%This is the first version. \documentstyle[12pt]{amsart} \input psfig \begin{document} %********************************** %Sign of interior points \def\II{\mathaccent'27I} \def\1{\char'401} \def\TT{\mathaccent'27T} \def\1{\char'401} \def\DD{\mathaccent'27D} \def\1{\char'401} \def\CC{\mathaccent'27C} \def\1{\char'401} \def\BB{\mathaccent'27B} \def\1{\char'401} \def\KK{\mathaccent'27K} \def\1{\char'401} \def\XX{\mathaccent'27X} \def\1{\char'401} \def\YY{\mathaccent'27Y} \def\1{\char'401} %******************************************* \title[Local connectivity of the Mandelbrot set at certain points]{Local connectivity of the Mandelbrot set at certain infinitely renormalizable points} \thanks{1991 Mathematics Subject Classification, Primary : 58F23; Secondary: 30C10} \author{Yunping Jiang} \thanks{Partially supported by NSF grant and PSC-CUNY award} \address{Department of Mathematics, Queens College of CUNY, 65-30 Kissena Blvd,\hfil\break Flushing, NY 11367} \maketitle \begin{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. \end{abstract} \vskip50pt \centerline{\bf Contents} \vskip10pt \noindent 1. Introduction \vskip5pt \noindent 2. Construction of bases of connected neighborhoods \vskip 48pt \noindent {\bf 1. Introduction} \vskip5pt 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 $O(z) =\{ P^{\circ n}_{c}(z)\}_{n=0}^{\infty}$ is bounded. The Julia set $J_{c}$ of $P_{c}$ is the boundary of $K_{c}$. The Mandelbrot set ${\cal M}$ is the set of parameters $c$ such that the critical orbit $O(c)=\{ P^{\circ n}_{c}(0)\}_{n=0}^{\infty}$ is bounded (see Fig. 1). Equivalently, ${\cal M}$ is the set of parameters $c$ such that $J_{c}$ is connected. For a parameter $c$ in the complement of the Mandelbrot set ${\cal M}$, the Julia set $J_{c}$ is a Cantor set. Douady and Hubbard (see \cite{dh1}) proved that the Mandelbrot set ${\cal M}$ is connected. \begin{figure} \centerline{\psfig{figure=pic.dir/mandpic1.ps,height=3in}} \caption{Computer generating picture of the boundary of} \centerline{the Mandelbrot set and some equipotential curves} \end{figure} \vskip5pt 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 \cite{pr,sh}), 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 \cite{su}), and that the Hausdorff dimension $HD(J_{c})$ of $J_{c}$ is $1+|c|/(4\log 2) +\hbox{o}(|c|)$ (see \cite{ru}). 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 ${\cal HP}$ be the set of parameters $c$ such that $P_{c}$ is hyperbolic. The main conjecture in this direction is that \vskip5pt \proclaim Conjecture 1. The set ${\cal HP}$ is open and dense in ${\cal M}$. \vskip5pt It is easy to verify that ${\cal HP}$ is open but to prove that ${\cal HP}$ is dense in ${\cal M}$ is a hard problem. \vskip5pt Douady and Hubbard \cite{dh2} proved that this conjecture follows from \vskip5pt \proclaim