\rhogin{document}

\tildeitle{Non-recurrent parameter rays of the Mandelbrot set\footnote{\varepsilonmph{Keywords: }Mandelbrot sets; non-recurrent. \ \ \varepsilonmph{AMS[2010]Classification:} 37F45 }}

\title{Non-recurrent parameter rays of the Mandelbrot set\footnote{\emph{Keywords: }
\rhogin{abstract}
In this paper, we prove that any parameter ray at a non-recurrent angle $\tildeheta$ lands at a non-recurrent parameter $c$ with $\tildeheta$ a characteristic angle of $f_c$; and conversely, every non-recurrent parameter $c$ is the landing point of one or two parameter rays at non-recurrent angles, and these angles are exactly the characteristic angles of $f_c$.

\varepsilonnd{abstract}

\section{Introduction}
  The quadratic family $\{f_c:z\mapsto z^2+c\}$ exhibits rich dynamics, when iterated.  The Mandelbrot set
\[{\mathcal M}:=\{c\in\mbox{$\mathbb C$}\mid f_c^n(c)\not\tildeo\infty\tildehetaxt{ as }n\tildeo\infty\}\]
organizes the space of quadratic polynomials up to conjugacy and has a beautiful structure.
It has been a very active area of research in the past few decades.
The importance of the Mandelbrot set is due to the fact that it is the simplest non-trival parameter space of analytic families of iterated holomorphic maps, and because of its universality as explained in \cite{DH1, Mc}

Much of the topological and combinatorial structures of the Mandelbrot set has been discovered by the work of Douady and Hubbard \cite{DH2}. A fundamental result in \cite{DH2} is to describe the landing behavior of the rational parameter rays.  One can see also \cite{Mil2,S,PR} for alternative approaches.

In this article, we study the landing property of irrational, precisely the \varepsilonmph{non-recurrent}, parameter rays.

Set ${\mathbb T}:=\mbox{$\mathbb R$}/\mbox{$\mathbb Z$}$ and $\tildeau:{\mathbb T}\tildeo{\mathbb T}$ the angle doubling map, i.e., $\tildeau(\tildeheta)=2\tildeheta\ {\rm mod}\mbox{$\mathbb Z$},\tildeheta\in{\mathbb T}$.  By abuse of notations, we  identify ${\mathbb T}$ with the unit circle under the correspondence $t\mapsto e^{2\pi it}$.

An angle $\tildeheta\in{\mathbb T}$ is called \varepsilonmph{non-recurrent} if  $\tildeau^n(\tildeheta)\not=\tildeau^m(\tildeheta)$ for any $0\lambdaeq m<n$, and there exists  $\deltaeltalta>0$ such that $|\tildeheta-\tildeau^n(\tildeheta)|>\deltaeltalta$ for all $n\gammaeq 1$. A quadratic polynomial $f_c$, or the parameter $c$, is called \varepsilonmph{non-recurrent} if $c\in {\mathcal M}$, all periodic points of $f_c$ are repelling and $|f^n_c(0)|>\deltaeltalta$ for all $n\gammaeq1$ and  a positive constance $\deltaeltalta$.

It is known that the Julia sets of non-recurrent quadratic polynomials are connected and locally-connected. The angle of an external ray landing at the critical value $c$ is said to be a \varepsilonmph{characteristic angle} of $f_c$. Refer to Section {\rm Re}f{preliminary} for the definitions of  external rays and parameter rays. The following is our main theorem.

\rhogin{theorem}\lambdaambdabel{main}
Any parameter ray at a non-recurrent angle lands at a non-recurrent parameter $c$ such that $\tildeheta$ is a characteristic angle of $f_c$. Conversely, every non-recurrent parameter $c$ is the landing point of one or two parameter rays at non-recurrent angles, and these angles are exactly the characteristic angles of $f_c$.
\varepsilonnd{theorem}

 Our proof is based on Kiwi's Combinatorial Continuity Theorem \cite[Theorem 1]{Ki2} and Yoccoz Rigidity Theorem \cite[Theorem III]{H} (or \cite[Theorem 4.1]{Ze}). We will review some background of polynomial dynamics and fix  notations in Section {\rm Re}f{preliminary}, and introduce the concept of \varepsilonmph{real lamination} in Section {\rm Re}f{p-l}. In Section {\rm Re}f{key-lemma}, we verify two combinatorial results used in the proof of the main theorem, and the proof of Theorem {\rm Re}f{main} is left in Section {\rm Re}f{proving}.

