is renormalizable.

In general, let us consider a small open disk D(α) centered at the separate fixed point α of P_{c} such that

\overline{D}(α) ⊂ D′(α) = P_{c}^{∘n_{1}}(D(α))

and such that

D′(α) ∩ {P_{c}^{∘i}(0)}_{i=0}^{n_{1}+N} = ∅.

Thicken C_{0} and B_{0i} for 1 ≤ i ≤ q − 1 as follows. Suppose C_{0} (respectively, B_{0i}) is bounded by two external rays R_{θ_{1}} and R_{θ_{2}} of angles θ_{1} and θ_{2}. Let ϵ > 0 be a small number such that the domains

U_{1} = ∪_{θ_{1}−ϵ<θ<θ_{1}+ϵ}(R_{θ} \ D_{α}) and U_{2} = ∪_{θ_{2}−ϵ<θ<θ_{2}+ϵ}(R_{θ} \ D_{α})

are disjoint from {P_{c}^{∘i}(0)}_{i=0}^{n_{1}+N}. Let

C̃_{0} = (U_{1}∪C_{0}∪U_{2}∪D(α))∩U_{r} (respectively, B̃_{0i} = (U_{1}∪B_{0i}∪U_{2}∪D(α))∩U_{r})

where U_{r} is the domain bounded by the equipotential curve s_{r}. Let

η̃_{0} = {C̃_{0}, B̃_{01}, …, B̃_{0(q−1)}}

and let

η̃_{n} = P^{−n}(η̃_{n}) = {C̃_{n}, B̃_{n1}, …, B̃_{nk_{n}}}

for 1 ≤ n ≤ n_{1} + N. The diffeomorphism g = P_{c}^{−(n_{1}−1)} : C_{N} → P_{c}(C_{n_{1}+N}) can be extended to C̃_{N}. Let B′ be the image of C̃_{N} under g. Then C̃_{n_{1}+N} = P_{c}^{−1}(B′). Let ˜C̊_{n} denote the interior of C̃_{n} for 0 ≤ n ≤ n_{1} + N. Then C̃_{n_{1}+N} ⊂ ˜ C̊_{N}. Thus

P_{c}^{∘n_{1}} : ˜ C̊_{n_{1}+N} → ˜ C̊_{N}

is a quadratic-like map and ˜ C̊_{n_{1}+N} is a renormalization about n_{1}. This proves the “only if” part.

Now suppose P_{c} is renormalizable. Let U_{1} be a renormalization about n_{1}, that is, (U_{1}, V_{1}, f_{1}) is a quadratic-like map with connected filled-in Julia set K_{f_{1}} where f_{1} = P_{c}^{∘n_{1}}|U_{1} and V_{1} = f_{1}(U_{1}). The map f_{1} has two fixed points β_{1} and α_{1} in U_{1}. Let

α_{1} ∈ ⋯ ⊆ D_{n}(α_{1}) ⊆ D_{n−1}(α_{1}) ⊆ ⋯ ⊆ D_{1}(α_{1}) ⊆ D_{0}(α_{1})

be an α_{1}-end. There is a D_{k}(α_{1}) such that K_{f_{1}} ⊂ D_{k}(α_{1}) and D_{k}(α_{1}) = C_{k}. Since

K_{f_{1}} ⊆ f_{1}^{−1}(U_{1} ∩ C_{k}) ⊆ U_{1} ∩ C_{k},

then P_{c}^{∘n_{1}} sends C_{k+in_{1}} to C_{k+(i−1)n_{1}} for all i > 0. Thus T (0) is periodic. It is the “if” part.

We define a function τ on the set N of natural numbers by using the critical tableau T (0) = (a_{nm}^{0})_{n≥0,m≥0} as follows: τ(n) = m if a_{(n−i)i}^{0} = 0 for 0 < i < n − m and if a_{m(n−m)}^{0} = 1; if there is no such integer m > 0, then τ(n) = −1.

If the critical tableau T (0) is periodic, then there are integers n_{1} > 0 and N ≥ 0 such that a_{in_{1}} = 1 for all i ≥ 0 and such that a_{ij} = 0 for i ≥ N and 0 < j < n_{1}. Thus τ(n) = n − n_{1} for n ≥ N + n_{1}.

If the critical tableau T (0) is non-recurrent, then there is the smallest integer N ≥ 0 such that a_{nm} = 0 for all n ≥ N and m > 0. Thus the image τ(N) is contained in the finite set {−1, 0, 1, …, N − 1}.

If the critical tableau T (0) is not periodic and is recurrent, then every row of T (0) has infinitely many 1’s and every column (except for the 0^{th}-column) has a 0. An integer n ≥ 0 is noble if for every entry a_{nk}^{0} such that a_{nk}^{0} = 1, we have a_{(n+1)k}^{0} = 1.

Lemma 5. If the critical tableau T (0) is not periodic and is recurrent, then the function τ satisfies the following properties:

(i) For any integer m ≥ 0, τ^{−1}(m) is not empty.

(ii) If m ≥ 0 is noble, then τ^{−1}(m) contains at least two different integers.

(iii) If τ(n) = m and if m is noble, then n is also noble.

(iv) If τ^{−1}(m) contains only one integer n, then n is noble.

Proof. We prove (i) first. Consider any m-row in T (0) for m ≥ 0. Let k > 0 be the integer such that a_{mi}^{0} = 0 for 0 < i < k and such that a_{mk}^{0} = 1. From (T1), a_{(m+k−i)i}^{0} = 0 for 0 < i < k. Thus τ(m + k) = m.

To prove (ii), suppose m ≥ 0 is noble. Let k be the same integer as that in the proof of (i). Let m_{1} be the integer such that a_{m_{1}k}^{0} = 1 and such that a_{ik}^{0} = 0 for all i > m_{1}. Consider a_{(m_{1}−k)(2k)}^{0}, a_{(m_{1}−2k)(3k)}^{0}, …, and a_{(m_{1}−(i−1)k)(ik)}^{0} where m_{1} − ik ≤ m < m_{1} − (i − 1)k. From the tableau rules (T1) and (T3), a_{(m_{1}−k+1)(2k)}^{0} = 0, a_{(m_{1}−2k+1)(3k)}^{0} = 0, …, a_{(m_{1}−(i−1)k+1)(ik)}^{0} = 0. If m = m_{1} − ik, from the tableau rules (T1) and (T3), a_{m(ik)}^{0} = 1. Since m is noble, a_{(m+1)(ik)}^{0} = 1. But from the tableau rules (T1) and (T3), a_{(m+1)(ik)}^{0} = 0. The contradiction implies that m > m_{1} − ik.

Now from the tableau rule (T2), a_{m(k+m_{1}−m)}^{0} = 0. Let k_{1} > k + m_{1} − m be the integer such that a_{mi}^{0} = 0 for k + m_{1} − m < i < k_{1} and such that a_{mk_{1}}^{0} = 1. Then a_{(m+k_{1}−i)i}^{0} = 0 for 0 < i < k_{1} and a_{mk_{1}}^{0} = 1. This says that τ(m + k_{1}) = m.

To prove (iii), suppose τ(n) = m where m is noble. For any a_{nk}^{0} = 1, since a_{(n−i)i}^{0} = 0 for 0 < i < n − m and a_{m(n−m)}^{0} = 1 and since the tableau rule (T1), we have a_{(n−i)(k+i)}^{0} = 0 for 0 < i < n − m and a_{m(k+n−m)}^{0} = 1. Since m is noble, a_{(m+1)(n−m)}^{0} = 1. Assume a_{(n+1)k}^{0} = 0. From the tableau rule (T3), a_{(m+1)(k+n−m)}^{0} = 0. This contradicts to that m is noble. Thus a_{(n+1)k}^{0} = 1. This means that n is noble.

Now we prove (iv). Suppose n > 0 is the only integer such that τ(n) = m. We first consider a_{(m+1)(n−m)}^{0}. If a_{(m+1)(n−m)}^{0} = 0, then we would have an integer k > n − m such that a_{mi}^{0} = 0 for n − m < i < k and such that a_{mk}^{0} = 1. From the tableau rule (T1), a_{(m+k−i)i}^{0} = 0 for 0 < i < k. This would imply that τ(m + k) = m, which contradicts the assumption. Thus, a_{(m+1)(n−m)}^{0} = 1. If there is an entry a_{nk_{1}}^{0} = 1 with a_{(n+1)k_{1}}^{0} = 0, then k_{1} > n − m and, from the tableau rule (T3), a_{m(n−m+k_{1})}^{0} = 1 and a_{(m+1)(n−m+k_{1})}^{0} = 0. Consider the smallest integer k_{2} > n − m + k_{1} such that a_{mi}^{0} = 0 for k_{1} + n − m < i < k_{2} and a_{mk_{2}}^{0} = 1. From the tableau rule (T1), a_{(m+k_{2}−i)i}^{0} = 0 for k_{1} + n − m < i < k_{2}. So we can find another integer n_{0} ≥ k_{2} − k_{1} + m > n such that τ(n_{0}) = m. This would contradict the assumption.

Theorem 10 [YOC]. Suppose P_{c}(z) = z^{2} + c is a recurrent quadratic polynomial. The critical tableau T (0) is periodic if and only if L_{0} contains more than one point.

Proof. We use the same notation as in the proof of Theorem 9 and the proof of Lemma 5. Suppose T (0) is periodic. Then for n > N + n_{1}

P_{c}^{∘n_{1}} : C_{n+1} → C_{n−n_{1}+1}

is a degree two proper holomorphic branch covering map. Replacing C_{n+1} by C̃_{n+1} if it is necessary, we may assume that this map is a quadratic-like map. Since L_{0} is the filled-in Julia set of this map, it contains more than one point. This is the “only if” part.

To prove the “if” part, suppose T (0) is not periodic. We will prove that L_{0} contains only one point. Since P_{c} is recurrent, there are infinitely many 1’s in every row of T (0), that is, T (0) is recurrent. Consider the first partition

η_{1} = {C_{1}, B_{11}, …, B_{1(q−1)}, B_{01}, … B_{0(q−1)}},

where B_{0i} = B_{0,i} for 1 ≤ i < q and where B_{1i} ⊆ C_{0} and P_{c}(B_{1i}) = B_{0,i} for 1 ≤ i < q. (Remember that η_{0} = {C_{0}, B_{0,1}, …, B_{o,q−1}} is the original partition.) Let c(n) = P_{c}^{∘n}(0). If the critical orbit CO = {c(n)}_{n=0}^{∞} is contained in the union

C_{1} ∪ B_{01} ∪ … ∪ B_{0(q−1)},

then T (0) is periodic of period q. Hence there must be one critical value c(n) in B_{11} ∪ ⋯ ∪ B_{1(q−1)}. Let c(n) be in B_{1i}. The annulus A_{0n}(0) = C_{0} \ B̊_{1i} is nondegenerate. Pull back A_{0n}(0) by P_{c} along A_{i(n−i)}(0) for 0 ≤ i ≤ n; we get a nondegenerate annulus A_{n0}(0).

Now consider τ^{−k}(n). For each m in τ^{−k}(n),

mod(A_{m0}(0)) ≥ \frac{mod(A_{n0}(0))}{2^{k}}.

If the number of τ^{−k}(n) is greater than or equal to 2^{k} for every k > 0, then

mod(C_{0} \ L_{0}) ≥ ∑_{m=1}^{∞} mod(A_{m0}(0)) ≥ ∑_{k=1}^{∞} ∑_{m∈τ^{−k}(n)} mod(A_{m0}(0)) ≥ ∑_{k=1}^{∞} mod(A_{n0}(0))

= (mod(A_{n0}(0))) · ∑_{k=1}^{∞} 1 = ∞.

So L_{0} = {0}.

If there is an integer k > 0 such that the number of τ^{−k}(n) is less than 2^{k}, then there are pre-images m > q of n under iterates of τ such that m is the only pre-image of q under τ. From (iii), m is noble. Hence τ^{−k}(m) are all noble and contain at least 2^{k} different integers. Moreover

mod(A_{p0}(0)) = \frac{mod(A_{m0}(0))}{2^{k}}

for every p in τ^{−k}(m). Therefore,

mod(C_{0} \ L_{0}) ≥ ∑_{k=1}^{∞} mod(A_{k0}(0)) ≥ ∑_{k=1}^{∞} ∑_{p∈τ^{−k}(m)} mod(A_{p0}(0)) ≥ ∑_{k=1}^{∞} mod(A_{m0}(0))

= (mod(A_{m0}(0))) · ∑_{k=1}^{∞} 1 = ∞.

