LOCAL CONNECTIVITY OF THE MANDELBROT SET AT CERTAIN INFINITELY RENORMALIZABLE POINTS

YUNPING JIANG

Abstract. We construct a subset of the Mandelbrot set which is dense on the boundary of the Mandelbrot set and which consists of only infinitely renormalizable points such that the Mandelbrot set is locally connected at every point of this subset. We prove the local connectivity by finding bases of connected neighborhoods directly.

Contents

1. Introduction

2. Construction of bases of connected neighborhoods

1. Introduction

Consider a quadratic polynomial P_{c}(z) = z^{2} + c, where c is a complex parameter. The filled-in Julia set K_{c} of P_{c} is the set of points z such that the orbit O(z) = {P_{c}^{∘n}(z)}_{n=0}^{∞} is bounded. The Julia set J_{c} of P_{c} is the boundary of K_{c}. The Mandelbrot set M is the set of parameters c such that the critical orbit O(c) = {P_{c}^{∘n}(0)}_{n=0}^{∞} is bounded (see Fig. 1). Equivalently, M is the set of parameters c such that J_{c} is connected. For a parameter c in the complement of the Mandelbrot set M, the Julia set J_{c} is a Cantor set. Douady and Hubbard (see [2]) proved that the Mandelbrot set M is connected.

The Julia set of P_{0}(z) = z^{2} is the round circle S^{1}. The map P_{0} restricted to S^{1} is an expanding self-map. From the Structure Stability Theorem in smooth dynamical systems (see [10, 12]), the dynamical system (P_{c}, J_{c}) for any |c| small enough is topologically conjugate to (P_{0}, S^{1}). Thus J_{c} for any |c| small enough is a Jordan curve. It is also known that J_{c} for any |c| small enough is a quasi-circle (see [13]), and that the Hausdorff dimension HD(J_{c}) of J_{c} is 1 + |c|/(4 log 2) + o(|c|) (see [11]).

1991 Mathematics Subject Classification, Primary : 58F23; Secondary: 30C10

Partially supported by NSF grant and PSC-CUNY award

Figure 1. Computer generating picture of the boundary of the Mandelbrot set and some equipotential curves

The techniques in the study of the dynamics of P_{c} for any |c| small enough can be used to the study of the dynamics of a hyperbolic quadratic polynomial which is a quadratic polynomial having 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 main conjecture in this direction is that

Conjecture 1. The set HP is open and dense in M.

It is easy to verify that HP is open but to prove that HP is dense in M is a hard problem.

Douady and Hubbard [3] proved that this conjecture follows from

Conjecture 2. The Mandelbrot set M is locally connected.

In order to study the dynamics of quadratic polynomials more penetrating, Douady and Hubbard [4] introduced the concept of a quadratic-like map which is a proper, degree two, analytic branch cover f : U → V where U and V are open domains isomorphic to a disc and \overline{U} ⊂ V. A quadratic polynomial P_{c} is a quadratic-like map when restricted to any domain bounded by an equipotential curve (see [8]).

For a quadratic-like map f : U → V, let 0 be the unique branch point of f. The filled-in Julia set K_{f} of f is defined as

K_{f} = ∩_{n=0}^{∞}f^{−n}(U).

Suppose K_{f} is connected (otherwise it is a Cantor set). The map f is said to be once n_{1}-renormalizable if there are subdomains 0 ∈ U_{1} ⊂ U and V_{1} ⊂ V isomorphic to a disc such that

f_{1} = f^{∘n_{1}} : U_{1} → V_{1}

is a quadratic-like map with connected filled-in Julia set K_{f_{1}}. Otherwise, f is called non-renormalizable. If f_{1} : U_{1} → V_{1} is also n_{2}-renormalizable, then we call f is twice (n_{1}, n_{2})-renormalizable. So on one can define a k-times (n_{1}, n_{2}, ⋯, n_{k})- renormalizable quadratic-like map as well as an infinitely renormalizable quadraticlike map.