\section{Polynomial dynamics}\lambdaambdabel{preliminary}
One can refer to \cite{DH2} for the details of the content in this section.

Let $f_c(z)=z^2+c,z\in\mbox{$\mathbb C$}$ be a quadratic polynomial. The set of all points
which remain bounded under all iterations of $f_c$ is called the \varepsilonmph{Filled-in Julia set} $K_c$. The boundary of the Filled-in Julia set is defined to be the \varepsilonmph{Julia set} $J_c$ and the complement of the Julia set is defined to be its \varepsilonmph{Fatou set} $F_c$.

If $c\in {\mathcal M}$, the filled-in Julia set $K_c$ is \varepsilonmph{simply-connected}, i.e., $\mbox{$\mathbb C$}\setminus K_c$ is connected. There is then a unique biholomorphic map $\phi_c$ from $\mbox{$\mathbb C$}\setminus K_c$ onto $\mbox{$\mathbb C$}\setminus\overline{\mathbb D}$, called the \varepsilonmph{B\"{o}ettcher coordinate}, such that $\lambdaim_{z\tildeo\infty}\phi_c(z)/z=1$ and $\phi_c\circ f_c(z)=\phi_c(z)^2$ for $z\in\mbox{$\mathbb C$}\setminus K_f$.
 The preimage of $(1,\infty)e^{2\pi it}$ under $\phi_c$, denoted $R_c(t)$, is called  the {\it external ray at angle} $t$. We say that the external ray $R_c(t)$ \varepsilonmph{lands} if $\overline{R_c(t)}\cap K_c$ is a singleton, and this point, denoted by $\gamma_c(t)$, is called the \varepsilonmph{landing point} of $R_c(t)$.

In the parameter plane, the Mandelbrot set ${\mathcal M}$ is simply-connected, and there is a biholomorphic map ${\mathbb P}^1hi$ from $\mbox{$\mathbb C$}\setminus {\mathcal M}$ onto $\mbox{$\mathbb C$}\setminus\overline{\mathbb D}$. The \varepsilonmph{parameter ray at angle $\tildeheta$} is defined as the set $R_{\mathcal M}(\tildeheta):=\{{\mathbb P}^1hi^{-1}(re^{2\pi i\tildeheta}),r>1\}$. Similarly, if $\overline{R_{\mathcal M}(\tildeheta)}\cap {\mathcal M}$ is a singleton, we say that $R_{\mathcal M}(\tildeheta)$ \varepsilonmph{lands}.

\section{The impression and real lamination of polynomials}\lambdaambdabel{p-l}

Let $f_c$ be a quadratic polynomial with connected Julia set.
If $J_c$ is locally connected, the map $\phi_c^{-1}$ can be continuously extended to $\mbox{$\mathbb C$}\setminus\mathbb D$ and each external ray $R_c(t)$ lands at a  Julia point. The map $\gamma_c:{\mathbb T}\tildeo J_f$, $t\mapsto \gamma_c(t)$, is continuous and surjective. In this case, the landing pattern of external rays for $f_c$ induces an equivalence relation $\lambdaambdambda(c)$ on ${\mathbb T}$ such that $t\overlineerset{\lambdaambdambda(c)}{\tildehicksim} s$ if and only if $\gamma_c(t)=\gamma_c(s)$.

Kiwi \cite{Ki2} generalized the definition of such an equivalence relation to a class of non locally-connected  case, with the concept {\it impression} instead of the landing point in the locally-connected case.