Again we have L_{0} = {0}. This completes the “if” part.

Theorem 11 [YOC]. If P_{c}(z) = z^{2} + c is a non-recurrent or recurrent non-renorm-alizable quadratic polynomial, then its filled-in Julia set K_{c} is locally connected.

Proof. Let α be the separate fixed point of P_{c}. Construct the two-dimensional Yoccoz puzzle for P_{c}. For any x in K_{c}, let

x ∈ ⋯ ⊆ D_{n}(x) ⊆ D_{n−1}(x) ⊆ ⋯ ⊆ D_{1}(x) ⊆ D_{0}(x)

be an x-end. If P_{c} is non-recurrent, then the critical tableau is non-recurrent. Lemma 3 and Lemma 4 imply that the diameter d(D_{n}(x)) tends to zero as n goes to infinity. If P_{c} is recurrent and non-renormalizable, then T (0) is recurrent and is not periodic. Lemma 4 and Theorem 10 imply that the diameter d(D_{n}(x)) tends to zero as n goes to infinity.

If x is not a preimage of α under any iterate of P_{c}, then x is an interior point of D_{n}(x) for all n ≥ 0. From Lemma 2, {D_{n}(x)} is a basis of connected neighborhoods at x. If x is a preimage of α under some iterate of P_{c}, then there are q different x-ends,

x ∈ ⋯ ⊆ D_{i,n}(x) ⊆ D_{i,(n−1)}(x) ⊆ ⋯ ⊆ D_{i,1}(x) ⊆ D_{i,0}(x)

where q is the period of the external rays landing at α. Let D̃_{n}(x) = ∪_{i=1}^{q}D_{i,n}(x). Then x is an interior point of D̃_{n}. Since K_{c} ∩ D_{1,n}(x), …, K_{c} ∩ D_{q,n}(x) have a common point x, from Lemma 2, K_{c} ∩ D̃_{n}(x) is connected. So {D̃_{n}(x)}_{n=0}^{∞} is a basis of connected neighborhoods at x.

From Theorem 4, all arguments in this section apply to a quadratic-like map. Suppose that (U, V, f) is a quadratic-like map and that its filled-in Julia set K_{f} is connected. Suppose two fixed points β and α of f are repelling. Let β be the nonseparate fixed point of f, that is, K_{f} \ {β} is connected, and let α be the separate fixed point of f, that is, K_{f} \{α} is disconnected. Since (U, V, f) is hybrid equivalent to a quadratic polynomial P_{c}, there is a quasiconformal homeomorphism H defined on V such that

H ∘ f = P_{c} ∘ H

on U. We call e_{θ,f} = H^{−1}(e_{θ} ∩ H(U)) the external ray of angle θ of f where e_{θ} is the external ray of P_{c} of angle θ.

Two points H(β) and H(α) are non-separate and separate fixed points of P_{c}, respectively. Suppose Γ is the union of a cycle of periodic external rays landing at H(α). Let Γ′ = H^{−1}(Γ ∩ H(U)). The set Γ′ cuts the domain U into q domains. Each of them contains points in the filled-in Julia set K_{c}. Let C_{0} be the domain containing 0 and let B_{0,i} be the domain containing f^{∘i}(0) for 1 ≤ i < q. The partition

η_{0} = {C_{0}, B_{0,1}, …, B_{0,q−1}}

is called the original partition for f. Let Γ′_{n} = f^{−n}(Γ′) and U_{n} = f^{−n}(U). Then Γ′_{n} cuts U_{n} into finitely many domains. Let C_{n} be the domain containing 0 and B_{n,i} for 1 ≤ i ≤ k_{n} be others. Then

η_{n} = {C_{n}, B_{n,1}, …, B_{n,k_{n}}}

is called the n^{th}-partition for f. We use f^{−n}(η_{0}) to denote η_{n}, i.e., η_{n} = f^{−n}(η_{0}), for 1 ≤ n < ∞. We have that f(C_{n}) and f(B_{n,i}) for 1 ≤ i ≤ k_{n} are in η_{n−1} for n > 0 (set k_{0} = q − 1). We call η = {η_{n}}_{n=0}^{∞} the two-dimensional Yoccoz puzzle of f. Let Λ = ∪_{n=0}^{∞}f^{−n}(α). Let L_{0} = ∩_{n=0}^{∞}C_{n} be the connected component of K_{f} \Λ containing 0. We state Theorems 10 and 11 in the following form. Theorem 12 [YOC]. Suppose (U, V, f) is a recurrent quadratic-like map. Then (U, V, f) is renormalizable if and only if L_{0} contains more than one point. Moreover, if (U, V, f) is non-renormalizable, then any connected component of K_{f} \ Λ consists of only one point and K_{f} is locally connected.

5. Infinitely Renormalizable Quadratic Julia Sets and Three-Dimensional Yoccoz Puzzles

Suppose (U, V, f) is a renormalizable quadratic-like map with connected filled-in Julia set K_{f}. Let η = {η_{n}}_{n=0}^{∞} be the two-dimensional Yoccoz puzzle for f. From the previous section, the critical tableau T (0) = (a_{nm}^{0})_{n≥0,m≥0} is periodic of period n_{1}. Let

0 ∈ ⋯ ⊆ C_{n} ⊆ C_{n−1} ⊆ ⋯ ⊆ C_{1} ⊆ C_{0}

be the critical end. There is an integer N > 0 such that a_{ij}^{0} = 0 for all i ≥ N and for all 0 < j < n_{1}. Let f_{0} = f^{∘n_{1}}|C̊_{N+n_{1}}. Then f_{0} : C̊_{N+n_{1}} → C̊_{N} is a proper, holomorphic branch cover of degree two. Assume C_{N+n_{1}} ⊂ C̊_{N}. (Otherwise, we can replace C_{n} with C̃_{n} (see Theorem 9)). Then (C̊_{N+n_{1}}, C̊_{N}, f_{0}) is a quadratic-like map and its filled-in Julia set is L_{0} = ∩_{n=0}^{∞}C_{n}.

Theorem 13. Suppose (U, V, f) is a renormalizable quadratic-like map with connected filled-in Julia set K_{f}. For any renormalization U_{1} about n_{1}, let f_{1} = f^{∘n_{1}}|U_{1} and let V_{1} = f_{1}(U_{1}). Then the filled-in Julia set K_{f_{1}} (or the Julia set J_{f_{1}}) of (U_{1}, V_{1}, f_{1}) is always L_{0} (or ∂L_{0}).

Proof. Let U′ = C̊_{N+n_{1}} ∩ U_{1} and let U′′ be the connected component of U′ containing 0. Let f_{2} = f^{∘n_{1}}|U′′ and V′′ = f_{2}(U′′). Then f_{2} : U′′ → V′′ is a degree two branch covering. It is also proper because

f_{2}^{−1}(K) = f_{0}^{−1}(K) ∩ f_{1}^{−1}(K)

for any compact set K of V′′. Since C_{N+n_{1}} ⊂ C̊_{N} and \overline{U}_{1} ⊂ V_{1}, then

C_{N+n_{1}} ∩ \overline{U}_{1} ⊂ C̊_{N} ∩ V_{1}.

Thus \overline{U′′} ⊂ V′′ because they are the connected components of U′ and C̊_{N} ∩ V_{1} containing 0. Both U′′ and V′′ are simply connected and isomorphic to a disc because they are intersections of simply connected domains each of which is isomorphic to a disc. Therefore, (U′′, V′′, f_{2}) is a quadratic-like map. Let K_{f_{2}} be the filled-in Julia set of (U′′, V′′, f_{2}). Since a filled-in Julia set is completely invariant and since 0 is in U′′, the two inverse images of 0 under f_{2} are in K_{f_{2}}. But these two points are also inverse images of 0 under f_{0} and under f_{1}. Therefore, they are both in L_{0} and in K_{f_{1}}. Using this argument, the set Ξ of all inverse images of 0 under iterates of f_{2} is contained in K_{f_{2}} and is also contained in L_{0} and in K_{f_{1}}. Therefore,

K_{f_{1}} = K_{f_{2}} = L_{0} (or J_{f_{1}} = J_{f_{2}} = ∂L_{0})

because each of ∂L_{0}, J_{f_{1}} = ∂K_{f_{1}}, and J_{f_{2}} = ∂K_{f_{2}} is the closure of Ξ (see Theorem 3).

As we saw in §2, the definition of the filled-in Julia set of a renormalization about n_{1} depends on choices of domains in renormalization. But the renormalized filled-in Julia set is actually canonical; it is independent of choices of domains in renormalization and is the limiting piece containing 0 in the two-dimensional Yoccoz puzzle from Theorem 12.

Suppose (U_{1}, V_{1}, f_{1}) is a recurrent renormalizable quadratic-like map with connected filled-in Julia set K_{1}. We call K_{1} (or J_{1}) a quadratic filled-in Julia set (or quadratic Julia set). It is renormalizable if the corresponding quadratic-like map is renormalizable. Let β_{1} and α_{1} be the non-separate and separate fixed points of f_{1}, i.e., K_{1}\{β_{1}} is still connected and K_{1}\{α_{1}} is disconnected. Let Λ_{1} = ∪_{n=0}^{∞}f_{1}^{−n}(α_{1}). Let K_{2} = L_{0} be the connected component of K_{1} \Λ_{1} containing 0. From Theorem 12, K_{1} is renormalizable if and only if K_{2} contains more than one point. The quadratic filled-in Julia set K_{2} is called the renormalization of K_{1}.

Inductively, let K_{i} be the renormalization of K_{i−1}. Let f_{i} = f_{i−1}^{∘n_{i−1}} for i ≥ 2, where n_{i−1} is the period of the critical tableau T^{i−1}(0) = (a_{nm}^{0}(i − 1))_{n≥0,m≥0} of the twodimensional Yoccoz puzzle for (U_{i−1}, V_{i−1}, f_{i−1}). Let β_{i} and α_{i} be the non-separate and the separate fixed points of f_{i}, i.e., K_{i} \ {β_{i}} is still connected and K_{i} \ {α_{i}} is disconnected. Let Λ_{i} = ∪_{n=0}^{∞}f_{i}^{−n}(α_{i}) and let K_{i+1} be the connected component of K_{i} \ Λ_{i} containing 0. Then K_{i} is renormalizable if and only if K_{i+1} contains more than one point. Here K_{i}, for i > 1, is called the i^{th}-renormalization of K_{1}. Theorem 12 can be generalized as follows.

Theorem 14 [YOC]. Suppose that (U_{1}, V_{1}, f_{1}) is a recurrent quadratic-like map and that K_{1} is its filled-in Julia set. The quadratic Julia set K_{1} is finitely renormalizable if and only if there is an integer m ≥ 1 such that K_{1}, …, K_{m} contains more than one point and such that K_{m+1} contains only the point 0. Moreover, if K_{1} is finitely renormalizable, then K_{1} is locally connected.

Proof. The first part of the theorem follows directly from Theorem 12. We prove the second part. Let α_{m} be the separate fixed point of f_{m}. Let Γ_{m} be a cycle of periodic external rays of f_{1} landing at α_{m} (refer to the end of §3). Using Γ_{m}, we can construct the two-dimensional Yoccoz puzzle: let η_{0}^{m} be the set consisting of the closures of the connected components of V_{1} \ Γ_{m}. Let η_{n}^{m} = f_{1}^{−n}(η_{0}^{m}) for n ≥ 1. Let C_{n}^{m} be the member of η_{n}^{m} containing 0. Since f_{m} is non-renormalizable, we use a proof similar to that of Theorem 10 to show that ∑_{n=0}^{∞} mod(A_{n0}^{m}(0)) = ∞, where A_{n0}^{m}(0) = C_{n}^{m} \ C̊_{n+1}^{m}. Applying Lemma 4, for every x-end

x ∈ ⋯ ⊆ D_{n}^{m}(x) ⊆ D_{n−1}^{m}(x) ⊆ ⋯ ⊆ D_{1}^{m}(x) ⊆ D_{0}^{m}(x),

L_{x} = ∩_{n=0}^{∞}D_{n}^{m}(x) contains only x. By using a similar argument to the proof of Theorem 11, we can now show that K_{1} is locally connected.