Conjecture 2. The Mandelbrot set ${\cal M}$ is locally connected. \vskip5pt In order to study the dynamics of quadratic polynomials more penetrating, Douady and Hubbard \cite{dh3} introduced the concept of a quadratic-like map which is a proper, degree two, analytic branch cover $f: U\rightarrow V$ where $U$ and $V$ are open domains isomorphic to a disc and $\overline{U}\subset V$. A quadratic polynomial $P_{c}$ is a quadratic-like map when restricted to any domain bounded by an equipotential curve (see \cite{mi1}). For a quadratic-like map $f: U\rightarrow V$, let $0$ be the unique branch point of $f$. The filled-in Julia set $K_{f}$ of $f$ is defined as $$ K_{f} =\cap_{n=0}^{\infty} f^{-n}(U). $$ 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 $0\in U_{1} \subset U$ and $V_{1}\subset V$ isomorphic to a disc such that $$ f_{1} =f^{\circ n_{1}} : U_{1} \rightarrow V_{1} $$ is a quadratic-like map with connected filled-in Julia set $K_{f_{1}}$. Otherwise, $f$ is called non-renormalizable. If $f_{1}: U_{1}\rightarrow V_{1}$ is also $n_{2}$-renormalizable, then we call $f$ is twice $(n_{1}, n_{2})$-renormalizable. So on one can define a $k$-{\sl times} $(n_{1}, n_{2}, \cdots, n_{k})$-renormalizable quadratic-like map as well as an infinitely renormalizable quadratic-like map. Suppose $f: U\rightarrow V$ is a quadratic-like map. A point $p$ is called a periodic point of period $n\geq 1$ of $f$ if $f^{\circ n}(p)=p$ and $f^{\circ i}(p) \neq p$ for $1\leq i1$; attractive if $0< |\lambda_{p}|<1$; super-attractive if $\lambda_{p}=0$. For a quadratic polynomial $P_{c}(z)=z^{2}+c$, let $PCO(c)=\{ P_{c}^{\circ n}(0)\}_{n=1}^{\infty}$ be its post-critical orbit. We classify the parameters $c$ in ${\cal M}\setminus {\cal HP}$ as $$ {\cal P}= \{ c\in {\cal M}\; |\; P_{c} \hbox{ has a parabolic periodic point }\},$$ $$ {\cal I}=\{ c\in {\cal M} \; |\: P_{c} \hbox{ has an irrationally indifferent periodic point }\}, $$ $${\cal N}=\{ c\in {\cal M} \; |\; c \hbox{ is not in } {\cal P}\cup {\cal I} \hbox{ and } 0\not\in \overline{PCO(c)}\; \}. $$ $${\cal FR} =\{ c\in {\cal M}\; |\; c \hbox{ is not in } {\cal P}\cup {\cal I}\cup {\cal N} \hbox{ and } P_{c} \hbox{ is not infinitely renormalizable }\},$$ and $$ {\cal IR} =\{ c \in {\cal M} \; |\; P_{c} \hbox{ is infinitely renormalizable }\}. $$ In particular, a parameter $c$ in ${\cal N}$ is called Misiurewicz if $PCO(c)$ 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 \cite{cg}). A point $c$ in ${\cal IR}$ is called an infinitely renormalizable. Recent work of Yoccoz (see \cite{hu}) shows that the Mandelbrot set ${\cal M}$ is locally connected at all points which are not infinitely renormalizable. (We would like to note that the local connectivity of the Mandelbrot set ${\cal M}$ at all points in ${\cal P}$ is assured by many experts \cite{hu,sd} 