We still assume that $f_c$ has the connected Julia set. Consider an argument $t\in {\mathbb T}$. We say that $z\in J_f$ belongs to the {\it impression} of $t$, written ${\rm Imp}_c(t)$, if and only if there exists a sequence  $\{z_n\in R_c(t_n)\}$ converging to $z$, with $\{t_n\}\subset {\mathbb T}$ converging to $t$. Note that $f_c(\tildehetaxt{Imp}_c(t))=\tildehetaxt{Imp}_c(\tildeau(t))$, and  ${\rm Imp}_c(t)=\gamma_c(t)$ for each $t\in {\mathbb T}$ if $J_c$ is locally connected.

Similar to the locally-connected case, we have the following two facts about the impressions, which will be used in the proof of Proposition {\rm Re}f{part1}

\rhogin{lemma}\lambdaambdabel{impression}
\rhogin{enumerate}
\item If there exists a sequence $\{z_n\in {\rm Imp}_c(t_n)\}$ with $z_n\tildeo z$ and $t_n\tildeo t$ as $n\tildeo\infty$, then $z\in {\rm Imp}_c(t)$.
\item For any $z\in J_c$, there exists an argument $t\in {\mathbb T}$ with $z\in{\rm Imp}_c(t)$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\rhogin{proof}
1. For each $n$, since $z_n\in\tildehetaxt{Imp}_c(t_n)$, we can choose an argument $s_n$ and a point $w_n\in R_c(s_n)$ such that $|z_n-w_n|<|z_n-z|$ and $|t_n-s_n|<|t_n-t|$. Then we have $|w_n-z|<2|z_n-z|\tildeo 0$ and $|s_n-t|<2|t_n-t|\tildeo 0$ as $n\tildeo \infty$. It means that $w_n\in R_c(s_n)$ converges to $z$ and $s_n$ converges to $t$, hence $z\in \tildehetaxt{Imp}_c(t)$.

\noindent 2. Let $z\in J_c$. Since $J_c$ is the boundary of the basin $\mbox{$\mathbb C$}\setminus K_c$, there exists a sequence $\{z_n\}\subset \mbox{$\mathbb C$}\setminus K_c$ with $z_n$ converging to $z$. Each $z_n$ belongs to an external ray of argument $t_n$. By picking a subsequence, we  assume $t_n\tildeo t$ as $n\tildeo \infty$. It follows from the definition that $z\in\tildehetaxt{Imp}_c(t)$.
\varepsilonnd{proof}

Let $f_c$ be a quadratic polynomial with connected Julia set and without irrational neutral cycles. Following Kiwi (see \cite[Definition 2.2]{Ki2}),
 the {\it real lamination} of $f_c$ is the smallest equivalence relation $\lambdaambdambda(c)$ in ${\mathbb T}$ which identities $s$ and $t$ whenever ${\rm Imp}_c(s)\cap {\rm Imp}_c(t)\not=\varepsilonmptyset$.  For a $\lambdaambdambda(c)$-class $A$, we denote ${\rm Imp}_c(A)$ the union of the impressions ${\rm Imp}_c(t)$ for all $t\in A$.

\section{The equivalence relation generated by angles}\lambdaambdabel{key-lemma}

For any angle $\tildeheta\in{\mathbb T}$, its preimages $\{\tildeheta/2,(\tildeheta+1)/2\}$ under $\tildeau$ divide ${\mathbb T}$ into two closed half circles, which are denoted by $L_0,L_1$ respectively. Then we can endow each angle $t\in{\mathbb T}$ two itineraries $\iota^{\pm}_\tildeheta(t)$ with respect to $\tildeheta$ such that $\iota^{\pm}_\tildeheta(t)= i_0i_1\lambdadots$ if, for each $n\gammaeq0$, there exists $\varepsilonsilon>0$ with $(\tildeau^n(t),\tildeau^n(t)\pm\varepsilonsilon)\subset L_{i_n}$.