Now let us consider an infinitely renormalizable quadratic-like map (U_{1}, V_{1}, f_{1}). Let K_{1} be the filled-in Julia set of f_{1}. Let K_{i} be the i^{th}-renormalization of K_{1}. Then K = {K_{i}}_{i=1}^{∞} is a sequence of renormalizations of K_{1}. Let {(U_{i}, V_{i}, f_{i})}_{i=1}^{∞} be a sequence of renormalizations with filled-in Julia set K_{i} where f_{i} = f_{i−1}^{∘n_{i−1}} and where n_{i−1} is the period of the critical tableau T^{i−1}(0) of the two-dimensional Yoccoz puzzle for (U_{i−1}, V_{i−1}, f_{i−1}), for i ≥ 2. Suppose U_{i} is a renormalization of (U_{i−1}, V_{i−1}, f_{i−1}).

We describe (U_{1}, V_{1}, f_{1}) as (n_{1}, n_{2}, …)-infinitely renormalizable. The grid {T^{i}(0)}_{i=1}^{∞} is called the three-dimensional critical tableau for (U_{1}, V_{1}, f_{1}). Let c(n) = f_{1}^{∘n}(0). The critical orbit of f_{1} is CO = {c(n)}_{n=0}^{∞}. Let GCO = ∪_{k=0}^{∞} ∪_{n=0}^{∞} f_{1}^{−k}(c(n)) be the grand critical orbit of f_{1}.

Definition 2. An infinitely renormalizable quadratic-like map (U_{1}, V_{1}, f_{1}) has a priori complex bounds if there are a constant λ > 0 and a sequence of renormalizations {(U_{i_{k}}, V_{i_{k}}, f_{i_{k}})}_{k=1}^{∞} of f_{1} such that

mod(V_{i_{k}} \ U_{i_{k}}) ≥ λ

for all k ≥ 1.

Theorem 15. Suppose (U_{1}, V_{1}, f_{1}) is an infinitely renormalizable quadratic-like map having the a priori complex bounds. Its filled-in Julia set K_{1} is locally connected at every point in GCO.

Proof. Suppose, without loss of generality, that {U_{i}, V_{i}, f_{i}}_{i=1}^{∞} is the sequence of renormalizations in Definition 2. Let λ > 0 be the constant in Definition 2. Then {U_{i}}_{i=1}^{∞} is a sequence of nested domains containing 0. Consider the annulus A_{i} = \overline{U}_{i} \ U_{i+1} for i ≥ 1. For each i ≥ 1, let cv_{i} = f_{i}(0) and let γ_{i} be a curve in V_{i} \ f_{i}^{∘2}(U_{i+1}) connecting cv_{i} and a point on the boundary of V_{i}. Let 0 ∈ U′_{i} ⊂ U_{i} be the connected component of the pre-image of V_{i}\γ_{i} under f_{i}^{∘2}. Then f_{i}^{∘2} : U′_{i} → V_{i}\γ_{i} is a degree two branch covering. Moreover, f_{i} : U′_{i} → f_{i}(U′_{i}) ⊂ U_{i} is also a degree two branch covering map. Thus f_{i} : U_{i} \ U′_{i} → V_{i} \ f(U′_{i}) is a degree two branch covering map. This implies that

mod(U_{i} \ U′_{i}) = \frac{1}{2} · mod(V_{i} \ f_{i}(U′_{i})).

But V_{i} \ U_{i} is a sub-annulus of V_{i} \ f_{i}(U′_{i}). So

mod(U_{i} \ U′_{i}) ≥ \frac{1}{2} · mod(V_{i} \ U_{i}) > \frac{λ}{2}.

Remember that U_{i+1} is the domain of the renormalization f_{i+1} = f_{i}^{n_{i}} where n_{i} ≥ 2. We have U_{i+1} ⊂ U′_{i}. Hence U_{i} \ U′_{i} is a sub-annulus A_{i}. So

mod(A_{i}) > \frac{λ}{2}.

Let A_{∞} = ∩_{i=1}^{∞}U_{i}. Since U_{1} \ A_{∞} = ∪_{i=1}^{∞}A_{i},

mod(U_{1} \ A_{∞}) ≥ ∑_{i=1}^{∞} mod(A_{i}) = ∞.

Thus, A_{∞} = {0}. This implies that the diameter d(U_{i}) tends to 0 as i goes to infinity.

Let α_{i} be the separate fixed point of f_{i}. Let Γ_{i} be a cycle of periodic external rays of f_{1} landing at α_{i} (refer to the end of §3). Let η_{0}^{i} be the set consisting of the closures of the connected components of U_{1} \Γ_{i} and let η_{n}^{i} = f_{1}^{−n}(η_{1}^{i}) for n ≥ 1. Then {η_{n}^{i}}_{n=0}^{∞} is a puzzle for f_{1}. Let C_{n}^{i} be the member of η_{n}^{i} containing 0. Consider the critical end

0 ∈ ⋯ ⊆ C_{n}^{i} ⊆ C_{n−1}^{i} ⊆ ⋯ ⊆ C_{1}^{i} ⊆ C_{0}^{i}

and the critical tableau T^{i}(0) = (a_{nm}^{0}(i))_{n≥0,m≥0}. Since (U_{1}, V_{1}, f_{1}) is (n_{1}, n_{2}, …)- renormalizable, T^{i}(0) is periodic of period m_{i} = ∏_{j=1}^{i} n_{j}. Thus there is an integer N > 0 such that a_{nm}^{0}(i) = 0 for n ≥ N and 0 < i < m_{i}. Thus

f_{1}^{∘m_{i}} = f_{i}^{∘n_{i}} : C̊_{m_{i}+N}^{i} → C̊_{N}^{i}

is a degree two proper holomorphic map. We may assume that C_{m_{i}+N}^{i} ⊂ C̊_{N}^{i} (otherwise, we can modify C_{n}^{i} as the proof of Theorem 9). Therefore,

f_{i+1} = f_{1}^{∘m_{i}} = f_{i}^{∘n_{i}} : C̊_{m_{i}+N}^{i} → C̊_{N}^{i}

is a quadratic-like map. Since

K_{i+1} = ∩_{j=0}^{∞}f_{1}^{−jm_{i}}(C_{N}^{i}) = ∩_{j=0}^{∞}C_{jm_{i}+N}^{i},

there is a C_{k(i)}^{i} contained in U_{i}. The diameter d(C_{k(i)}^{i}) of C_{k(i)}^{i} tends to zero as i goes to infinity. From Lemma 2, C_{k(i)}^{i} ∩ K_{1} is connected. So {C_{k(i)}^{i}}_{i=1}^{∞} is a basis of connected neighborhoods at 0.

For any x = f_{1}^{∘n}(0), consider {D_{i,n}(x) = f_{1}^{∘n}(C_{k(i)}^{i})}_{i=1}^{∞}. It is a basis of connected neighborhoods at x. For any y in f_{1}^{−m}(x) (where 0 is not in f^{−n}(x) for 0 < n ≤ m), there is an open neighborhood W of y such that f_{1}^{∘m} : W → f^{∘m}(W) is a homeomorphism. Let g be its inverse. Then {g(D_{i,n}(x))}_{i=1}^{∞} is a basis of connected neighborhoods at y.

Suppose (U_{1}, V_{1}, f_{1}) is an (n_{1}, n_{2}, …)-infinitely renormalizable quadratic-like map. We call the puzzle {{η_{n}^{i}}_{n=0}^{∞}}_{i=1}^{∞} constructed in Theorem 15, the three-dimensional Yoccoz puzzle for (U_{1}, V_{1}, f_{1}). Let m_{i} = ∏_{j=1}^{i} n_{j} and {K_{i}}_{i=1}^{∞} be the sequence of renormalizations of K_{1}. Let c(j) = f^{∘j}(0) for j ≥ 0, and let CO = {c(j)}_{j=0}^{∞} be the critical orbit of f_{1}.

Definition 3. An infinitely renormalizable quadratic-like map (U_{1}, V_{1}, f_{1}) is unbranched if there are a constant λ > 0 and a sequence of domains {W_{k}}_{k=1}^{∞} such that W_{k} ⊃ K_{i_{k}}, such that

mod(W_{k} \ K_{i_{k}}) ≥ λ,

and such that W_{k} ∩ CO = {c(jm_{i_{k}})}_{j=0}^{∞} for all k ≥ 1 Theorem 16. Suppose (U_{1}, V_{1}, f_{1}) is an infinitely renormalizable unbranched quadratic-like map having the a priori complex bounds. Then its filled-in Julia set K_{1} is locally connected.

Proof. Suppose, without loss of generality, that k = i and i_{k} = i in Definition 3, and that {U_{i}, V_{i}, f_{i}}_{i=1}^{∞} is a sequence of renormalizations in Definition 2. Let λ > 0 be a constant satisfying Definitions 2 and 3.

Let {{η_{n}^{i}}_{n=0}^{∞}}_{i=1}^{∞} be the three-dimensional Yoccoz puzzle for (U_{1}, V_{1}, f_{1}). Let {C_{k(i)}^{i}}_{i=1}^{∞} be the basis of connected neighborhoods constructed in Theorem 15. By choosing k(i) large enough, we can have

mod(W_{i} \ C_{k(i)}^{i}) ≥ \frac{λ}{2}

for all i ≥ 1.

If x = 0, Theorem 15 says that K_{1} is locally connected at x. For each x ≠ 0 in K_{1}, there are two cases: either (1) x is non-recurrent, which means that there is an integer i ≥ 1 such that {f_{1}^{∘n}(x)}_{n=0}^{∞} ∩ C_{k(i)}^{i} = ∅; or else (2) x is recurrent.

In case (1), we prove it by applying the results in hyperbolic geometry. Let x be a non-recurrent point. Then there is a C_{k(i)}^{i} such that the orbit O(x) = {f_{1}^{∘n}(x)}_{n=0}^{∞} is disjoint from the interior of C_{k(i)}^{i}. Let r be an external ray of f_{1} cutting U_{1} \ f_{1}(C_{k(i)}^{i}) into a simply connected domain Ω. Consider two inverse branches Q_{1} and Q_{2} of f_{1}|Ω.

Let Ω̃ = V_{1} \ (f_{1}(C_{k(j)}^{j}) ∪ r) for some j > i such that \overline{Ω} ⊂ Ω̃. Let Q̃_{1} and Q̃_{2} be the inverses of f_{1}|Ω̃ and let Ω̃_{1} and Ω̃_{2} be the images of Ω̃ under Q̃_{1} and Q̃_{2}, respectively. Consider Ω̃, Ω̃_{1}, and Ω̃_{2} as hyperbolic Riemann surfaces. Then Q_{1} : Ω → Ω_{1} and Q_{2} : Ω → Ω_{2} strictly contract the hyperbolic distances of Ω̃, Ω̃_{1}, and Ω̃_{2}. Since Ω contains finite number of connected components of K_{1} \ f_{1}(C_{k(i)}^{i}), we can cut Ω into finite number of simply connected domains Ω′_{1}, …, Ω′_{l} such that Ω′_{k} ∩K_{1} is connected for each 1 ≤ k ≤ l. Since the orbit O(x) is contained in Ω, from the images of Ω′_{1}, …, Ω′_{l} under the semi-group generated by Q_{1} and Q_{2}, we can get a basis of connected neighborhoods at x. Therefore, K_{1} is locally connected at x.

In case (2), f_{1}^{∘n}(x) enters C_{k(i)}^{i} infinitely many times. For each i ≥ 1, consider the puzzle η^{i} = {η_{n}^{i}}_{n=0}^{∞}. For the x-end,

x ∈ ⋯ ⊆ D_{n}^{i}(x) ⊆ D_{n−1}^{i}(x) ⊆ ⋯ ⊆ D_{1}^{i}(x) ⊆ D_{0}^{i}(x),

let T^{i}(x) = (a_{nm}(i))_{n≥0,m≥0} be the corresponding tableau. Let q_{i} be the integer such that a_{k(i)q} = 0 for 0 ≤ q < q_{i} and such that a_{k(i)q_{i}} = 1. Let p_{i} = k(i) + q_{i}. Then g_{i,x} = f_{1}^{∘q_{i}} : D_{p_{i}}^{i}(x) → C_{k(i)}^{i} is a proper holomorphic diffeomorphism. Since there is no critical point in W_{i} \ C_{k(i)}^{i}, this diffeomorphism can be extended to a proper holomorphic diffeomorphism on W_{i}. Let x_{i} = f_{1}^{∘q_{i}}(x). We can modify W_{j} such that

\frac{λ}{2} ≤ mod(W_{j} \ C_{k(j)}^{j}) ≤ 2λ.