Suppose f : U → V is a quadratic-like map. A point p is called a periodic point of period n ≥ 1 of f if f^{∘n}(p) = p and f^{∘i}(p) ≠ p for 1 ≤ i < n. The number λ_{p} = (f^{∘n})′(p) is called the multiplier of f at p. A periodic point p of period n of f is called parabolic if λ_{p} = exp(2πip/q) where p and q are integers; irrationally indifferent if λ_{p} = exp(2πθi) where θ is irrational; repelling if |λ_{p}| > 1; attractive if 0 < |λ_{p}| < 1; super-attractive if λ_{p} = 0. For a quadratic polynomial P_{c}(z) = z^{2} + c, let P CO(c) = {P_{c}^{∘n}(0)}_{n=1}^{∞} be its post-critical orbit. We classify the parameters c in M \ HP as

P = {c ∈ M | P_{c} has a parabolic periodic point }, I = {c ∈ M | P_{c} has an irrationally indifferent periodic point }, N = \{c ∈ M | c is not in P ∪ I and 0 ∉ \overline{P CO(c)} \}.

FR = {c ∈ M | c is not in P ∪ I ∪ N and P_{c} is not infinitely renormalizable }, and

IR = {c ∈ M | P_{c} is infinitely renormalizable }.

In particular, a parameter c in N is called Misiurewicz if P CO(c) consists of only finitely many points. It is known that the set of all Misiurewicz points is dense on the boundary of the Mandelbrot set (see [1]). A point c in IR is called an infinitely renormalizable.

Recent work of Yoccoz (see [6]) shows that the Mandelbrot set M is locally connected at all points which are not infinitely renormalizable. (We would like to note that the local connectivity of the Mandelbrot set M at all points in P is assured by many experts [6, 15] but a proof is still needed). There remains the many infinitely renormalizable points for which to complete the proof of Conjecture 2. However, we prove the following theorem to further confirm Conjecture 2. Main Theorem. There is a subset Υ of the Mandelbrot set M which is dense on the boundary ∂M of the Mandelbrot set M and which consists of only infinitely renormalizable points such that the Mandelbrot set M is locally connected at every point c of Υ.

The proof of Main Theorem is in the next section. The talks with Professors Tan Lei, C. Petersen, and A. Douady are very helpful. The article of J. Hubbard [6] and the book of Carleson and Gamelin [1] are helpful during my understanding of the structure of the Mandelbrot set M and the work of Yoccoz. I would like to thank them very much. I also would like to thank Professor M. Shishikura for pointing out the work of Eckmann and Epstein [5]. Research is partially supported by NSF grant # DMS-9400974 and PSC-CUNY award # 6-65348. Research at MSRI is supported in part by NSF grant # DMS-9022140.

2. Construction of bases of connected neighborhoods

We give the proof of Main Theorem now. Let us start the construction of the subset Υ in Main Theorem around the simplest point −2 ∈ M. Consider the polynomial P (z) = z^{2} − 2 and its Julia set J = [−2, 2]. It has a non-separate fixed point β = 2 and a separate fixed point α, this menas that J \ {β} is still connected and J \ {α} is disconnected. Let γ be a fixed equipotential curve of P. Two external rays land at α. Let Γ be the union of these two external rays. Let D be the closed domain bounded by γ. Let D_{0} be the domain containing 0 and bounded by γ and P^{−1}(Γ) (see Fig. 2). The restriction P |(C \ (−∞, −2]) has two inverse branches

G_{0} : C \ (−∞, −2] → LH = {z = x + yi ∈ C | x < 0}

and

G_{1} : C \ (−∞, −2] → RH = {z = x + yi ∈ C | x > 0}.

Let D_{n} = G_{1}^{∘n}(D_{0}) and let B_{n} = G_{0}(D_{n−1}) for n ≥ 1 (see Fig. 2). Since 2 is an expanding fixed point of P and P (−2) = 2, the diameter diam(B_{n}) tends to zero exponentially as n goes to infinity.

Fig. 2 can be constructed as well for c in a small neighborhood about −2 because β, α, and Γ are structurally stable. Let 0 ∉ U_{0} be a small neighborhood about −2 with diam(U_{0}) ≤ 1 such that Fig. 2 is preserved for c in U_{0}. Let β(c), α(c), Γ(c), D_{n}(c) for 0 ≤ n < ∞, and B_{n}(c) for 1 ≤ n < ∞ be the corresponding points, sets, and domains for P_{c}, c ∈ U_{0}. Let \overline{β}(c) ≠ β(c) be another pre-image of β(c) under P_{c}. Then the diameter diam(B_{n}) tends to 0 and the set B_{n}(c) approaches to \overline{β}(c) as n goes to infinity uniformly on U_{0}.