From the definition, we can see that if $t$ is not an iterated preimage of $\tildeheta$, then $\iota_\tildeheta^+(t)=\iota^-_\tildeheta(t)$. In particular, we have $\iota^+_\tildeheta(\tildeheta)=\iota^-_\tildeheta(\tildeheta)$ for any non-periodic $\tildeheta$. In this case, the sequence $\iota^+_\tildeheta(\tildeheta)=\iota^-_\tildeheta(\tildeheta)$ is called the {\it kneading sequence of $\tildeheta$}, written $\nu(\tildeheta)$. It is called {\it aperiodic} if it is not a periodic
symbol sequence under the shift map.

By Kiwi \cite[Definition 4.5]{Ki2}, the {\it equivalence relation generated by} $\tildeheta\in{\mathbb T}$, denoted by $\lambdaambdambda(\tildeheta)$, is defined as the smallest equivalence relation
such that if $\iota_\tildeheta^+(t)=\iota_\tildeheta^-(s)$, then $s$ and $t$ are equivalent.

From now on, we always assume that $\tildeheta\in{\mathbb T}$ is non-recurrent. The following is a key Lemma in our proof.

\rhogin{lemma}\lambdaambdabel{aperiodic}
If $\tildeheta$ is non-recurrent, then $\nu(\tildeheta)$ is aperiodic.
\varepsilonnd{lemma}
\rhogin{proof}
On the contrary, we assume that $\nu(\tildeheta)$ is periodic of period $p\gammaeq1$. For each integer $0\lambdaeq k\lambdaeq p$, set
\[B_k:=\{\tildeau^{k+np}(\tildeheta)\mid n\gammaeq0\}.\]
Then we have $\tildeau(\overline{B_k})\subset\overline{B_{k+1}},k\in\{0,\lambdadots,p-1\},$ and $\overline{B_p}\subset \overline{B_0}$. Note that all elements of $B_k$ have a common itinerary, then each $B_k$ is contained in a component of ${\mathbb T}\setminus\{\tildeheta/2,(\tildeheta+1)/2\}$. Moreover, by the non-recurrent property, the closures $\overline{B_k}$ are disjoint from $\{\tildeheta/2, (\tildeheta+1)/2\}$. It follows that for each $k\in\{0,\lambdadots,p-1\}$, the map $\tildeau:\overline{B_k}\tildeo \overline{B_{k+1}}$ is injective, and hence $\tildeau^p:\overline{B_0}\tildeo \overline{B_0}$ is injective. According to \cite[Lemma 18.8]{Mi1}, the set $\overline{B_0}$ is finite, a contradiction.
\varepsilonnd{proof}

Since $\nu(\tildeheta)$ is aperiodic, by \cite[Proposition 4.7]{Ki2}, we get that
\rhogin{proposition}\lambdaambdabel{property}
 The equivalence relation $\lambdaambdambda(\tildeheta)$ is closed and satisfies that
\rhogin{enumerate}
\item  each $\lambdaambdambda(\tildeheta)$-class  is a finite subset of ${\mathbb T}$;
\item  if $A$ is a $\lambdaambdambda(\tildeheta)$-class, then $\tildeau(A)$ is a $\lambdaambdambda(\tildeheta)$-class;
\item  for any two different $\lambdaambdambda(\tildeheta)$-classes $A,B$, the convex hulls of $A$ and $B$ are disjoint.
\varepsilonnd{enumerate}
\varepsilonnd{proposition}
Combining the fact that $\tildeheta$ is non-recurrent, we can obtain more information about $\lambdaambdambda(\tildeheta)$. Since $\tildeheta$ is not periodic, then $\iota^+_\tildeheta(\tildeheta/2)=\iota^-_\tildeheta((\tildeheta+1)/2)$, and hence $\tildeheta/2$ and $(\tildeheta+1)/2$ are $\lambdaambdambda(\tildeheta)$-equivalent. We call the $\lambdaambdambda(\tildeheta)$-class containing $\{\tildeheta/2,(\tildeheta+1)/2\}$ the {\it critical class}, denoted by $C_\tildeheta$; and the one containing $\tildeheta$ the {\it characteristic class}, denoted by $A_\tildeheta$.