From the previous theorem, the diameter d(C_{k(j)}^{j}) tends to zero as j goes to infinity. We can find an integer j > i such that W_{j} ⊂ C_{k(i)}^{i}. Let

x_{j} = f_{1}^{∘q_{j}}(x) = f_{1}^{∘(q_{j}−q_{i})}(x_{i}).

Consider the puzzle η^{j} = {η_{n}^{j}}_{j=1}^{∞}. Let

x_{i} ∈ ⋯ ⊆ D_{n}^{j}(x_{i}) ⊆ D_{n−1}^{j}(x_{i}) ⊆ ⋯ ⊆ D_{1}^{j}(x_{i}) ⊆ D_{0}^{j}(x_{i})

be the x_{i}-end and let T^{j}(x_{i}) = (b_{nm}(j))_{n≥0,m≥0} be the corresponding tableau. Then one can check that f_{1}^{∘(q_{j}−q_{i})} is a proper holomorphic diffeomorphism from

D_{k(j)+q_{j}−q_{i}}^{j}(x_{i}) → C_{k(j)}^{j}.

In other words, b_{k(j)m} = 0 for 0 ≤ m < q_{j} − q_{i}, but b_{k(j)(q_{j}−q_{i})} = 1. Let g_{ij} be the inverse of f_{1}^{∘(q_{j}−q_{i})} : D_{k(j)+q_{j}−q_{i}}^{j}(x_{i}) → C_{k(j)}^{j}. Then g_{ij} can be extended to W_{j}. Since C_{k(i)}^{i} is bounded by external rays landing at some pre-iamges of α_{i} under iterations of P_{c} and is a part of an invariant net under f_{1}, W_{ij} = g_{ij}(W_{j}) is contained in C_{k(i)}^{i}. Thus

mod(W_{i} \ W_{ij}) ≥ \frac{λ}{2}.

Consider X_{i} = g_{i,x}(W_{i}) and X_{j} = g_{j,x}(W_{j}) = g_{i,x}(W_{ij}). Then

mod(X_{i} \ X_{j}) ≥ \frac{λ}{2}.

Therefore, inductively, we can find a sequence of domains {X_{i_{s}}}_{s=1}^{∞} such that

mod(X_{i_{s}} \ X_{i_{s+1}}) ≥ \frac{λ}{2}

for s ≥ 1. Thus the diameter of X_{i_{s}} tends to zero as s goes to infinity. For each puzzle η^{i_{s}} = {η_{n}^{i_{s}}}_{n=0}^{∞}, consider both the x-end,

x ∈ ⋯ ⊆ D_{n}^{i_{s}}(x) ⊆ D_{n−1}^{i_{s}}(x) ⊆ ⋯ ⊆ D_{1}^{i_{s}}(x) ⊆ D_{0}^{i_{s}}(x)

and the corresponding tableau T^{i_{s}}(x) = (a_{nm}(i_{s}))_{n≥0,m≥0}. For each k(i_{s}), there is an integer q_{i_{s}} such that a_{k(i_{s})q}^{i} = 0 for 0 ≤ q < q_{i_{s}} and such that a_{k(i_{s})q_{is}}^{i} = 1. Then for p_{i_{s}} = k(i_{s}) + q_{i_{s}}, f_{1}^{∘q_{is}} : D_{p_{is}}(x) → C_{k(i_{s})} is a proper holomorphic diffeomorphism. This implies that

D_{p_{is}}^{i_{s}}(x) ⊆ X_{i_{s}}.

So the diameter d(D_{p_{is}}^{i_{s}}(x)) tends to zero as s goes to infinity. From Lemma 2, {D_{p_{is}}^{i_{s}}(x)}_{s=1}^{∞} forms a basis of connected neighborhoods at x.

6. A Generalized Sullivan’s Sector Theorem

Let I = [0, 1] be the closed unit interval. Let E_{0} be the set of all functions G such that

(1) G : I_{G} ⊇ I → G(I_{G}) is a homeomorphism and G(0) = 0, G(1) = 1, and

(2) G can be extended to be a schlicht function g on C_{G} = (C\R^{1})∪I̊_{G} preserving upper- and lower-half planes. We assume I_{G} is the maximum interval satisfies (1) and (2) for each G in E_{0} and call it the definition interval of G. We will not distinguish g and G anymore. Take

S_{γ}(z) = r^{\frac{1}{γ}} e^{\frac{θ}{γ} i} : C \ \{x < 0\} → C

as the standard γ-root where z = re^{θi} for r > 0, −π < θ < π, and γ > 1. For every a ≤ 0, we call L_{a}(z) = ES_{γ}(z −a)+F a γ-root at a where E = 1/((1−a)^{\frac{1}{γ}} −(−a)^{\frac{1}{γ}} )

and F = −(−a)^{\frac{1}{γ}} E are determined by L_{a}(0) = 0 and L_{a}(1) = 1. Then L_{a} is an element in E_{0} whose definition interval is [a, ∞).

Suppose that L_{a} is a γ-root at a and that G is an element in E_{0} whose definition interval is I_{G}. We say that L_{a} and G are compatible if [a, 1] ⊂ G(I_{G}). For a compatible pair L_{a} and G, let a′ = G^{−1}(a), J = [a′, 1], L ∪ R = I_{G} \ J, and b = min{|L|, |R|}.

We consider several disks related to a compatible pair L_{a} and G (see Fig. 3). Set D^{1} to be the closed disk centered at the middle point (1 + a′)/2 of [a′, 1] with diameter 1 + |a′| + 2b. Set D^{2} to be the closed disk centered at the middle point (1 + a′)/2 of [a′, 1] with diameter k = min{1 + |a′| + b, 2(1 + |a′|)}. Set D^{3} to be the maximum closed disk centered at a′ and contained in D^{2}. Then the radius of D^{3} is d = min{b/2, (1 + |a′|)/2}. We note that

D^{3} ⊂ D^{2} ⊂ D^{1}.

The map G is a schlicht function on D^{1}. Let μ = (1 + |a′|)/b and ν = (2μ + 3)^{4}. From Koebe’s distortion theorem (see [BIE]) for any ξ and η in D^{2},

ν^{−1} ≤ \frac{|G′(ξ)|}{|G′(η)|} ≤ ν.

Using the fact that G(I) = I, there is at least one η in D^{2} such that |G′(η)| = 1. Hence

ν^{−1} ≤ |G′(ξ)| ≤ ν

for all ξ in D^{2}. This implies that

\frac{|a′|}{ν} ≤ |a| ≤ |a′| · ν.

Let UH = {z = x + yi ∈ C | y > 0} be the upper-half plane. For any z in UH, let θ(z) = arg(z).

Theorem 17. Suppose that L = L_{a} and G are a pair of compatible elements in E_{0}. There is a constant 0 < θ < π depending only on μ such that the image L(G(UH)) of the upper-half plane under L ∘ G contains an open triangle Δ based on [L(a), 0] whose angle at L(a) is π/γ and whose angle at 0 is θ.

Proof. The image G(D^{3}) of D^{3} under G contains the closed disk D^{4} centered at a with radius d/ν. Similarly, for any a′ ≤ x ≤ 1, consider the closed disk D(x) centered at x with radius d. Then D(x) ⊂ D^{2} and G(D(x)) contains the closed disk centered at G(x) with radius d/ν. This implies that the convex-hull X of {0} ∪ D^{4} is contained in G(D^{2}). Either X = D^{4} or X ∩ UH has an angle at 0. In the later case, the angle φ of X ∩ UH at 0 has sin φ = d/(|a|ν).

Since L is a γ-root for γ > 1, the convex set L(X) contains a triangle Λ based on [L(a), 0] whose angle at L(a) is π/γ and whose angle ω at 0 can be calculated from

\frac{sin ω}{sin(\frac{π}{γ} + ω)} = (\frac{d}{|a|ν} )^{\frac{1}{γ}}

through the law of sines. Because d = min{b/2, (1 + |a′|)/2},

\frac{d}{|a|} ≥ \frac{d}{|a′|ν} ≥ min\{\frac{1}{2μν} , \frac{1}{2ν} \}.

Hence Λ contains a triangle Δ based on [L(a), 0] whose angle at L(a) is still π/γ and whose angle θ at 0 is calculated from

\frac{sin θ}{sin(\frac{π}{γ} + θ)} = ( min\{\frac{1}{2μν} , \frac{1}{2ν} \})^{\frac{1}{γ}} .

Suppose Q = (L∘G)^{−1}(\overline{Δ}), where Δ is the triangle obtained in Theorem 17. Then

Q ⊂ D^{2} ⊂ D^{1}.

Suppose {(L_{i}, G_{i})}_{i=0}^{n} is a sequences of compatible pairs in E_{0} where L_{i} is a γ-root at a_{i}. Let a′_{i}, b_{i}, k_{i}, d_{i}, μ_{i}, ν_{i}, and θ_{i} be the numbers, and D_{i}^{1}, D_{i}^{2}, D_{i}^{3}, Δ_{i}, and Q_{i} be the sets corresponding to each compatible pair L_{i} and G_{i}. Let

L = L_{n} ∘ G_{n} ∘ ⋯ ∘ L_{i} ∘ G_{i} ∘ ⋯ ∘ L_{0} ∘ G_{0}.

Then L is a schlicht function defined on C_{I} = (C \ R^{1}) ∪ I̊.

Definition 4. We call L a root-like map if there are constants C > 0 and λ > 1 such that

(i) a_{0} = 0 and a_{1} ≥ 1/C,

(ii) |a_{j}| ≥ max{(λ^{j−i}/C) · |a_{i}|, (1 + (λ − 1)/C) · |a_{i}|} for all 1 ≤ i < j ≤ n, and

(iii) μ_{i} < C for all 1 ≤ i ≤ n.

Theorem 18. Suppose L is a root-like map. There is a constant θ > 0 depending only on λ and C such that the image of the upper-half plane under L is contained in the sector

Sec_{θ} = {z ∈ C | 0 ≤ arg(z) ≤ π − θ}.

Before we prove this theorem, we introduce some basic results in hyperbolic geometry. Let C_{I} = (C \ R^{1}) ∪ I̊ be a plane domain. Then q(z) = −z^{2}/(1 − z^{2}) is a diffeomorphism from the upper-half plane UH onto C_{I}. Consider UH to be a hyperbolic plane with Poincaré metric d_{H,UH}s = |dz|/y for z = x + yi. This metric induces a hyperbolic metric d_{H,C_{I}}s = q_{∗}(|dz|/y) on C_{I}. The plane domain C_{I} under this metric is a hyperbolic Riemann surface. Let d = d_{H,C_{I}} be the induced hyperbolic distance. We note that q maps the positive imaginary line in UH onto the interval I and maps the real line, which is the boundary of UH, onto the set R^{1} \ I̊.

Lemma 6. A hyperbolic neighborhood Φ(r) = {z ∈ C_{I} | d(z, I) < r} is the union of two Euclidean disks D^{+} and D^{−}, symmetric to each other with respect to I, centered at c^{+} and at c^{−} = −c^{+}, with the same radius R^{+} = R^{−}. Moreover,

R^{+} = \frac{1}{2 sin β} and c^{+} = \frac{1}{2} + \frac{cot β}{2} i

where

β = 4 cot^{−1}(e^{r})

is the angle at 0 between ∂D^{+} and the negative real line (and the angle at 1 between ∂D^{+} and the ray [1, ∞)).

Proof. Consider the pre-image Φ′ = q^{−1}(Φ(r)). It is a hyperbolic neighborhood in UH and consists of all points in UH whose hyperbolic distances to the half-line l_{+} = {z = yi | y > 0} are less than r. The boundary ∂Φ′ consists of two rays starting from 0. Thus Φ′ is a sector, symmetric with respect to the half-line l_{+}. Suppose β/2 is the outer angle of this sector (with respect to the real line). Since a geodesic in UH is a semi-circle or half-line perpendicular to the real line, it is easy to check that

log ( cot \frac{β}{4} ) = r.

Therefore, Φ(r) is the union of two disks D^{+} and D^{−} symmetric with respect to I. The angle between ∂D^{+} (or ∂D^{−}) and the negative real line is β. Moreover, every point z in ∂D^{+} (or ∂D^{−}) views I under the same angle β, that is, every triangle △(0z1) has the angle β at z. Now consider the point u such that the segment \overline{1u} is a diameter of D^{+}. The triangle △(u01) is a right triangle. We can calculate the length 2R^{+} of the segment 1u and length |u| of the segment \overline{0u} as follows:

2R^{+} = \frac{1}{sin β} , |u| = cot β.