Let

W_{n} = {c | c ∈ B_{n}(c)}.

Because W_{n} is bounded by external rays and an equipotential curve of the Mandelbrot set M,

M̃_{n} = W_{n} ∩ M

is connected. Since the equation P_{c}(0)−\overline{β}(c) = 0 has a unique solution −2 in U_{0}, the Rouché Theorem assures that P_{c}(0) − x = 0 has a unique solution for any x in B_{n}(c)

LOCAL CONNECTIVITY OF THE MANDELBROT SET AT CERTAIN POINTS 5

and large n and the solution is close to −2. Therefore, there is an integer N_{0} > 0 such that for n ≥ N_{0},

W_{n} ⊂ U_{0}.

That is,

diam(W_{n}) ≤ 1

for n ≥ N_{0} (see Fig. 3).

For each W_{n} where n ≥ N_{0}, since c ∈ B_{n}(c), C_{n}(c) = P_{c}^{−1}(B_{n}(c)) is a simply connected domain containing 0. It is also a subdomain of D_{0}(c). Let

F_{n,c} = P_{c}^{∘(n+1)} : C̊_{n}(c) → D̊_{0}(c)

is a quadratic-like map (see Fig. 4). The family

{F_{n,c} : C̊_{n}(c) → D̊_{0}(c) | c ∈ W_{n}}

is full (see [4]). So W_{n} contains a copy M_{n} of the Mandelbrot set M (see [4]). For c ∈ M_{n}, the Julia set J_{F_{n,c}} of F_{n,c} : C̊_{n}(c) → D̊_{0}(c) is connected. Therefore, for

c ∈ Υ_{1} = ∪_{n≥N_{0}}^{∞}M_{n},

P_{c} is once renormalizable.

For a fixed integer i_{0} ≥ N_{0}, consider W_{i_{0}} and M_{i_{0}}; there is a parameter c_{i_{0}} ∈ M_{i_{0}} such that

F_{i_{0}} = F_{i_{0},c_{i_{0}}} : C_{i_{0}} = C_{i_{0}}(c_{i_{0}}) → D_{i_{0}} = D_{0}(c_{i_{0}})

is hypbrid equivalent (see [4]) to P (z) = z^{2} − 2. The quadratic-like map F_{i_{0}} : C_{i_{0}} → D_{i_{0}} has the non-separate fixed point β_{i_{0}} and the separate fixed point α_{i_{0}}. Let \overline{β}_{i_{0}} be another pre-image of β_{i_{0}} under F_{i_{0}}. Let Γ_{i_{0}} be the external rays of P_{c_{i_{0}}} landing at α_{i_{0}}. Let D_{i_{0}0} be the domain containing 0 and bounded by ∂C_{i_{0}} and F_{i_{0}}^{−1}(Γ_{i_{0}}). Let \overline{β}_{i_{0}} ∈ E_{i_{0}0} and β_{i_{0}} ∈ E_{i_{0}1} be the components of the closure of C_{i_{0}} \ D_{i_{0}0}. Let G_{i_{0}0} and G_{i_{0}1} be the inverses of F_{i_{0}}|E_{i_{0}0} and F_{i_{0}}|E_{i_{0}1}. Let

D_{i_{0}n} = G_{i_{0}1}^{∘n}(D_{i_{0}0})

and let

B_{i_{0}n} = G_{i_{0}0}(D_{i_{0}(n−1)})

for n ≥ 1. Since β_{i_{0}}, α_{i_{0}}, and Γ_{i_{0}} are structurally stable, we can find a small neighborhood U_{i_{0}} about c_{i_{0}} with diam(U_{i_{0}}) ≤ 1/2 such that the corresponding domains B_{i_{0}}(c) can be constructed for P_{c}, c ∈ U_{i_{0}}. Let

W_{i_{0}n} = {c ∈ C | F_{i_{0},c}(0) ∈ B_{i_{0}n}(c)}.