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. \vskip5pt \proclaim Main Theorem. There is a subset $\Upsilon$ of the Mandelbrot set ${\cal M}$ which is dense on the boundary $\partial {\cal M}$ of the Mandelbrot set ${\cal M}$ and which consists of only infinitely renormalizable points such that the Mandelbrot set ${\cal M}$ is locally connected at every point $c$ of $\Upsilon$. \vskip5pt 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 \cite{hu} and the book of Carleson and Gamelin \cite{cg} are helpful during my understanding of the structure of the Mandelbrot set ${\cal M}$ 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 \cite{ee}. 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. \vskip48pt \noindent {\bf 2. Construction of bases of connected neighborhoods} \vskip5pt We give the proof of Main Theorem now. Let us start the construction of the subset $\Upsilon$ in Main Theorem around the simplest point $-2\in {\cal M}$. Consider the polynomial $P(z) =z^{2}-2$ and its Julia set $J=[-2, 2]$. It has a non-separate fixed point $\beta=2$ and a separate fixed point $\alpha$, this menas that $J\setminus \{ \beta \}$ is still connected and $J\setminus \{ \alpha\}$ is disconnected. Let $\gamma$ be a fixed equipotential curve of $P$. Two external rays land at $\alpha$. Let $\Gamma$ be the union of these two external rays. Let $D$ be the closed domain bounded by $\gamma$. Let $D_{0}$ be the domain containing $0$ and bounded by $\gamma$ and $P^{-1}(\Gamma)$ (see Fig. 2). The restriction $P|({\bf C}\setminus (-\infty,-2])$ has two inverse branches $$ G_{0}: {\bf C}\setminus (-\infty,-2] \rightarrow {\bf LH}=\{ z=x+yi \in {\bf C} \; |\; x<0\} $$ and $$ G_{1}: {\bf C}\setminus (-\infty,-2] \rightarrow {\bf RH}=\{ z=x+yi \in {\bf C} \; |\; x>0\}. $$ Let $D_{n} =G_{1}^{\circ n}(D_{0})$ and let $B_{n} =G_{0}(D_{n-1})$ for $n\geq 1$ (see Fig. 2). Since $2$ is an expanding fixed point of $P$ and $P(-2) =2$, the diameter $\hbox{diam}(B_{n})$ tends to zero exponentially as $n$ goes to infinity. \addtocounter{figure}3 \begin{figure} \centerline{\psfig{figure=pic.dir/mandpic2.ps}} \end{figure} Fig. 2 can be constructed as well for $c$ in a small neighborhood about $-2$ because $\beta$, $\alpha$, and $\Gamma$ are structurally stable. Let $0\not\in U_{0}$ be a small neighborhood about $-2$ with $\hbox{diam}(U_{0}) \leq 1$ such that Fig. 2 is preserved for $c$ in $U_{0}$. Let $\beta (c)$, $\alpha (c)$, $\Gamma (c)$, $D_{n}(c)$ for $0\leq n< \infty$, and $B_{n}(c)$ for $1\leq n< \infty$ be the corresponding points, sets, and domains for $P_{c}$, $c\in U_{0}$. Let $\overline{\beta}(c)\neq \beta (c)$ be another pre-image of $\beta (c)$ under $P_{c}$. Then the diameter $\hbox{diam}(B_{n})$ tends to $0$ and the set $B_{n}(c)$ approaches to $\overline{\beta}(c)$ as $n$ goes to infinity