Therefore,

R^{+} = \frac{1}{2 sin β} and c^{+} = \frac{1}{2} + \frac{cot β}{2} i.

Lemma 7. Let z be a point in C_{I} and let Φ(r) = D^{+}∪D^{−} be the smallest hyperbolic neighborhood containing z. The Euclidean radius R^{+} of D^{+} (and D^{−}) is

\frac{|z − 1|}{2 sin(arg(z))}.

Proof. Let u be the point in ∂D^{+} such that the segment \overline{1u} is a diameter of D^{+}. The angle of the triangle △(0z1) at z and the angle of the triangle △(0u1) at u are both β (see the previous lemma). Now applying the law of sines,

\frac{sin(arg(z))}{|1 − z|} = sin β = \frac{1}{|1 − u|}.

Therefore,

2R^{+} = \frac{|z − 1|}{sin(arg(z))}.

Suppose L is a root-like map. Then there is a constant angle σ and a constant C_{0} > 0 depending only on C such that θ_{i} ≥ σ and ν_{i} ≤ C_{0} for all 0 ≤ i ≤ n.

For each 1 ≤ i ≤ n − 1, let A_{i+1} = Q_{i+1} \ Δ_{i}. Let A_{1} = Q_{1} \ {z| ∈ C, 0 ≤ arg(z) < π − π/γ}. Suppose Φ(r_{i}) = D_{i}^{+} ∪ D_{i}^{−} is the smallest hyperbolic neighborhood in C_{I} containing A_{i} ≠ ∅ and let D_{i}^{0} be the smallest disk centered at 1/2 containing Φ(r_{i}). Let R_{i}^{+} be the Euclidean radius of D_{i}^{+} and let R_{i} be the Euclidean radius of D_{i}^{0}. Since z and 1 are in D_{i}^{2}, from Lemmas 6 and 7,

R_{i} ≤ 2 · R_{i}^{+} ≤ \frac{k_{i}}{sin θ_{i}} ≤ \frac{2 · (1 + |a′_{i}|)}{sin θ_{i}} ≤ \frac{2 · (1 + C) · |a′_{i}|}{sin σ} = C_{1} · |a′_{i}|

where k_{i} = min{1 + |a′_{i}| + b_{i}, 2(1 + |a′_{i}|)} is the diameter of D_{i}^{2} and where C_{1} = 2 · (1 + C)/ sin σ is a positive constant depending only on C (see Fig. 4).

For every 1 ≤ i ≤ n, the definition interval of ϕ_{i} = L_{i} ∘ G_{i} is [a′_{i}, a′′_{i}] = [a′_{i}, ∞) ∩ I_{G_{i}}. According to (iii) of Definition 4, the right-endpoint a′′_{i} satisfies

a′′_{i} ≥ \frac{1 + |a′_{i}|}{C} + 1 ≥ 1 + \frac{|a′_{i}|}{C}.

For every 1 ≤ i < j ≤ n,

|a′_{j}| ≥ \frac{λ^{j−i}}{CC_{0}^{2}} |a′_{i}|.

Therefore, if τ = min{1, 1/(C · C_{0})^{2}}, we have

[a′_{j}, a′′_{j}] ⊃ I_{i,τ} = [λ^{j−i}τa′_{i}, λ^{j−i}τ|a′_{i}| + 1].

In other words, ϕ_{j} = L_{j} ∘ G_{j} are schlicht functions on C_{I_{i,τ}} = (C \ R^{1}) ∪ I_{i,τ} for all 1 ≤ i < j ≤ n.

Lemma 8. There is a fixed integer n_{0} > 0 depending only on C and on λ such that for any 0 ≤ i < n and any j ≥ i + n_{0},

|a′_{j}| + \frac{1}{2} > λ^{j−i−n_{0}}R_{i} and a′′_{j} − \frac{1}{2} > λ^{j−i−n_{0}}R_{i}.

Proof. Take m_{0} as the biggest integer such that

m ≤ \frac{log(\frac{C_{1}}{τ})}{log λ}.

From the above estimates, one can see that n_{0} = max{m_{0} + 1, 0} is the integer satisfying the lemma.

Lemma 9. There is a constant C_{2} > 0 depending only on λ such that for any 0 ≤ i < n − n_{0}, let Π_{i} = L_{n} ∘ G_{n} ∘ ⋯ ∘ L_{j} ∘ G_{j} ∘ ⋯ ∘ L_{i+n_{0}+1} ∘ G_{i+n_{0}+1}, the distortion of Π_{i} on Φ(r_{i}) is bounded by C_{2}, more precisely,

\frac{|Π′_{i}(ξ)|}{|Π′_{i}(η)|} ≤ C_{2}

for all ξ and η in Φ(r_{i}).

Proof. Consider the disk D_{j}^{5} centered at 1/2 with radius t_{j} = min{|a′_{j}| + 1/2, a′′_{j} − 1/2}. The composition ϕ_{j} = L_{j} ∘ G_{j} is a schlicht function on D_{j}^{5}. Let r_{ij} = R_{i}/t_{j} be the ratio of the radii of D_{i}^{0} and D_{j}^{5}. Then

r_{ij} ≤ λ^{i+n_{0}−j}

for i + n_{0} + 1 ≤ j ≤ n. From Koebe’s distortion theorem (see [BIE]),

\frac{|ϕ′_{j}(ξ)|}{|ϕ′_{j}(η)|} ≤ (\frac{1 + r_{ij}}{1 − r_{ij}} )^{4} ≤ (\frac{1 + λ^{−k}}{1 − λ^{−k}} )^{4}

for 0 < k = j − i − n_{0} ≤ n − i − n_{0} and for all ξ and η in D_{i}^{0}. Since ϕ_{j} is a schlicht function on C_{I}, it contracts the hyperbolic distance d on C_{I}. Thus

ϕ_{j}(Φ(r_{i})) ⊂ Φ(r_{i}) ⊂ D_{i}^{0}.

Therefore, by the chain rule,

\frac{|Π′_{i}(ξ)|}{|Π′_{i}(η)|} = ∏_{j=n_{0}+1}^{n} \frac{|ϕ′_{j}(ξ)|}{|ϕ′_{j}(η)|} ≤ C_{2} = ( ∏_{k=1}^{∞} \frac{1 + λ^{−k}}{1 − λ^{−k}} )^{4}

for all ξ and η in Φ(r_{i}).

Lemma 10. There is a constant C_{3} > 0 depending only on C such that for 0 ≤ i ≤ n, for i ≤ j ≤ i + n_{0}, for ϕ_{j} = L_{j} ∘ G_{j}, and for all ξ and η in D_{i}^{+} (or D_{i}^{−}),

\frac{|ϕ′_{j}(ξ)|}{|ϕ′_{j}(η)|} ≤ C_{3}.

Proof. Suppose x is a real number. Let α = α(x) = min{|x − z| | z ∈ D_{i}^{+} ∩ UH}. Suppose that c = c_{i}^{+} and that R = R_{i}^{+} are the center and the radius of D_{i}^{+}. Suppose h = h_{i}^{+} is the length of the segment \overline{c\frac{1}{2}} (the straight line connecting c and 1/2). Since the two triangles △(x\frac{1}{2}c) and △(0\frac{1}{2}c) are both right triangles, then

(α + R)^{2} = (\frac{1}{2} − x)^{2} + h^{2}

and

R^{2} = (\frac{1}{2})^{2} + h^{2}.

Therefore,

\frac{α + R}{R} = \sqrt{1 + \frac{x^{2} − x}{R^{2}}}.

This implies that there is a constant 0 < C_{4} < 1 depending only on C such that for x = a′_{j} or x = a′′_{j},

\frac{R}{α + R} ≤ C_{4}.

Now consider the largest disk D_{j}^{6} centered at c such that ϕ_{j} is a schlicht function on it. Then the radius of D_{j}^{6} is greater than or equal to min{R + α(a′_{j}), R + α(a′′_{j})}. From Koebe’s distortion theorem (see [BIE]),

\frac{|ϕ′_{j}(ξ)|}{|ϕ′_{j}(η)|} ≤ C_{3} = (\frac{1 + C_{4}}{1 − C_{4}} )^{4}

for any ξ and η in D_{i}^{+}.

Combining Lemmas 9 and 10, we obtained the following estimate:

Lemma 11. There is a constant C_{5} > 0 depending on λ and on C such that for any 0 ≤ i < n, let Σ_{i} = L_{n} ∘ G_{n} ∘ ⋯ ∘ L_{j} ∘ G_{j} ∘ ⋯ ∘ L_{i} ∘ G_{i}, the distortion of Σ_{i} on D_{i}^{+} (or D_{i}^{−}) is bounded by C_{5}, more precisely,

\frac{|Σ′_{i}(ξ)|}{|Σ′_{i}(η)|} ≤ C_{5}

for all ξ and η in D_{i}^{+} (or D_{i}^{−}).

Proof. Since each ϕ_{j} is a schlicht function on C_{I}, it contracts the hyperbolic distance d on C_{I}. So

ϕ_{j}(D_{i}^{+}) ⊂ D_{i}^{+}

for all i ≤ j ≤ n. If n − i ≤ n_{0}, then from Lemma 10 and the chain rule,

\frac{|Σ′_{i}(ξ)|}{|Σ′_{i}(η)|} ≤ C_{3}^{n_{0}}

for all ξ and η in D_{i}^{+}.

Now we consider n − i > n_{0} and write Σ_{i} = Π_{i} ∘ Θ_{i} where Θ_{i} = L_{i+n_{0}} ∘ G_{i+n_{0}} ∘ ⋯ ∘ L_{i} ∘ G_{i} and where Π_{i} = L_{n} ∘ G_{n} ∘ ⋯ ∘ L_{i+n_{0}+1} ∘ G_{i+n_{0}+1}. From Lemma 10,

\frac{|Θ′_{i}(ξ)|}{|Θ′_{i}(η)|} ≤ C_{3}^{n_{0}}

for all ξ and η in D_{i}^{+}, and from Lemma 9,

\frac{|Π′_{i}(ξ)|}{|Π′_{i}(η)|} ≤ C_{2}

for all ξ and η in D_{i}^{+}. Again, because Θ_{i} is a schlicht function on C_{I} and contracts the hyperbolic distance d on C_{I}, we have Θ_{i}(D_{i}^{+}) ⊂ D_{i}^{+}. Therefore, from the chain rule,

\frac{|Σ′_{i}(ξ)|}{|Σ′_{i}(η)|} ≤ C_{5} = C_{2} · C_{3}^{n_{0}}

for all ξ and η in D_{i}^{+}.

Lemma 12. Suppose G is in E_{0} and D is a closed simply connected convex domain with I ⊂ D ⊂ C_{G}. Then for all z = x + yi with y > 0 in D

sin ( arg(G(z))) ≥ \frac{sin(arg(z))}{N_{0}}

where N_{0} = sup_{ξ,η∈D} |G′(ξ)/G′(η)| measures the distortion of G on D.

Proof. Since G maps UH into itself, it contracts the hyperbolic metric dz/y on UH. Suppose z = x + yi with y > 0 and G(z) = X + Y i. Then |G′(z)|y ≤ Y. Therefore,

sin(arg(z)) = \frac{y}{|z|} ≤ \frac{Y}{|z| · |G′(z)|} = sin ( arg(G(z))) · \frac{|G(z)|}{|z| · |G′(z)|}.

So

\frac{sin(arg(z))}{N_{0}} ≤ sin ( arg(G(z)))

for all z = x + yi with y > 0 in D.

Now we complete the proof of Theorem 18 as follows.

Proof of Theorem 18. For any z_{0} in UH, let z_{i+1} = L_{i}(G_{i}(z_{i})) for 0 ≤ i ≤ n. Since 0 ≤ arg(z_{1}) ≤ π/γ, the smallest positive integer i such that z_{i} lies in Δ_{i} must be either bigger than zero or not exist. If such a positive integer does not exist, then 0 ≤ arg(z_{n+1}) ≤ π − θ_{n} ≤ π − σ. Now let us suppose that this smallest number exists and is i_{0} + 1. Then z_{i_{0}} is in A_{i_{0}} ≠ ∅ which is a subset of D_{i_{0}}^{+}. Since 0 < arg(z_{i_{0}}) ≤ π−θ_{i}, Lemmas 11 and 12 assure us there is a constant angle 0 < θ ≤ σ such that 0 < arg(z_{n+1}) ≤ π − θ.