\rhogin{lemma}\lambdaambdabel{equivalence} Let $A_\tildeheta$ be the characteristic class of $\lambdaambdambda(\tildeheta)$. Then we have
\rhogin{enumerate}
\item $A_\tildeheta$ is \varepsilonmph{wandering}, i.e., $\tildeau^n(A_\tildeheta)\cap \tildeau^m(A_\tildeheta)=\varepsilonmptyset$ for each $0\lambdaeq m<n$;
\item  $A_\tildeheta$ contains at most two angles;
\item  $A_\tildeheta$ is non-recurrent, i.e,  $\varepsilonxists~\deltaeltalta>0$ s.t $dist(A_\tildeheta,\tildeau^n(A_\tildeheta))>\deltaeltalta$ for all $n\gammaeq 1$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\rhogin{proof}
{\it 1.} On the contrary, without loss of generality, we assume that $A_\tildeheta$ is periodic. Then the fact of $\#A_\tildeheta<\infty$ implies that $\tildeheta$ is eventually periodic, a contradiction.

{\noindent \it 2.} Since $A_\tildeheta$ is wandering, its orbit does not contain $C_\tildeheta$. Note that each $\lambdaambdambda(\tildeheta)$-class except  $C_\tildeheta$ is contained in one component of ${\mathbb T}\setminus \{\tildeheta/2,(\tildeheta+1)/2\}$ (by 3 of Proposition {\rm Re}f{property}), it follows that $\#A_\tildeheta=\#\tildeau^n(A_\tildeheta)$ for all $n\gammaeq0$. Using Thurston's No Wandering Polygon Theorem (\cite[Theorem II.5.2]{T}), the conclusion holds.

{\noindent \it 3.} If $A_\tildeheta$ contains one angle, since $\tildeheta$ is non-recurrent, the set $A_\tildeheta$ is naturally non-recurrent. So, by assentation (2), we just need to prove the case that $\#A_\tildeheta=2$.

For any $\la phalphapha\not=\rhota\in{\mathbb T}$, we define the arc $(\la phalphapha,\rhota)\subset {\mathbb T}$ as the closure of the connected component of ${\mathbb T}\setminus\{\la phalphapha,\rhota\}$ that consists of the angles we traverse if we move on ${\mathbb T}$ in the counterclockwise direction from $\la phalphapha$ to $\rhota$. The length of an arc $S\subset {\mathbb T}$ is denoted by $|S|$.
 We define a map $\sigmagma$ on all arcs in ${\mathbb T}$  such that $\sigmagma(\la phalphapha,\rhota)$ equals to $(\tildeau(\la phalphapha),\tildeau(\rhota))$ if $\tildeau(\la phalphapha)\not=\tildeau(\rhota)$, and equals to $\tildeau(\la phalphapha)$ otherwise.

It is apparent that $|\sigmagma(S)|=2|S|$ if $|S|<1/2$ and $|\sigmagma(S)|=2|S|-1$ otherwise.

Let $A_\tildeheta=\{\tildeheta,\varepsilonta\}$.  Then it divides ${\mathbb T}$ into two closed arcs. We denote the shorter one by $S^+_1$ and the longer one by $S_1^-$. For $n\gammaeq1$, set $$A_n:=\tildeau^{n-1}(A_\tildeheta),\ S^+_n:=\sigmagma^{n-1}(S^+_1)\tildehetaxt{ and }S^-_n:=\sigmagma^{n-1}(S^-_1).$$
According to the proof of assentation 2, each $A_n$ is a $\lambdaambdambda(\tildeheta)$-class containing two angles. And it divides ${\mathbb T}$ into $S^+_n$ and $S^-_n$.
By 2 of Proposition {\rm Re}f{property}, the critical class $C_\tildeheta$ is equal to $\{\frac{\tildeheta}{2},\frac{\tildeheta+1}{2},\frac{\varepsilonta}{2},\frac{\varepsilonta+1}{2}\}$. It divides ${\mathbb T}$ into four arcs. We denote the two shorter ones by $S^+_0,-S^+_0$, and the two longer ones by $S^-_0,-S^-_0$. It is clear that $|S^+_0|=|-S^+_0|<|S^-_0|=|-S^-_0|<1/2$ and $\sigmagma(\pm S^\deltaeltalta_0)=S^\deltaeltalta_1$ for $\deltaeltalta\in\{+,-\}$.