The diameter diam(B_{i_{0}n}(c)) tends to zero and the set B_{i_{0}n}(c) approaches to \overline{β_{i_{0}}}(c) uniformly on U_{i_{0}} as n goes to infinity. Since the equation F_{i_{0},c}(0) − \overline{β}_{i_{0}}(c) = 0 has a unique solution c_{i_{0}}, the Rouché Theorem implies that F_{i_{0},c}(0) − x = 0 has a unique solution near c_{0} for any x ∈ B_{i_{0}n}(c) and any n large. Thus, there is an integer N_{i_{0}} ≥ 0 such that for n ≥ N_{i_{0}}, W_{i_{0}n} ⊆ U_{i_{0}}. Thus diam(W_{i_{0}n}) ≤ 1/2. Since W_{i_{0}n} for n ≥ N_{i_{0}} is bounded by external rays and an equipotential curve of M, M̃_{i_{0}n} = W_{i_{0}} ∩ M is connected. For each c in W_{i_{0}n}, n ≥ N_{i_{0}}, let C_{i_{0}n}(c) = F_{i_{0},c}^{−1}(B_{i_{0}n}(c)). Then

F_{i_{0}n,c} = F_{i_{0},c}^{∘(n+1)} : C̊_{i_{0}n}(c) → D̊_{i_{0}0}(c)

is a quadratic-like map. Moreover,

{F_{i_{0}n,c} | c ∈ W_{i_{0}n}}

is a full family. Thus W_{i_{0}n} contains a copy M_{i_{0}n} of the Mandelbrots set M. For

c ∈ Υ_{2} = ∪_{i_{0}≥N_{0}} ∪_{i_{1}≥N_{i_{0}}} M_{i_{0}i_{1}},

P_{c} is twice renormalizable.

We use the induction to complete the construction of the subset around −2. 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}}. Let v = i_{0} … i_{k−2}. There is a parameter c_{w} ∈ M_{w} such that

F_{w} = F_{w,c_{w}} : C_{w} = C_{w}(c_{w}) → D_{w} = D_{v0}(c_{w})

is hypbrid equivalent to P (z) = z^{2} − 2. The quadratic-like map F_{w} : C_{w} → D_{w} has the non-separate fixed point β_{w} and the separate fixed point α_{w}. Let \overline{β}_{w} be another pre-image of β_{w} under F_{w}. Let Γ_{w} be the external rays of P_{c_{w}} landing at α_{w}. Let D_{w0} be the domain containing 0 and bounded by ∂C_{w} and F_{w}^{−1}(Γ_{w}). Let \overline{β}_{w} ∈ E_{w0} and β_{w} ∈ E_{w1} be the components of the closure of C_{w} \ D_{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 let

B_{wn} = G_{w0}(D_{w(n−1)})

for n ≥ 1. Since β_{w}, α_{w}, and Γ_{w} are structurally stable, we can find a small neighborhood U_{w} about c_{w} with diam(U_{w}) ≤ 1/2^{k} such that the corresponding domains D_{wn}(c) and B_{wn}(c) can be constructed for P_{c}, c ∈ U_{w}. Let

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

The diameter diam(B_{wn}(c)) tends to zero and the set B_{wn}(c) approaches to \overline{β_{w}}(c) uniformly on U_{w} as n goes to infinity. Since the equation F_{w,c}(0) − \overline{β}_{w}(c) = 0 has a unique solution c_{w}, the Rouché Theorem implies that F_{w,c}(0) − x = 0 for any x in B_{wn}(c) and for n large has a unique solution which is close to c_{w}. Therefore, there is an integer N_{w} ≥ 0 such that for n ≥ N_{w}, W_{wn} ⊆ U_{w}. Thus diam(W_{wn}) ≤ 1/2^{k}. Because W_{wn} for n ≥ N_{i_{1}} is bounded by external rays and an equipotential curve of M,

M̃_{wn} = W_{wn} ∩ M

is connected.