uniformly on $U_{0}$. \begin{figure} \centerline{\psfig{figure=pic.dir/mandpic3.ps}} \end{figure} Let $$ W_{n} =\{ c \; |\; c\in B_{n}(c)\}. $$ Because $W_{n}$ is bounded by external rays and an equipotential curve of the Mandelbrot set ${\cal M}$, $$ \tilde{{\cal M}}_{n}=W_{n}\cap {\cal M} $$ is connected. Since the equation $P_{c} (0)-\overline{\beta} (c)=0$ has a unique solution $-2$ in $U_{0}$, the Rouch\'e Theorem assures that $P_{c} (0)-x=0$ has a unique solution for any $x$ in $B_{n}(c)$ and large $n$ and the solution is close to $-2$. Therefore, there is an integer $N_{0}>0$ such that for $n\geq N_{0}$, $$ W_{n}\subset U_{0}. $$ That is, $$ \hbox{diam}(W_{n})\leq 1 $$ for $n\geq N_{0}$ (see Fig. 3). For each $W_{n}$ where $n\geq N_{0}$, since $c\in B_{n}(c)$, $C_{n}(c)=P_{c}^{-1}(B_{n}(c))$ is a simply connected domain containing $0$. It is also a subdomain of $D_{0}(c)$. Let $$ F_{n,c}=P_{c}^{\circ (n+1)}: \CC_{n}(c) \rightarrow \DD_{0}(c) $$ is a quadratic-like map (see Fig. 4). The family $$ \{ F_{n, c}: \CC_{n}(c) \rightarrow \DD_{0}(c) \; |\; c\in W_{n} \} $$ is full (see \cite{dh3}). So $W_{n}$ contains a copy ${\cal M}_{n}$ of the Mandelbrot set ${\cal M}$ (see \cite{dh3}). For $c\in {\cal M}_{n}$, the Julia set $J_{F_{n,c}}$ of $F_{n,c}: \CC_{n}(c) \rightarrow \DD_{0}(c)$ is connected. Therefore, for $$c \in \Upsilon_{1} =\cup_{n\geq N_{0}}^{\infty} {\cal M}_{n},$$ $P_{c}$ is once renormalizable. \begin{figure} \centerline{\psfig{figure=pic.dir/mandpic4.ps}} \end{figure} For a fixed integer $i_{0}\geq N_{0}$, consider $W_{i_{0}}$ and ${\cal M}_{i_{0}}$; there is a parameter $c_{i_{0}} \in {\cal M}_{i_{0}}$ such that $$F_{i_{0}}=F_{i_{0}, c_{i_{0}}}: C_{i_{0}}=C_{i_{0}}(c_{i_{0}}) \rightarrow D_{i_{0}}=D_{0}(c_{i_{0}})$$ is hypbrid equivalent (see \cite{dh3}) to $P(z)=z^{2}-2$. The quadratic-like map $F_{i_{0}}: C_{i_{0}}\rightarrow D_{i_{0}}$ has the non-separate fixed point $\beta_{i_{0}}$ and the separate fixed point $\alpha_{i_{0}}$. Let $\overline{\beta}_{i_{0}}$ be another pre-image of $\beta_{i_{0}}$ under $F_{i_{0}}$. Let $\Gamma_{i_{0}}$ be the external rays of $P_{c_{i_{0}}}$ landing at $\alpha_{i_{0}}$. Let $D_{i_{0}0}$ be the domain containing $0$ and bounded by $\partial C_{i_{0}}$ and $F_{i_{0}}^{-1}(\Gamma_{i_{0}})$. Let $\overline{\beta}_{i_{0}} \in E_{i_{0}0}$ and $\beta_{i_{0}} \in E_{i_{0}1}$ be the components of the closure of $C_{i_{0}}\setminus D_{i_{0}0}$. 