We claim that in the set $\{S^{\pm}_n\mid n\gammaeq1\}$, the arc $S^+_1$ has the shortest length. If not, suppose that $k\gammaeq2$ is the first integer such that $S^+_k$ or $S^-_k$, say $S^+_k$, has a shorter length than $S^+_1$. Then $|S_{k-1}^+|>1/2$. It follows from 3 of Proposition {\rm Re}f{property} that the arc $S_{k-1}^+$ contains $C_\tildeheta$, and hence contains three of the arcs $\pm S^+_0,\pm S^-_0$. Its image $\sigmagma(S_{k-1}^+)=S_k^+$ therefore contains either $\tildeau(S^+_0)=S^+_1$ or $\tildeau(S^-_0)=S^-_1$. It implies $|S_k^+|\gammaeq\min\{|S_1^+|,|S_1^-|\}=|S_1^+|$, a contradiction to the assumption that $|S^+_k|<|S_1^+|$.

Set $\tildeheta_n:=\tildeau^n(\tildeheta)$ and $\varepsilonta_n:=\tildeau^n(\varepsilonta)$ for all $n\gammaeq1$. We now start to prove point 3 by contradiction. We can assume that $\tildehetaxt{dist}(A_{n_k},A_\tildeheta)\tildeo 0$, $\tildeheta_{n_k}\tildeo\tildeheta'$, and $\varepsilonta_{n_k}\tildeo\varepsilonta'$ as $k\tildeo\infty$ by passing to a subsequence if necessary. Since $\lambdaambdambda(\tildeheta)$ is closed, $\varepsilonta', \tildeheta'$ are in the same $\lambdaambdambda(\tildeheta)$-class. Notice that $|S_{n_k}^{\pm}|\gammaeq |S_1^+|$ as explained in the claim above. Then $\varepsilonta'\neq\tildeheta'$. Since $\tildehetaxt{dist}(A_{n_k},A_\tildeheta)\tildeo 0$ and $\# A_\tildeheta=2$, we have $\tildeheta'=\varepsilonta$ and $\varepsilonta'=\tildeheta$.

Hence for any $\varepsilonsilon>0$, there exist $k_1$ and $k_2$ such that $|\varepsilonta_{n_{k_1}}-\tildeheta|<\varepsilonsilon/2$ and
$$2^{n_{k_1}}|\tildeheta_{n_{k_2}}-\varepsilonta|=|\tildeau^{n_{k_1}}(\tildeheta_{n_{k_2}})-\tildeau^{n_{k_1}}(\varepsilonta)|<\varepsilonsilon/2.$$
It follows that $|\tildeau^{n_{k_1}+n_{k_2}}(\tildeheta)-\tildeheta|<\varepsilonsilon$ for any $\varepsilonsilon>0$, which is impossible.
\varepsilonnd{proof}

\section{Proof of Theorem {\rm Re}f{main}}\lambdaambdabel{proving}

The proof of Theorem {\rm Re}f{main} is based on the following two propositions.

\rhogin{proposition}\lambdaambdabel{part1}
Given a non-recurrent angle $\tildeheta$, the parameter ray $R_{\mathcal M}(\tildeheta)$ lands at a non-recurrent parameter $c$ such that $f_c$ takes $\tildeheta$ as a characteristic angle.
\varepsilonnd{proposition}
\rhogin{proof}
We set ${\rm Acc}_{\mathcal M}(\tildeheta)$ the accumulation set of $R_{\mathcal M}(\tildeheta)$ on ${\mathcal M}$. By Lemma {\rm Re}f{aperiodic} and \cite[Theorem 1]{Ki2}, we have that  all cycles of $f_c$ are repelling and $\lambdaambdambda(c)=\lambdaambdambda(\tildeheta)$ for all $c\in {\rm Acc}_{\mathcal M}(\tildeheta)$. Let $c\in {\rm Acc}_{\mathcal M}(\tildeheta)$. We will show that $f_c$ is non-recurrent with a characteristic angle $\tildeheta$.