For each c in W_{wn}, n ≥ N_{w}, let C_{wn}(c) = F_{w,c}^{−1}(B_{wn}(c)). Then

F_{wn,c} = F_{w,c}^{∘(n+1)} : C̊_{wn}(c) → D̊_{w0}(c)

is a quadratic-like map. Moreover,

{F_{wn,c} : C̊_{wn}(c) → D̊_{w0}(c) | c ∈ W_{wn}}

is a full family. Thus W_{wn} contains a copy M_{wn} ⊂ M̃_{wn} of the Mandelbrot set M. For

c ∈ Υ_{k+1} = ∪_{w} ∪_{i_{k}≥N_{w}} M_{wi_{k}},

P_{c} is (k + 1)-times renormalizable where w = i_{0}i_{1} … i_{k−1} runs over all sequences of integers of length k in the induction.

We have thus constructed a subset Υ(−2) = ∩_{k=1}^{∞}Υ_{k} such that for each c ∈ Υ(−2), P_{c} is infinitely renormalizable and such that −2 is a limit point of Υ(−2). For each c ∈ Υ(−2), there is a corresponding sequence w_{∞} = i_{0}i_{1} … i_{k} … of integers such that

{c} = ∩_{k=0}^{∞}W_{i_{0}...i_{k}}.

Since

M̃_{i_{0}...i_{k}} = W_{i_{0}...i_{k}} ∩ M

is connected,

{W_{i_{0}...i_{k}}}_{k=0}^{∞}

is a basis of connected neighborhoods of the Mandelbrot M at c. In other words, the Mandelbrot set M is locally connected at c.

We prove further, after understood the above construction, that for any Misiurewicz point, there is a subset Υ(c_{0}) such that every c in Υ(c_{0}) is infinitely renormalizable, such that the Mandelbrot set M is locally connected at every point c in Υ(c_{0}), and such that c_{0} is a limit point of Υ(c_{0}) as follows.

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 [7])). 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 can construct the two-dimensional Yoccoz puzzle as follows (see [7] for the notation). Let C_{−1} be the domain bounded by γ. Then 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 for n ≥ 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 integer such that

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

is a homeomorphism, where C_{r_{0}} is the domain containing 0 in η_{r_{0}}, r_{0} ≥ 0. There is an integer r > r_{0} such that r + q > l and such that B_{0} = f^{−q}(C_{r}) does not contain p. Thus

P_{c_{0}}^{∘q} : B_{0} → C_{r}

is a homeomorphism. Define

B_{n} = (P_{c_{0}}^{∘nk}|D_{l+nk}(p))^{−1}(B_{0}) ⊆ D_{l+nk}(p)

for n ≥ 1. Note that B_{n} is in η_{r+q+nk}. Then

P_{c_{0}}^{∘(q+nk)} : B_{n} → C_{r}

is a homeomorphism.

Since α, p, Γ, C_{r}, D_{n}, and B_{n}, for n ≥ 0, are structurally stable, they can be constructed for P_{c} as long as c near c_{0}. Let U_{0} be a neighborhood about c_{0} with

Figure 5. A small copy of the Mandelbrot set and hairs around it

diam(U_{0}) ≤ 1 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 ≥ 0, 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)}.

Since the equation P_{c}^{∘m}(0) − p(c) = 0 has a unique solution c_{0} in U_{0} (see [3]), the Rouché Theorem implies that the equation P_{c}^{∘m}(0) − x = 0 has a unique solution for any x in B_{n}(c) and large n and that the solution is close to c_{0}. Therefore, there is an integer N_{0} = N_{0}(c_{0}) > 0 such that W_{n} ⊂ U_{0}. Thus diam(W_{n}) ≤ 1 for all n ≥ N_{0}. Since W_{n} is bounded by external rays and an equipotential curve of the Mandelbrot set M, then

M̃_{n} = M ∩ W_{n}

is connected (see Fig. 5).