7. Feigenbaum-Like Quadratic-Like Maps

In this section, we discuss Feigenbaum-like quadratic-like maps and prove the result of Sullivan which says that a Feigenbaum-like quadratic-like map has the a priori complex bounds and is unbranched.

Let us first recall some facts about infinitely renormalizable folding mappings. Let (U, V, f) be a real quadratic-like map, that is, f(U ∩ R^{1}) ⊆ V ∩ R^{1}. Conjugating by a linear fraction transformation, we may assume that f(−1) = f(1) = −1 and that f|[−1, 1] is a folding map of [−1, 1] with a unique quadratic critical point 0. For example, P_{t}(z) = t−(1+t)z^{2} for 0 ≤ t ≤ 1 is a real quadratic-like map whenever it is restricted to any domain bounded by an equipotential curve. Furthermore, suppose (U, V, f) is infinitely renormalizable. Then the filled-in Julia set K_{f} is connected. Let β_{0} = −1 and let α_{0} be the fixed point of f in (−1, 1). Then β_{0} is the non-separate fixed point, α_{0} is the separate fixed point, and K_{f} ∩ R^{1} = [−1, 1]. The mapping f_{0} = f|[−1, 1] is an infinitely (n_{1}, n_{2}, …, n_{k}, …)-renormalizable folding mapping where {n_{k}}_{k=1}^{∞} is the maximum sequence of integers such that f_{0} is renormalizable about m_{k} for m_{k} = ∏_{i=1}^{k} n_{i}. Let I_{k} = [−a_{k}, a_{k}] be the maximal interval containing 0 (set m_{0} = 0 and I_{0} = [−1, 1]) such that

(a) f_{0}^{∘m_{k}} is monotone when restricted to [−a_{k}, 0] and to [0, a_{k}],

(b) f_{0}^{∘m_{k}}(I_{k}) ⊂ I_{k},

(c) I_{k}, f_{0}(I_{k}), …, f_{0}^{∘(m_{k}−1)}(I_{k}) have pairwise disjoint interiors, and

(d) f_{0}^{∘m_{k}} has exactly two fixed points β_{k} and α_{k} in I_{k} where a_{k} = |β_{k}|. There is a domain U_{k} ⊇ I_{k} such that (U_{k}, V_{k}, f^{∘m_{k}}) is a quadratic-like map with connected filled-in Julia set K_{k}, where K_{k} is the k^{th}-renormalization of K_{1}. Then β_{k} is the non-separate fixed point and α_{k} is the separate fixed point of this quadratic-like map and then K_{k} ∩ R^{1} = I_{k}. An infinitely renormalizable real quadratic-like map is called Feigenbaum-like if n_{k} = 2 for all k.

Suppose that c(i) = f^{∘i}(0) is the i^{th} critical value of f and that J_{k}(i) is the interval bounded by c(i) and c(m_{k} + i) for k ≥ 0 and 0 ≤ i < m_{k}. We note that J_{k}(0) = J_{k}(m_{k}). Then f_{0} : J_{k}(0) → J_{k}(1) is folding for all k ≥ 0 and f_{0} : J_{k}(i) → J_{k}(i + 1) is a homeomorphism for every k ≥ 1 and 1 ≤ i < m_{k}. Let ζ_{k} = {J_{k}(i)}_{0≤i<m_{k}} for k ≥ 0. Let I_{k}(i) = f_{0}^{∘i}(I_{k}), 0 ≤ i < m_{k}, and let ξ_{k} = {I_{k}(i)}_{0≤i<m_{k}}. Note that J_{k}(i) ⊆ I_{k}(i). We use LI_{k}(i) and RI_{k}(i) to denote the intervals in ξ_{k} adjacent to I_{k}(i) and on the left and right sides of I_{k}(i), respectively (there is only LI_{k}(1) or RI_{k}(2) in ξ_{k}). Let LI_{k}^{+}(i) be the smallest interval containing LI_{k}(i) and the left end-point of I_{k}(i) and let RI_{k}^{+}(i) be the smallest interval containing RI_{k}(i) and the right end-point of I_{k}(i), for i = 0 or 3 ≤ i < m_{k}. Let LI_{k}^{+}(2) = [−1, c(2)] and RI_{k}^{+}(1) = [c(1), 1]. Similarly, we can define LJ_{k}(i) and RJ_{k}(i) and LJ_{k}^{+}(i) and RJ_{k}^{+}(i) for 0 ≤ i < m_{k}. The following theorem is due to Sullivan.

Theorem 19 [SU2]. There is a constant C > 0 such that

min{|LI_{k}^{+}(i)|, |RI_{k}^{+}(i)|} ≥ C · |I_{k}(i)|,

and such that

min{|LJ_{k}^{+}(i)|, |RJ_{k}^{+}(i)|} ≥ C · |J_{k}(i)|.

for all k ≥ 0 and 0 ≤ i < m_{k}.

Consider the slit domain V_{0} = V \[c(1), ∞). The map f|V_{0} has two inverse branches (see Fig. 5)

g_{0} : V_{0} → U_{0,0} = U ∩ {z = x + yi ∈ C | x < 0}

and

g_{1} : V_{0} → U_{0,1} = U ∩ {z = x + yi ∈ C | x > 0}.

For each k ≥ 1, let g_{s(i)} : J_{k}(i + 1) → J_{k}(i) for 1 ≤ i < m_{k} where s(i) = 0 if J_{k}(i) is contained in the negative half of the real line, and otherwise s(1) = 1. Then

A_{k} = g_{s(1)} ∘ g_{s(2)} ∘ ⋯ ∘ g_{s(m_{k}−1)} : J_{k}(m_{k}) = J_{k}(0) → J_{k}(1)

is a homeomorphism and can be extended homeomorphically to the maximum closed interval

T_{k}(m_{k}) ⊇ LJ_{k}^{+}(m_{k}) ∪ J_{k}(m_{k}) ∪ RJ_{k}^{+}(m_{k}).

Furthermore, A_{k} can be extended analytically to V_{k} = (V \ R^{1}) ∪ T̊_{k}(m_{k}). Let us continue to use A_{k} to denote this extension. Let U′_{k} = A_{k}(V_{k}), and let U_{k} = f^{−1}(U′_{k}) be the pre-image of U′_{k} under f. Since f has no attractive and no parabolic periodic point, then g_{0}(A_{k}(T_{k}(m_{k}))) and g_{1}(A_{k}(T_{k}(m_{k}))) are contained strictly in T_{k}(m_{k}). Thus, \overline{U}_{k} ⊂ V_{k} and

f^{∘m_{k}} : U_{k} → V_{k}

is a quadratic-like map (see Fig. 6).

Let T_{k}(i) = g_{s(i)}(T_{k}(i + 1)) for i = m_{k} − 1, m_{k} − 2, …, 1. Let T_{k}(0) = f_{0}^{−1}(T_{k}(1)). Then

J_{k}(0) = J_{k}(m_{k}) ⊆ I_{k} ⊆ T_{k}(0) ⊂ T_{k}(m_{k}).

The interval T_{k}(m_{k}) is bounded by two critical values c(q(k)) and c(r(k)) of f; one of them is a maximum value and the other is a minimum value of f^{∘m_{k}}. Suppose T_{k}(1) = [d_{1}, e_{1}] where d_{1} < c(1) < e_{1}. Let d_{i} = f^{∘i}(d_{1}) and let e_{i} = f^{∘i}(e_{1}) for 1 ≤ i ≤ m_{k}. Note that d_{m_{k}} and e_{m_{k}} are c(r(k)) and c(q(k)). We normalize T_{k}(i) into the unit interval [0, 1]: let l_{i} : T_{k}(i) → [0, 1] be the linear map such that l_{i}(d_{i}) = 0 and l_{i}(e_{i}) = 1 and let

g̃_{s(i)} = l_{i} ∘ g_{s(i)} ∘ l_{i+1}^{−1}

for i = m_{k} − 1, …, 2, 1. The map g̃_{s(i)} fixes 0 and 1 and is a univalent function defined on

C_{[0,1],V} = ((C \ R^{1}) ∪ (0, 1)) ∩ V.

The restriction g̃_{s(i)}|[0, 1] is a homeomorphism of [0, 1]. Let

L_{k} = g̃_{s(1)} ∘ g̃_{s(2)} ∘ ⋯ ∘ g̃_{s(m_{k}−1)}

for k ≥ 1.

Lemma 13. Suppose (U, V, f) is a Feigenbaum-like quadratic-like map. Then

{L_{k}}_{k=3}^{∞}

are uniform root-like maps.

Proof. Since f is a Feigenbaum-like map, we have m_{k} = 2^{k} and T_{k}(m_{k}) = J_{k−2}(0) for 3 ≤ k < ∞. For any k ≥ 3, consider the homeomorphism f^{∘(2^{k−1}−1)} : J_{k}(2^{k−1} + 1) → J_{k}(2^{k}) = J_{k}(0). Its inverse g_{s(2^{k−1}+1)} ∘ ⋯ ∘ g_{s(2^{k}−1)} : J_{k}(2^{k}) → J_{k}(2^{k−1} + 1) can be extended to T_{k−1}(2^{k−1}), that is, we can consider

g_{s(2^{k−1}+1)} ∘ ⋯ ∘ g_{s(2^{k}−1)} : T_{k−1}(2^{k−1}) → T_{k−1}(1).

Let

G_{0} = g̃_{s(2^{k−1}+1)} ∘ ⋯ ∘ g̃_{s(2^{k}−1)}

and let

L_{a_{0}} = g̃_{s(2^{k−1})}.

Then G_{0} is a univalent map (or holomorphic embedding) of C_{[0,1],V} such that G_{0}(0) = 0 and G_{0}(1) = 1 and such that G_{0}|[−1, 1] is a homeomorphism of [0, 1]. The map L_{a_{0}} is a square root at 0 (see Fig. 7).

Consider the homeomorphism f^{∘(2^{k−2}−1)} : J_{k}(2^{k−2} + 1) → J_{k}(2^{k−1}). Its inverse g_{s(2^{k−2}+1)} ∘ ⋯ ∘ g_{s(2^{k−1}−1)} : J_{k}(2^{k−1}) → J_{k}(2^{k−2} + 1) can be extended to T_{k−2}(2^{k−2}), that is, we can consider

G′_{1} = g_{s(2^{k−2}+1)} ∘ ⋯ ∘ g_{s(2^{k−1}−1)} : T_{k−2}(2^{k−2}) → T_{k−2}(1).

Let

G_{1} = g̃_{s(2^{k−2}+1)} ∘ ⋯ ∘ g̃_{s(2^{k−1}−1)}

and let

L_{a_{1}} = g̃_{s(2^{k−2})}.

The map G_{1} is a univalent map of C_{[0,1],V} and G_{1}|[−1, 1] is a homeomorphism of [0, 1]. The map L_{a_{1}} is a square root at a_{1}. The pre-image of c(1) under G′_{1} is c(2^{k−2}). One of the end-points of T_{k}(2^{k−1}) is 0; the other one is between J_{k−1}(0) and one of LJ_{k−1}(0) or RJ_{k−1}(0). From Theorem 19, two components of T_{k−2}(2^{k−2})\T_{k}(2^{k−1}) have lengths greater than a constant C (obtained from Theorem 19) times the length of T_{k}(2^{k−1}). Thus (L_{a_{1}}, G_{1}) is a compatible pair and satisfies (i) and (iii) of Definition 4 (see Fig. 7).

Next we consider the homeomorphism f^{∘(2^{k−3}−1)} : J_{k}(2^{k−3} + 1) → J_{k}(2^{k−2}). Its inverse g_{s(2^{k−3}+1)} ∘ ⋯ ∘ g_{s(2^{k−2}−1)} : J_{k}(2^{k−2}) → J_{k}(2^{k−3} + 1) can be extended to T_{k−3}(2^{k−3}); that is, we can consider

G′_{2} = g_{s(2^{k−3}+1)} ∘ ⋯ ∘ g_{s(2^{k−2}−1)} : T_{k−3}(2^{k−3}) → T_{k−3}(1).

Let

G_{2} = g̃_{s(2^{k−3}+1)} ∘ ⋯ ∘ g̃_{s(2^{k−2}−1)}

and let

L_{a_{2}} = g̃_{s(2^{k−3})}.