By Lemma {\rm Re}f{impression} (2), we choose an angle $t_0\in {\mathbb T}$ with $0\in {\rm Imp}_c(t_0)$. Note that the B\"{o}ettcher coordinate $\phi_c$ satisfies that $\phi_c(-z)=-\phi_c(z)$ for $z\in \mbox{$\mathbb C$}\setminus K_c$, then the sets ${\rm Imp}_c(t_0)$ and ${\rm Imp}_c(t_0+1/2)$ are symmetric about the origin. It follows that $0\in {\rm Imp}_c(t_0)\cap {\rm Imp}_c(t_0+1/2)$, and hence $t_0,t_0+1/2$ are in a common $\lambdaambdambda(\tildeheta)$-class (because $\lambdaambdambda(c)=\lambdaambdambda(\tildeheta)$). By Proposition {\rm Re}f{property} (3), we know that $t_0,t_0+1/2$ are contained in the critical class $C_\tildeheta$.
Then the impression of the characteristic class ${\rm Imp}_c(A_\tildeheta)$ contains the critical value $c$.

Set $c_n:=f^{n-1}_c(c)$ and recall that $A_n:=\tildeau^{n-1}(A_\tildeheta)$. Then $c_n$ belongs to ${\rm Imp}_c(A_n)$ for each $n\gammaeq1$. The non-recurrent property of $f_c$ is equivalent to that the accumulation set of $\{c_n,n\gammaeq1\}$ is disjoint from the critical value $c$. We continue the argument by contradiction and assume that the sequence $\{c_{n_k}\in \tildehetaxt{Imp}_c(t_{n_k})\}$ with $t_{n_k}\in A_{n_k}$ converges to $c$ as $n\tildeo \infty$.

By passing to a subsequence if necessary, we  assume that $t_{n_k}\tildeo t$ as $k\tildeo\infty$. From Lemma {\rm Re}f{impression}(1), we see that $c\in{\rm Imp}_c(t)$, and hence $t\in A_\tildeheta$. This contradicts Lemma {\rm Re}f{equivalence} (3). Thus $f_c$ is non-recurrent.
Since the Julia sets of non-recurrent quadratic polynomials are locally connected,   the set ${\rm Imp}_c(A_\tildeheta)$ reduces to one point $c$. So $R_c(\tildeheta)$ lands at $c$.

We have seen that all $f_c$ with $c\in {\rm Acc}_{\mathcal M}(\tildeheta)$ are non-recurrent and have a common  real lamination $\lambdaambdambda(\tildeheta)$. Due to Yoccoz Rigidity Theorem \cite[Theorem III]{H} or  \cite[Theorem 4.1]{Ze}, we have that ${\rm Acc}_{\mathcal M}(\tildeheta)$ reduces to one point. So the parameter ray $R_{\mathcal M}(\tildeheta)$ lands.
\varepsilonnd{proof}

\rhogin{proposition}\lambdaambdabel{part2}
Let $f_c$ be a non-recurrent quadratic polynomial. Then it has at most two characteristic angles, and the parameter rays at these angles land at $c$.
\varepsilonnd{proposition}
\rhogin{proof}
Let $f_c$ be a non-recurrent quadratic polynomial. Since $J_c$ is locally connected, there exists a characteristic angle $\tildeheta$ with $R_c(\tildeheta)$ landing at $c$. Clearly $\tildeheta$ is non-recurrent. By Proposition {\rm Re}f{part1}, the parameter ray $R_{\mathcal M}(\tildeheta)$ lands at a non-recurrent parameter $c'$ so that $f_{c'}$ has a characteristic angle $\tildeheta$. We then have that $\tildeheta/2$ and $(\tildeheta+1)/2$ are contained in both a $\lambdaambdambda(c)$-class and a $\lambdaambdambda(c')$-class. By \cite[Proposition 4.10]{Ki2}, we get $\lambdaambdambda(c)=\lambdaambdambda(c')=\lambdaambdambda(\tildeheta)$. Using again the Yoccoz Rigidity Theorem, it follows that $c=c'$. By Lemma {\rm Re}f{equivalence} (2), the cardinality of the $\lambdaambdambda(c)$-class that contains $\tildeheta$ is at most two. Hence $f_c$ has at most two characteristic angles.
\varepsilonnd{proof}