For any c ∈ W_{n}, n ≥ N_{0}, let R_{n}(c) be the pre-image of B_{n}(c) under the map

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

and let C_{m+r+q+nk}(c) = P_{c}^{−1}(R_{n}(c)). Then C_{m+r+q+nk}(c) is the domain containing 0 in η_{m+r+q+nk} and

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

LOCAL CONNECTIVITY OF THE MANDELBROT SET AT CERTAIN POINTS 11

is a quadratic-like map. Moreover,

{F_{n,c} : C̊_{m+r+q+nk}(c) → C̊_{r}(c) | c ∈ W_{n}}

is a full family. Thus W_{n} contains a copy M_{n} = M_{n}(c_{0}) of the Mandelbrot set M where M_{n} ⊂ M̃_{n}. For any c ∈ ∪_{n≥N_{0}}M_{n}, P_{c} is once renormalizable. Now repeat the induction we did for −2. We construct a subset Υ(c_{0}) such that every c ∈ Υ(c_{0}) is infinitely renormalizable, such that M is locally connected at every c ∈ Υ(c_{0}), and such that c_{0} is a limit point of Υ(c_{0}).

Let Υ = ∪_{c_{0}}Υ(c_{0}) where c_{0} runs over all Misiurewicz points in M. Then every c ∈ Υ is infinitely renormalizable and M is locally connected at every c ∈ Υ. Since the set of Misiurewicz points is dense in ∂M, the set Υ is dense in ∂M. This completes the proof of the theorem.

Remark 1. Eckmann and Epstein [5] and Douady and Hubbard [4] estimated the size of M_{n} in the construction. Since M̃_{n} \ M_{n} contains hairs (see Fig. 5) and may destroy the local connectivity of M, we must estimate the size of M̃_{n}.

Remark 2. For c in Υ, one can prove that the Julia set J_{c} is locally connected at the critical point 0 as well as at the grand critical orbit of P_{c}. But since we did not show that P_{c} is unbranched we could not conclude that J_{c} is locally connected from the result in [7]. However, we constructed a similar subset Υ̃ of the Mandelbrot set M in [7] which is also dense on the boundary of the Mandelbrot set M such that P_{c} is unbranched infinitely renormalizable and has the complex a priori bounds for every point c of this subset. Thus the Julia set J_{c} of P_{c} for c in Υ̃ is locally connected according to the result in [7].

References

1. L. Carleson and T. Gamelin, Complex Dynamics. Springer-Verlag, Berlin, Heidelberg, 1993.

2. A. Douady and J. H. Hubbard, Itération des polynômes quadratiques complexes. C.R. Acad. Sci. Paris, 294, 1982, pp. 123-126.

3. A. Douady and J. H. Hubbard, Etude dynamique des polynomes complexes I & II. Publ. Math. d’Orsay, 1984.

4. A. Douady and J. H. Hubbard, On the dynamics of polynomial-like maps. Ann. Sci. Éc. Norm. Sup., 18, 1985, pp. 287-344.

5. J. Eckmann and H. Epstein, Scaling of Mandelbrot sets by critical point preperiodicity. Comm. Math. Phys., 101, 1985.

6. J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J. -C. Yoccoz. Topological Methods in Modern Mathematics, A Symposium in Honor of John Milnor’s Sixtieth Birthday, 1993.

7. Y. Jiang, Renormalization method in quadratic-like maps. Prepint. (It is the revision of Infinitely renormalizable quadratic Julia sets, ETH, Zürich, Preprint).

8. J. Milnor, Dynamics in one complex variable: Introductory lectures. IMS preprint 1990/5, Stony Brook.

9. J. Milnor, Local connectivity of Julia sets: expository lectures. IMS preprint 1992/11, Stony Brook.

10. F. Przytycki, On U-stability and structural stability of endomorphisms satisfying Axiom A. Studia-Math., 60, 1977, no. 1, pp. 61–77.

11. D. Ruelle, Repellers for real analytic maps. Ergod. Th. & Dynam. Sys., 1982, 2, pp. 99-107.

12. M. Shub, Endomorphisms of compact differentiable manifolds. Amer. J. Math., 91, 1969.

13. D. Sullivan, Seminar on conformal and hyperbolic geometry. IHES Preprint, March, 1982.

14. L. Tan Voisinages connexes des points de Misiurewicz. Ann. Inst. Fourier, 1992, Grenoble 124.

15. Private conversations with Shishikura and Douady.

Department of Mathematics, Queens College of CUNY, 65-30 Kissena Blvd, Flushing, NY 11367