The map G_{2} is a univalent map of C_{[0,1],V} and G_{2}|[−1, 1] is a homeomorphism of [0, 1]. The map L_{a_{2}} is a square root at a_{2}. The preimage of c(1) under G′_{2} is c(2^{k−3}). The interval T_{k}(2^{k−2}) is contained in T_{k−1}(2^{k−1}). From Theorem 19, two components of T_{k−3}(2^{k−3})\T_{k}(2^{k−2}) have lengths greater than a constant C (obtained from Theorem 19) times the length of T_{k}(2^{k−2}). Thus (L_{a_{2}}, G_{2}) is a compatible pair and satisfies (iii) of Definition 4 (see Fig. 7).

In general, for 3 < i ≤ k − 1, consider the homeomorphism f^{∘(2^{k−i}−1)} : J_{k}(2^{k−i} + 1) → J_{k}(2^{k−i+1}). Its inverse g_{s(2^{k−i}+1)} ∘ ⋯ ∘ g_{s(2^{k−i+1}−1)} : J_{k}(2^{k−i+1}) → J_{k}(2^{k−i} + 1) can be extended to T_{k−i}(2^{k−i}); that is, we can consider

G′_{i+1} = g_{s(2^{k−i}+1)} ∘ ⋯ ∘ g_{s(2^{k−i+1}−1)} : T_{k−i}(2^{k−i}) → T_{k−i}(1).

Let

G_{i+1} = g̃_{s(2^{k−i}+1)} ∘ ⋯ ∘ g̃_{s(2^{k−i+1}−1)}

and let

L_{a_{i+1}} = g̃_{s(2^{k−i})}.

The map G_{i+1} is a univalent map of C_{[0,1],V} and G_{i+1}|[0, 1] is a homeomorphism. The map L_{a_{i+1}} is a square root at a_{i+1}. The preimage of c(1) under G′_{i+1} is c(2^{k−i}). The interval T_{k}(2^{k−i+1}) is contained in T_{k−i+2}(2^{k−i+2}). From Theorem 19, two components T_{k−i}(2^{k−i}) \ T_{k}(2^{k−i+1}) have lengths greater than a constant C times the length of T_{k}(2^{k−i+1}). Thus (L_{a_{i+1}}, G_{i+1}) is a compatible pair and satisfies (iii) of Definition 4.

One of the endpoints of T_{k}(2^{k−i} + 1) is to the left of c(2^{k−i} + 1); the other is in the interval in ζ_{k} which is adjacent to J_{k}(2^{k−i} + 1). The branch point of g_{s(2^{k−i})} is always c(1). From Theorem 19, we have a constant λ > 1 such that {a_{i}}_{i=1}^{k} satisfies (ii) of Definition 4 for C = 1.

40 YUNPING JIANG

For all k ≥ 3, we therefore decompose

L_{k} = L_{a_{k}} ∘ G_{k} ∘ L_{a_{k−1}} ∘ G_{k−1} ∘ ⋯ ∘ L_{a_{1}} ∘ G_{1} ∘ L_{a_{0}} ∘ G_{0}.

From the construction above, L_{k}, 3 ≤ k < ∞, is uniform root-like map. Let (U, V, f) be a Feigenbaum-like quadratic-like map. Let m_{k} = 2^{k}. Consider the renormalizations

f^{∘m_{k}} : U_{k} → V_{k}

for 1 ≤ k < ∞ where V_{k} = (V \ R^{1}) ∪ T̊k(m_{k}). Let U_{k} ∩ R^{1} = [−o_{k}, o_{k}]. Let w_{k,θ} be the the ray starting at o_{k} with slop tan θ for 0 < θ < π/2. Let 0 ∈ R_{k,θ} be the domain bounded by w_{k,θ}, −w_{k,θ}, \overline{w}_{k,θ}, and −\overline{w}_{k,θ}.

Lemma 14. Suppose (U, V, f) is a Feigenbaum-like quadratic-like map. Then there is a constant angle θ_{0} > 0 such that

U_{k} ⊆ R_{k,θ_{0}}

for all k ≥ 0 (see Fig. 8).

Proof. Consider f^{∘(m_{k}−1)} : J_{k}(1) → J_{k}(0). Its inverse has the maximum extension

A_{k} = g_{s(1)} ∘ g_{s(2)} ∘ ⋯ ∘ g_{s(m_{k}−1)} : V_{k} = (V \ R^{1}) ∪ T̊_{k}(m_{k}) → U′_{k}.

Remember that

L_{k} = l_{1} ∘ A_{k} ∘ l_{m_{k}}^{−1}

where l_{m_{k}} and l_{1} are the linear maps normalizing T_{m_{k}} and T_{k}(1), respectively, to [0, 1]. Let

w′_{k,θ} = {z ∈ C | arg(z − d_{1}) = θ, ℑ(z) > 0}

be the ray starting at d_{1} with angle 0 ≤ θ ≤ π where U′_{k} ∩ R^{1} = T_{k}(1) = [d_{1}, e_{1}] with d_{1} < c(1) < e_{1}. Let R′_{k,θ} be the sector containing c(1) bounded by w′_{k,θ} and \overline{w′}_{k,θ}. Applying Theorem 18, there is a constant angle 0 < θ_{1} ≤ π such that U′_{k} is contained in a sector domain R′_{k,θ_{1}} (see Fig. 8). Since f : U → V is a quadratic-like map, it is comparable with z ↦ z^{2} near 0. So there is a constant angle 0 < θ_{0} < π/2 depending on θ_{1} such that U_{k} = f^{−1}(U′_{k}) is contained in R_{k,θ_{0}} (see Fig. 8).

Take I_{0} = (−1, 1). Let C_{I_{0}} = (C \ R^{1}) ∪ I_{0}. Let d = d_{H,C_{I_{0}}} be the hyperbolic distance on C_{I_{0}}. Let Ω_{r} = {z ∈ C_{I_{0}} | d(z, I_{0}) < r} be a hyperbolic neighborhood. From Lemma 6, Ω_{r} is the union of two disks D_{β}^{+} and D_{β}^{−} centered at c_{β}^{+} = i cot β and c_{β}^{−} = −i cot β with radii R_{β}^{+} = R_{β}^{−} = 1/ sin β where β is the angle between ∂D_{β}^{+} and the line [1, ∞) at 1 (see Fig. 9). Using the law of cosines for the triangle Δ(c0z), for any z = re^{iϕ} in ∂D_{β}^{+},

r = cot β sin ϕ + \sqrt{csc^{2} β − cot^{2} β cos^{2} ϕ}.

Let q(z) = \sqrt{z} be the square root from C \ \{x < 0\} to the right half-plane RH. Let Π_{β}^{+} = q(D_{β}^{+}). For 0 < β < π/4, consider Ω_{r′} = D_{2β}^{+} ∪ D_{2β}^{−}. Let z_{0} = re^{iτ} ≠ 1 be the intersection point of ∂D_{2β}^{+} and Π_{β}^{+}. Then τ is the unique non-zero solution of the equation

\sqrt{cot β sin(2ϕ) + \sqrt{csc^{2} β − cot^{2} β cos^{2}(2ϕ)}}

= cot(2β) sin ϕ + \sqrt{csc^{2}(2β) − cot^{2}(2β) cos^{2} ϕ}.

Thus τ = τ(β) is a strictly increasing function and τ → 0 as β → 0. Let z_{0} −1 = r̃e^{iθ}. Then θ = θ(τ) is a strictly increasing function and θ → 0 as τ → 0. Let θ = θ ∘ τ(β). It is a strictly decreasing function and θ → 0 as β → 0. Let β = β(θ) be its inverse function.

For any 0 < θ_{0} < π/2, let 0 < β_{0} = β(θ_{0}) < π/2 and let

Q_{θ_{0}} = \{z ∈ Π_{β_{0}} | θ_{0} ≤ arg(z − 1) ≤ \frac{π}{2} \}.

Then Q_{θ_{0}} ⊆ D_{2β_{0}}^{+}. Let \overline{Q}_{θ_{0}} = \{\overline{z} | z ∈ Q_{θ_{0}}\} and let S_{θ_{0}} = Q_{θ_{0}} ∪(−Q_{θ_{0}})∪\overline{Q}_{θ_{0}} ∪(−\overline{Q}_{θ_{0}}). For a number 0 < ν_{0} < 1, let

ν_{0} · S_{θ_{0}} = {w = ν_{0} · z | z ∈ S_{θ_{0}}}.

Let A_{θ_{0}} = Ω_{r} \ (ν_{0}S_{θ_{0}}). From the above calculation, we have

Lemma 15. There is a constant C = C(θ_{0}, ν_{0}) > 0 depending on θ_{0} and on ν_{0} such that the modulus mod(A_{θ_{0}}) of the annulus A_{θ_{0}} is greater than C.

Proof. Let a = diam(ν_{0}S_{θ_{0}}) be the diameter of ν_{0}S_{θ_{0}} and let b = d(∂(ν_{0}S_{θ_{0}}), ∂Ω_{r}) be the distance between ∂(ν_{0}S_{θ_{0}}) and ∂Ω_{r}. Then a/b is bounded from above by a constant depending only on θ_{0} and on ν_{0}. This implies the lemma (by using Grötzsch argument (refer to [AH1])).

Suppose 0 < C_{0} < 1 is a constant. If we use q_{a}(z) = \sqrt{z − a}/\sqrt{1 − a} for |a| < C_{0} to replace q(z) in the above calculation, then Lemma 15 has a generalized version.

Lemma 15′. There is a constant C = C(θ_{0}, ν_{0}, C_{0}) > 0 depending on θ_{0}, ν_{0}, and C_{0} such that the modulus mod(A_{θ_{0}}) of the annulus A_{θ_{0}} is greater than C.

Theorem 20 [SU2]. A Feigenbaum-like quadratic-like map (U, V, f) has the a priori complex bounds and is unbranched.

Proof. We use the same notation as in the previous lemmas. Suppose (U, V, f) is a Feigenbaum-like quadratic-like map. Let m_{k} = 2^{k}. Consider the renormalizations

f^{∘m_{k}} : U_{k} → V_{k}

for 1 ≤ k < ∞ where V_{k} = (V \ R^{1}) ∪ T̊_{k}(m_{k}) and where U_{k} ∩ R^{1} = [−o_{k}, o_{k}]. From Lemma 14, there is a constant angle 0 < θ_{0} < π/2 such that

U_{k} ⊆ R_{k,θ_{0}}

for k > 0. Let β_{0} = β(θ_{0}).

Consider f^{∘(m_{k}−1)} : J_{k}(1) → J_{k}(0). Its inverse has the maximum extension

A_{k} = g_{s(1)} ∘ g_{s(2)} ∘ ⋯ ∘ g_{s(m_{k}−1)} : V_{k} → U′_{k}

where U′_{k} ∩ R^{1} = T_{k}(1) = [d_{1}, e_{1}] with d_{1} < c(1) < e_{1}. From Theorem 19 (and the bounded geometry property of the attractive Cantor set), there is a constant C_{0} > 0 such that C_{0}^{−1} ≤ |c(1) − d_{1}|/|e_{1} − c(1)| < C_{0} for all k > 0. We normalize T̊_{k}(m_{k}) = (d_{m_{k}}, e_{m_{k}}) to (−1, 1) by the linear map s_{1} such that s_{1}(d_{m_{k}}) = 1 and s_{1}(e_{m_{k}}) = −1. We normalize (0, o_{k}) to (0, 1) by the linear map s_{2} such that s_{2}(0) = 0 and s_{2}(o_{k}) = 1. We normalize T̊_{k}(1) = (d_{1}, e_{1}) to (−1, 1) by the linear map s_{3} such that s_{3}(e_{1}) = −1 and s_{3}(d_{1}) = 1. Let a = s_{3}(c(1)). Then there is a constant we still denote it as 0 < C_{0} < 1 such that |a| < C_{0} for all k > 0. There is an integer n_{0} > 0 such that for any k > n_{0}, Ω_{r} = D_{β_{0}}^{+} ∪D_{β_{0}}^{−} is contained in s_{1}(V). Let B_{k} = s_{3} ∘A_{k} ∘s_{1}^{−1} and let q_{a}(z) = s_{2} ∘ g_{s_{m_{k}}} ∘ s_{3}^{−1}. Then q_{a} is comparable with \sqrt{z − a}/\sqrt{1 − a}. Since B_{k} contracts the hyperbolic distance d_{H,C_{I_{0}}}, then B_{k}(Ω_{r}) ⊆ Ω_{r}. From Lemma 14,

X′_{k} = q_{a}(B_{k}(Ω_{r})) ⊆ S_{θ_{0}} = Q_{θ_{0}} ∪ (−Q_{θ_{0}}) ∪ \overline{Q}_{θ_{0}} ∪ (−\overline{Q}_{θ_{0}}).