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 $$ D_{i_{0}n}=G_{i_{0}1}^{\circ n}(D_{i_{0}0}) $$ and let $$ B_{i_{0}n} = G_{i_{0}0}(D_{i_{0}(n-1)}) $$ for $n\geq 1$. Since $\beta_{i_{0}}$, $\alpha_{i_{0}}$, and $\Gamma_{i_{0}}$ are structurally stable, we can find a small neighborhood $U_{i_{0}}$ about $c_{i_{0}}$ with $\hbox{diam}(U_{i_{0}})\leq 1/2$ such that the corresponding domains $B_{i_{0}}(c)$ can be constructed for $P_{c}$, $c\in U_{i_{0}}$. Let $$ W_{i_{0}n} =\{ c\in {\bf C} \; |\; F_{i_{0},c}(0) \in B_{i_{0}n}(c) \}. $$ The diameter $\hbox{diam}(B_{i_{0}n}(c))$ tends to zero and the set $B_{i_{0}n}(c)$ approaches to $\overline{\beta_{i_{0}}}(c)$ uniformly on $U_{i_{0}}$ as $n$ goes to infinity. Since the equation $F_{i_{0},c}(0)-\overline{\beta}_{i_{0}}(c)=0$ has a unique solution $c_{i_{0}}$, the Rouch\'e Theorem implies that $F_{i_{0}, c}(0)-x=0$ has a unique solution near $c_{0}$ for any $x\in B_{i_{0}n}(c)$ and any $n$ large. Thus, there is an integer $N_{i_{0}}\geq 0$ such that for $n\geq N_{i_{0}}$, $W_{i_{0}n}\subseteq U_{i_{0}}$. Thus $\hbox{diam}(W_{i_{0}n})\leq 1/2$. Since $W_{i_{0}n}$ for $n\geq N_{i_{0}}$ is bounded by external rays and an equipotential curve of ${\cal M}$, $\tilde{\cal M}_{i_{0}n}=W_{i_{0}}\cap {\cal M}$ is connected. For each $c$ in $W_{i_{0}n}$, $n\geq N_{i_{0}}$, let $C_{i_{0}n}(c) =F^{-1}_{i_{0},c}(B_{i_{0}n}(c))$. Then $$ F_{i_{0}n,c}=F^{\circ (n+1)}_{i_{0},c} : \CC_{i_{0}n}(c) \rightarrow \DD_{i_{0}0}(c) $$ is a quadratic-like map. Moreover, $$ \{ F_{i_{0}n,c}\; |\; c\in W_{i_{0}n}\} $$ is a full family. Thus $W_{i_{0}n}$ contains a copy ${\cal M}_{i_{0}n}$ of the Mandelbrots set ${\cal M}$. For $$ c\in \Upsilon_{2} = \cup_{i_{0}\geq N_{0}} \cup_{i_{1}\geq N_{i_{0}}} {\cal M}_{i_{0}i_{1}}, $$ $P_{c}$ is twice renormalizable. We use the induction to complete the construction of the subset around $-2$. Suppose we have constructed $W_{w}$ where $w=i_{0}i_{1}\ldots i_{k-1}$ and $i_{0}\geq N_{0}$, $i_{1}\geq N_{i_{1}}$, $\ldots$, $i_{k-1}\geq N_{i_{0}i_{1}\ldots i_{k-2}}$. Let $v=i_{0}\ldots i_{k-2}$. There is a parameter $c_{w} \in {\cal M}_{w}$ such that $$F_{w}=F_{w, c_{w}}: C_{w}=C_{w}(c_{w}) \rightarrow D_{w}=D_{v0}(c_{w})$$ is hypbrid equivalent to $P(z)=z^{2}-2$. The quadratic-like map $F_{w}: C_{w} \rightarrow D_{w}$ has the non-separate fixed point $\beta_{w}$ and the separate fixed point $\alpha_{w}$. Let $\overline{\beta}_{w}$ be another pre-image of $\beta_{w}$ under $F_{w}$. Let $\Gamma_{w}$ be the external rays of $P_{c_{w}}$ landing at $\alpha_{w}$. Let $D_{w0}$ be the domain containing $0$ and bounded by $\partial C_{w}$ and $F_{w}^{-1}(\Gamma_{w})$. Let $\overline{\beta}_{w} \in E_{w0}$ and $\beta_{w} \in E_{w1}$ be the components of the closure of $C_{w}\setminus D_{w0}$. Let $G_{w0}$ and $G_{w1}$ be the inverses of $F_{w}|E_{w0}$ and $F_{w}|E_{w1}$. Let $$ D_{wn}=G_{w1}^{\circ n}(D_{w0}) $$ and let $$ B_{wn} = G_{w0}(D_{w(n-1)}) $$ for $n\geq 1$. Since $\beta_{w}$, $\alpha_{w}$, and $\Gamma_{w}$ are structurally stable, we can find a small neighborhood $U_{w}$ about $c_{w}$ with $\hbox{diam}(U_{w})\leq 1/2^{k}$ such that the corresponding domains $D_{wn}(c)$ and $B_{wn}(c)$ can be constructed for $P_{c}$, $c\in U_{w}$. Let $$ W_{wn} =\{ c\in {\bf C} \; |\; F_{w,c}(0) \in B_{wn}(c) \}. $$ The diameter $\hbox{diam}(B_{wn}(c))$ tends to zero and the set $B_{wn}(c)$ approaches to $\overline{\beta_{w}}(c)$ uniformly on $U_{w}$ as $n$ goes to infinity. Since the equation $F_{w,c}(0)-\overline{\beta}_{w}(c)=0$ has a unique solution $c_{w}$, the Rouch\'e 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 $N_{w}\geq 0$ such that for $n\geq N_{w}$, $W_{wn}\subseteq U_{w}$. Thus $\hbox{diam}(W_{wn})\leq 1/2^{k}$. Because $W_{wn}$ for $n\geq N_{i_{1}}$ is bounded by external rays and an equipotential curve of ${\cal M}$, $$ \tilde{\cal M}_{wn}=W_{wn}\cap {\cal M}$$ is connected. For each $c$ in $W_{wn}$, $n\geq N_{w}$, let $C_{wn}(c) =F^{-1}_{w,c}(B_{wn}(c))$. Then $$ F_{wn,c}=F^{\circ (n+1)}_{w,c} : \CC_{wn}(c) \rightarrow \DD_{w0}(c) $$ is a quadratic-like map. Moreover, $$\{ F_{wn,c}: \CC_{wn}(c) \rightarrow \DD_{w0}(c) \; |\; c\in W_{wn}\}$$ is a full family. Thus $W_{wn}$ contains a copy ${\cal M}_{wn}\subset \tilde{\cal M}_{wn}$ of the Mandelbrot set ${\cal M}$. For $$ c\in \Upsilon_{k+1} = \cup_{w} \cup_{i_{k}\geq N_{w}} {\cal M}_{wi_{k}}, $$ $P_{c}$ is $(k+1)$-{\sl times} renormalizable where $w=i_{0}i_{1}\ldots i_{k-1}$ runs over all sequences of integers of length $k$ in the induction. We have thus constructed a subset $\Upsilon (-2) =\cap_{k=1}^{\infty} \Upsilon_{k}$ such that for each $c\in \Upsilon (-2)$, $P_{c}$ is infinitely renormalizable and such that $-2$ is a limit point of $\Upsilon (-2)$. For each $c\in \Upsilon (-2)$, there is a corresponding sequence $w_{\infty} =i_{0}i_{1}\ldots i_{k}\ldots$ of integers such that $$ \{c \} =\cap_{k=0}^{\infty}W_{i_{0}\ldots i_{k}}. $$ Since $$ \tilde{\cal M}_{i_{0}\ldots i_{k}}=W_{i_{0}\ldots i_{k}}\cap {\cal M} $$ is connected, $$ \{ W_{i_{0}\ldots i_{k}}\}_{k=0}^{\infty} $$ is a basis of connected neighborhoods of the Mandelbrot ${\cal M}$ at $c$. In other words, the Mandelbrot set ${\cal M}$ is locally connected at $c$. We prove further, after understood the above construction, that for any Misiurewicz point, there is a subset $\Upsilon (c_{0})$ such that every $c$ in $\Upsilon (c_{0})$ is infinitely renormalizable, such that the Mandelbrot set ${\cal M}$ is locally connected at every point $c$ in $\Upsilon (c_{0})$, and such that $c_{0}$ is a limit point of $\Upsilon(c_{0})$ as follows. Suppose $c_{0}$ is a Misiurewicz point in ${\cal M}$. Then there is an integer $m>1$ such that $p=P_{c_{0}}^{\circ m}(0)$ is a repelling periodic point of period $k\geq 1$. Let $\alpha$ 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 $\alpha$ as the separate fixed point of the last renormalization of $P_{c_{0}}$ (see \cite{ji})). Let $\Gamma$ be the union of a cycle of external rays landing at $\alpha$. Let $\gamma$ be a fixed equipotential