\rhogin{proof}[Proof of Theorem {\rm Re}f{main}]
It follows directly from Propositions {\rm Re}f{part1} and {\rm Re}f{part2}.
\varepsilonnd{proof}

\rhogin{thebibliography}{BKM}
\smallsetminusall{
\bibitem[DH1]{DH1} A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, \varepsilonmph{Ann. Scient. Ec. Norm. Sup.}, 18(1985), 287--343.

\bibitem[DH2]{DH2} A. Douady and J. H. Hubbard, \varepsilonmph{ Etude Dynamique Des Polynomes Complexes, I, II}, Orsay: Publ Math Orsay, 1984--1985.

\bibitem[Hu]{H} J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of
J.-C. Yoccoz, in \varepsilonmph{Topological methods in modern mathematics} (Stony Brook, NY, 1991),
Publish or Perish, Houston, TX, 1993, 467--511.

\bibitem[Ki1]{Ki1} J. Kiwi, Real laminations and the topological dynamics of complex polynomials, \varepsilonmph{Adv. in
Math.}, 184(2004), 207--267.
\bibitem[Ki2]{Ki2} J. Kiwi, Combinatorial continuity in complex polynomial dynamics, \varepsilonmph{Proc. London Math. Soc.}(3) 91 (2005) 215--248.

\bibitem[Mc]{Mc} C. McMullen, The Mandelbrot set is universal, \varepsilonmph{The Mandelbrot set, theme and variations} (ed. Tan Lei), London Mathematical Society Lecture Note Series 274 (Cambridge University Press, 2000), 1--17.

\bibitem[Mil1]{Mi1} J. Milnor, \varepsilonmph{ Dynamics in One Complex Variable}. Princeton University Press 2006.

\bibitem[Mil2]{Mil2} J. Milnor, {Periodic orbits, external rays and the mandelbrot set}, \varepsilonmph{Ast\'{e}risque}, 261:277¨C-333, 2000.

\bibitem[PR]{PR} C.Petersen and G.Ryd, {Convergence of rational rays in parameter spaces}, The Mandelbrot set, \varepsilonmph{theme and variations} (ed. Tan Lei), London Mathematical Society Lecture Note Series 274 (Cambridge University Press, 2000), 161--172.

\bibitem[Sch]{S}  D. Schleicher, {Rational parameter rays of the mandelbrot set}, \varepsilonmph{Ast\'{e}risque}, 261:405¨C-443, 2000.

\bibitem[Thu]{T} W. Thurston, {Polynomial dynamics from Combinatorics to Topology}, 1-109 in \varepsilonmph{Complex
Dynamics: Families and Friends}, ed. Dierk Schleicher, A K Peters, Wellesley, MA, 2009.

\bibitem[Ze]{Ze} J. Zeng, \varepsilonmph{Quasisymmetric rigidity, carpet Julia sets and the landing of dynamical and parameter rays,} Phd Thesis, Fudan University and Universit\'{e} d'Angers, 2015, \url{http://okina.univ-angers.fr/publications/ua13589}

}

\varepsilonnd{thebibliography}

\noindent Yan Gao, \\
Mathematical School  of Sichuan University, Chengdu 610064,
P. R. China. \\
Email: gyan@scu.edu.cn

\noindent Jinsong Zeng, \\
Academy of Mathematics and Systems Science, \\
Chinese
Academy of Sciences, Beijing 100190, P. R. China. \\
Email: zeng.jinsong@amss.ac.cn

\varepsilonnd{document}