Let X′′_{k} = s_{1} ∘ s_{3}^{−1}(X′_{k}). Then X′′_{k} ⊆ ν_{0}S_{θ_{0}} for all k > 0 where ν_{0} > 0 is a constant obtained from Lemma 4.2. Let Y′′_{k} = Ω_{r}. Then from Lemma 15′,

mod(Y′′_{k} \ X′′_{k}) > C

for all k > n_{0} where C > 0 is a constant.

Now let X_{k} = s_{1}^{−1}(X′′_{k}) and let Y_{k} = s_{1}^{−1}(Y′′_{k}). Then

f^{∘m_{k}} : X_{k} → Y_{k}

is quadratic-like map and mod(Y_{k} \X_{k}) > C for all k > n_{0}. This means that (U, V, f) has the a priori complex bounds.

Let W_{k} = Y_{k} \ (LJ_{k}(0) ∪ RJ_{k}(0)). Applying Theorem 19 and the above argument, there is a constant C′ > 0 such that mod(W_{k} \ K_{k}) > C′ for all k > n_{0} where K_{k} is the filled-in Julia set of f^{∘m_{k}} : X_{k} → Y_{k}. But W_{k} ∩ CO = {c(jm_{k})}_{j=0}^{∞}. So (U, V, f) is unbranched. This completes the proof.

Theorem 20 and Theorem 16 now give us that

Corollary 1. The filled-in Julia set K_{f} of a Feigenbaum-like quadratic-like map (U, V, f) is locally connected.

Sullivan [SU2] (see also [MEV]) also proved that any real infinitely renormalizable quadratic-like map f : U → V of bounded type has the a priori complex bounds. Thus it is unbranched and its filled-in Julia set is locally connected from Theorem 16.

8. The Local Connectivity of Certain Infinitely Renormalizable Quadratic Julia Sets

We prove in this section the following result.

Theorem 21. There is a subset Υ̃ in M such that (1) Υ̃ is dense in the boundary ∂M of the Mandelbrot set M, (2) for every c in Υ̃, P_{c} is unbranched, infinitely renormalizable and has the a priori complex bounds.

From Theorem 16, we have

Corollary 2. The filled-in Julia set K_{c} of P_{c} is locally connected for every c in Υ̃.

Proof of Theorem 21. Suppose c_{0} is a Misiurewicz point in M. Then there is an integer m > 1 such that p = P_{c_{0}}^{∘m}(0) is a repelling periodic point of period k ≥ 1. Let α be the separate fixed point of P_{c_{0}}. Without loss of generality, we assume that P_{c_{0}} is non-renormalizable. (If P_{c_{0}} is renormalizable, it must be finitely renormalizable. We would then take α as the separate fixed point of the last renormalization of P_{c_{0}} (see §4)). Let Γ be the union of a cycle of external rays landing at α. Let γ be a fixed equipotential curve of P_{c_{0}}. Using Γ and γ, we construct the two-dimensional Yoccoz puzzle as follows (see §3). Let C_{−1} be the domain bounded by γ. The set Γ cuts C_{−1} into a finite number of closed domains. Let η_{0} denote the set of these domains. Let η_{n} = P_{c_{0}}^{−n}(η_{0}). Let C_{n} be the member of η_{n} containing 0.

Let

p ∈ ⋯ ⊆ D_{n}(p) ⊆ D_{n−1}(p) ⊆ ⋯ ⊆ D_{1}(p) ⊆ D_{0}(p)

be a p-end, where D_{n}(p) ∈ η_{n}. Let

c_{0} ∈ ⋯ ⊆ E_{n}(c_{0}) ⊆ E_{n−1}(c_{0}) ⊆ ⋯ ⊆ E_{1}(c_{0}) ⊆ E_{0}(c_{0})

be a c_{0}-end, where E_{n}(c_{0}) ∈ η_{n}. We have P_{c_{0}}^{∘(m−1)}(E_{n+m−1}(c_{0})) = D_{n}(p).

Since the diameter diam(D_{n}(p)) tends to zero as n → ∞ and since p is a repelling periodic point, we can find an integer l ≥ m such that |(P_{c_{0}}^{∘k})′(x)| ≥ λ > 1 for all x ∈ D_{l}(p) and such that

P_{c_{0}}^{∘(m−1)} : E_{l+m−1}(c_{0}) → D_{l}(p)

is a homeomorphism. Let q > 0 be the smallest integer such that P_{c_{0}}^{∘q}(D_{l}(p)) contains 0, i.e., it is C_{r_{0}} in η_{r_{0}} where r_{0} ≥ 0. Then

f = P_{c_{0}}^{∘q} : D_{l}(p) → C_{r_{0}}

is a homeomorphism. Let r > r_{0} be an integer such that B_{0} = f^{−1}(C_{r}) ⊂ D_{l}(p) does not contain p where C_{r} is the member of η_{r} containing 0. Then

P_{c_{0}}^{∘q} : B_{0} → C_{r}

is a homeomorphism. The domain B_{0} is a member of η_{r+q}. Let B_{n} ⊆ D_{l+nk} be the pre-image of B_{0} under P_{c_{0}}^{∘nk}|D_{l+nk}(p) for n ≥ 1. The domain B_{n} is a member of η_{r+q+nk} and

P_{c_{0}}^{∘(q+nk)} : B_{n} → C_{r}

is a homeomorphism.

From the structural stability theorem (see [PRZ,SHU]), the points α and p and the sets Γ, C_{r}, D_{n} and B_{n} for n ≥ 1 are all preserved by a small perturbation c of c_{0} (refer to [JI4]). Therefore they can be constructed for P_{c} as long as c near c_{0}. Let U_{0} be a neighborhood about c_{0} such that the corresponding points α(c) and p(c) and the corresponding sets Γ(c), C_{r}(c), D_{n}(c), and B_{n}(c) for n ≥ 1 are all preserved for c ∈ U_{0}. Moreover, as n goes to infinity, the diameter diam(B_{n}(c)) tends to zero and the set B_{n}(c) approaches to p(c) uniformly on U_{0}. Let

W_{n} = W_{n}(c_{0}) = {c ∈ C | P_{c}^{m}(0) ∈ B_{n}(c)}.

Then from the result in [JI4], W_{n} ⊆ U_{0} for n large enough.

For any c ∈ W_{n}, Let R_{n}(c) be the preimage of B_{n}(c) under the map P_{c_{0}}^{∘(m−1)} : E_{l+m−1}(c) → D_{l}(p, c) and let C_{r+q+nk+m}(c) = P_{c}^{−1}(R_{n}(c)). The domain C_{r+q+nk+m}(c) is the member containing 0 in η_{r+q+m+nk} and

F_{n,c} = P_{c}^{∘(q+nk+m)} : X_{n}(c) = C̊_{r+q+nk+m}(c) → Y_{n}(c) = C̊_{r}(c)

is a quadratic-like map. Let A_{n}(c) = C̊_{r}(c) \ C_{r+q+nk+m}(c). Since the diameter diam(C_{r+q+nk+m})(c)) tends to zero as n goes to infinity uniformly in U_{0}, there is an integer N_{0} > 0 such that

mod(A_{n}(c)) ≥ 1

for all n ≥ N_{0} and c ∈ W_{n}. Since

{F_{n,c} : X_{n} → Y_{n} | c ∈ W_{n}}

is a full family of quadratic-like maps, W_{n} contains a copy M_{n} = M_{n}(c_{0}) of the Mandelbrot set M (see [DH3]). For any c ∈ M_{n}, P_{c} is once renormalizable and

CO(c) ∩ C_{r+q+nk+m}(c) = {c(j(q + nk + m))}_{j=0}^{∞}

where CO(c) = {c(i) = P_{c}^{∘i}(0)}_{i=0}^{∞}. Let

Υ̃_{1}(c_{0}) = ∪_{n=N_{0}}^{∞}M_{n}.

We use the induction to complete the construction of the subset Υ̃(c_{0}) around c_{0}. Suppose we have constructed W_{w} where w = i_{0}i_{1} … i_{k−1} and i_{0} ≥ N_{0}, i_{1} ≥ N_{i_{1}}, …, i_{k−1} ≥ N_{i_{0}i_{1}...i_{k−2}}. There is a parameter c_{w} ∈ M_{w} such that

F_{w} = F_{w,c_{w}} : X_{w} = X_{w}(c_{w}) → Y_{w} = Y_{w}(c_{w})

is hybrid equivalent (see §1) to P (z) = z^{2} − 2. For F_{w}, let β_{w} and α_{w} be its nonseparate and separate fixed points. Let β̃_{w} be another preimage of β_{w} under F_{w}. Let Γ_{w} be the external rays of P_{c_{w}} landing at α_{w}. Let Y_{w0} be the domain containing 0 and bounded by ∂X_{w} and F_{w}^{−1}(Γ_{w}). Let β̃_{w} ∈ E_{w0} and β_{w} ∈ E_{w1} be the components of the closure of X_{w} \ Y_{w0}. Let G_{w0} and G_{w1} be the inverses of F_{w}|E_{w0} and F_{w}|E_{w1}. Let

D_{wn} = G_{w1}^{∘n}(D_{w0})

and

B_{wn} = G_{w0}(D_{w(n−1)})

for n ≥ 1. From the structural stability theorem (see [PRZ,SHU]), the points β_{w} and α_{w} and the sets Γ_{w} are all preserved by a small perturbation c of c_{w}. Therefore we can find a small neighborhood U_{w} about c_{w} such that the corresponding domains D_{wn}(c) and B_{wn}(c) can be constructed for P_{c}, c ∈ U_{w} (refer to [JI4]). Let

W_{wn} = {c ∈ C | F_{w,c}(0) ∈ B_{wn}(c)}.

The diameter diam(B_{wn}(c)) → 0 as n → ∞ uniformly on U_{w}. From the result in [JI4], W_{wn} ⊆ U_{w} for n large.

For each c in W_{wn}, n ≥ N_{w}, let X_{wn}(c) = F_{w,c}^{−1}(B̊_{wn}(c)) and Y_{wn}(c) = Y̊_{w0}(c). Then

F_{wn,c} = F_{w,c}^{∘(n+1)} : X_{wn}(c) → Y_{wn}(c)

is a quadratic-like map. Let

A_{wn}(c) = X_{wn}(c) \ \overline{Y}_{wn}(c).

Since the diameter diam(Y_{wn}(c)) tends to zero as n goes to infinity uniformly in U_{w}. There is an integer N_{w} > 0 such that

mod(A_{wn}(c)) ≥ 1

for all n ≥ N_{w} and c ∈ W_{wn}. Since

{F_{wn,c} : X_{wn} → Y_{wn} | c ∈ W_{wn}}

is a full family of quadratic-like maps, W_{wn} contains a copy M_{wn} = M_{wn}(c_{0}) of the Mandelbrot set M (see [DH3]). For any c ∈ M_{wn}, P_{c} is k-times renormalizable and A_{wn}(c) contains no critical values of P_{c}. Let

Υ̃_{k}(c_{0}) = ∪_{w} ∪_{n=N_{w}}^{∞} M_{wn}

where w runs over all sequences of integers of length k in the induction.

We have thus constructed a subset Υ̃(c_{0}) = ∩_{k=1}^{∞}Υ̃_{k}(c_{0}) such that every c ∈ Υ̃(c_{0}) is infinitely renormalizable and such that c_{0} is a limit point of Υ̃(c_{0}). From the above construction, for every c ∈ Υ̃(c_{0}), P_{c} is unbranced and has the a priori complex bounds.

Let Υ̃ = ∪_{c_{0}} Υ̃(c_{0}) where c_{0} runs over all Misiurewicz points in M. Then for every c ∈ Υ̃, P_{c} is unbranched infinitely renormalizable and has the a priori complex bounds. Since the set of Misiurewicz points is dense in ∂M (see [CAG]), the set Υ̃ is dense in ∂M. It completes the proof of the theorem.

Remark 3. Douady (see [MI2]) constructed an example of an infinitely renormalizable quadratic polynomial whose filled-in Julia set is not locally connected.

References

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Department of Mathematics, Queens College of CUNY, Flushing, NY 11367 E-mail address: yunqc@yunping.math.qc.edu