curve of $P_{c_{0}}$. Using $\Gamma$ and $\gamma$, we can construct the two-dimensional Yoccoz puzzle as follows (see \cite{ji} for the notation). Let $C_{-1}$ be the domain bounded by $\gamma$. Then $\Gamma$ cuts $C_{-1}$ into a finite number of closed domains. Let $\eta_{0}$ denote the set of these domains. Let $\eta_{n} =P_{c_{0}}^{-n} (\eta_{0})$. Let $C_{n}$ be the member of $\eta_{n}$ containing $0$ for $n\geq 0$. Let $$ p\in \cdots \subseteq D_{n}(p) \subseteq D_{n-1}(p) \subseteq \cdots \subseteq D_{1}(p) \subseteq D_{0}(p) $$ be a $p$-end, where $D_{n}(p) \in \eta_{n}$. Let $$ c_{0}\in \cdots \subseteq E_{n}(c_{0}) \subseteq E_{n-1}(c_{0}) \subseteq \cdots \subseteq E_{1}(c_{0}) \subseteq E_{0}(c_{0}) $$ be a $c_{0}$-end, where $E_{n}(c_{0}) \in \eta_{n}$. We have $P_{c_{0}}^{\circ (m-1)}(E_{n+m-1}(c_{0})) =D_{n}(p)$. Since the diameter $\hbox{diam}(D_{n}(p))$ tends to zero as $n\rightarrow \infty$ and since $p$ is a repelling periodic point, we can find an integer $l\geq m$ such that $|(P_{c_{0}}^{\circ k})'(x)|\geq \lambda >1$ for all $x\in D_{l}(p)$ and such that $$ P_{c_{0}}^{\circ (m-1)}: E_{l+m-1}(c_{0}) \rightarrow D_{l}(p) $$ is a homeomorphism. Let $q\geq 0$ be the integer such that $$ f=P_{c_{0}}^{\circ q} : D_{l}(p) \rightarrow C_{r_{0}} $$ is a homeomorphism, where $C_{r_{0}}$ is the domain containing $0$ in $\eta_{r_{0}}$, $r_{0}\geq 0$. 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 $$ P_{c_{0}}^{\circ q} : B_{0} \rightarrow C_{r} $$ is a homeomorphism. Define $$ B_{n}= \Big( P_{c_{0}}^{\circ nk}|D_{l+nk}(p)\Big)^{-1}(B_{0}) \subseteq D_{l+nk}(p) $$ for $n\geq 1$. Note that $B_{n}$ is in $\eta_{r+q+nk}$. Then $$ P_{c_{0}}^{\circ (q+nk)} : B_{n} \rightarrow C_{r} $$ is a homeomorphism. Since $\alpha$, $p$, $\Gamma$, $C_{r}$, $D_{n}$, and $B_{n}$, for $n\geq 0$, 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 $\hbox{diam}(U_{0})\leq 1$ such that the corresponding points $\alpha(c)$ and $p(c)$ and the corresponding sets $\Gamma (c)$, $C_{r}(c)$, $D_{n}(c)$, and $B_{n}(c)$, for $n\geq 0$, are all preserved for $c\in U_{0}$. Moreover, as $n$ goes to infinity, the diameter $\hbox{diam}(B_{n}(c))$ tends to zero and the set $B_{n}(c)$ approaches to $p(c)$ uniformly on $U_{0}$. Let $$ W_{n}=W_{n}(c_{0})=\{ c \in {\bf C} \; |\; P_{c}^{m} (0) \in B_{n}(c)\}. $$ Since the equation $P^{\circ m}_{c}(0)-p(c)=0$ has a unique solution $c_{0}$ in $U_{0}$ (see \cite{dh2}), the Rouch\'e Theorem implies that the equation $P^{\circ m}_{c}(0)-x=0$ 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 $W_{n}\subset U_{0}$. Thus $\hbox{diam}(W_{n})\leq 1$ for all $n\geq N_{0}$. Since $W_{n}$ is bounded by external rays and an equipotential curve of the Mandelbrot set ${\cal M}$, then $$ \tilde{\cal M}_{n} ={\cal M}\cap W_{n} $$ is connected (see Fig. 5). \begin{figure} \centerline{\psfig{figure=pic.dir/mandpic5.ps,height=3in}} \caption{A small copy of the Mandelbrot set and hairs around it} \end{figure} \vskip5pt For any $c\in W_{n}$, $n\geq N_{0}$, let $R_{n}(c)$ be the pre-image of $B_{n}(c)$ under the map $$P_{c_{0}}^{\circ (m-1)}: E_{l+m-1}(c)\rightarrow D_{l}(p,c)$$ 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 $\eta_{m+r+q+nk}$ and $$F_{n,c}=P_{c}^{\circ (q+nk+m)}: \CC_{m+r+q+nk}(c)\rightarrow \CC_{r}(c)$$ is a quadratic-like map. Moreover, $$ \{ F_{n,c}: \CC_{m+r+q+nk}(c)\rightarrow \CC_{r}(c) \; |\; c\in W_{n}\}$$ is a full family. Thus $W_{n}$ contains a copy ${\cal M}_{n}={\cal M}_{n}(c_{0})$ of the Mandelbrot set ${\cal M}$ where ${\cal M}_{n}\subset \tilde{\cal M}_{n}$. For any $c\in \cup_{n\geq N_{0}} {\cal M}_{n}$, $P_{c}$ is once renormalizable. Now repeat the induction we did for $-2$. We construct a subset $\Upsilon (c_{0})$ such that every $c\in \Upsilon (c_{0})$ is infinitely renormalizable, such that ${\cal M}$ is locally connected at every $c\in \Upsilon (c_{0})$, and such that $c_{0}$ is a limit point of $\Upsilon (c_{0})$. Let $\Upsilon =\cup_{c_{0}}\Upsilon (c_{0})$ where $c_{0}$ runs over all Misiurewicz points in ${\cal M}$. Then every $c\in \Upsilon$ is infinitely renormalizable and ${\cal M}$ is locally connected at every $c\in \Upsilon$. Since the set of Misiurewicz points is dense in $\partial {\cal M}$, the set $\Upsilon$ is dense in $\partial {\cal M}$. This completes the proof of the theorem. \vskip5pt {\bf Remark 1.} Eckmann and Epstein \cite{ee} and Douady and Hubbard \cite{dh3} estimated the size of ${\cal M}_{n}$ in the construction. Since $\tilde{\cal M}_{n}\setminus {\cal M}_{n}$ contains hairs (see Fig. 5) and may destroy the local connectivity of ${\cal M}$, we must estimate the size of $\tilde{\cal M}_{n}$. \vskip5pt {\bf Remark 2.} For $c$ in $\Upsilon$, 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 \cite{ji}. However, we constructed a similar subset $\tilde{\Upsilon}$ of the Mandelbrot set ${\cal M}$ in \cite{ji} which is also dense on the boundary of the Mandelbrot set ${\cal M}$ 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 $\tilde{\Upsilon}$ is locally connected according to the result in \cite{ji}. \begin{thebibliography}{99} \bibitem{cg} L. Carleson and T. Gamelin, {\sl Complex Dynamics}. Springer-Verlag, Berlin, Heidelberg, 1993. \vskip5pt \bibitem{dh1} A. Douady and J. H. Hubbard, {\sl It\'eration des polyn\^omes quadratiques complexes}. C.R. Acad. Sci. Paris, {\bf 294}, 1982, pp. 123-126. \bibitem{dh2} A. Douady and J. H. Hubbard, {\sl Etude dynamique des polynomes complexes I \& II}. Publ. Math. d'Orsay, 1984. \bibitem{dh3} A. Douady and J. H. Hubbard, {\sl On the dynamics of polynomial-like maps}. Ann. Sci. \'Ec. Norm. 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