Let $f(z)$ be a rational map of degree $d=\deg f\geq2$ on the complex sphere $\mathbb{\overline{C}}$. The Julia set $J(f)$ of a rational function $f$ is defined to be the closure of all repelling periodic points of $f$, and its complement set is called Fatou set $F(f)$. It is known that $J(f)$ is a perfect set (so $J(f)$ is uncountable, and no point of $J(f)$ is isolated), and also that if $J(f)$ is disconnected, then it has infinitely many components.

Let $f:\mathbb{\overline{C}}\rightarrow \mathbb{\overline{C}}$ be a rational function. We call a compact forward invariant subset $X\subset J(f)$ (i.e. satisfying property $f(X)\subset X$) hyperbolic if there exists $n\geq 1$ such that

$ |(f^n)^{'}(x)|>1 $

for every $x\in X$ and $f^n$ is topologically conjugate to a subshift of finite type. If only condition $|(f^n)^{'}(x)|>1$ is satisfied, we call the map $f|_X$ expanding.

We call a rational function $f: J(f)\mapsto J(f)$ hyperbolic if there exists $n\geq 1$ such that

$ \inf\{(f^n)^{'}(z)|: z\in J(f)\}>1 . $

Denote $CV(f)$ the critical values of a rational function $f$. Let

$ PCV(f)=\bigcup\limits_{n\geq1}f^n(CV(f)) . $

It follows from [1, Theorem 2.2] that a rational function $f:J(f)\mapsto J(f)$ is hyperbolic if and only if

$ \overline{PCV(f)}\bigcap J(f)= \emptyset . $

Denote by $J(f)$ the Julia set of a rational function. A rational map $f$ is expansive if the Julia set $J(f)$ contains no critical points of $f$. It follows from [1] that each hyperbolic rational function is expansive and that a rational function is expansive but not hyperbolic if and only if the Julia set contains no critical points of $f$ but intersect the $\omega$-limit set of critical points.

We call expansive but not hyperbolic rational functions parabolic. It follows from [1] that a rational function $f:J(f)\mapsto J(f)$ is expansive but not hyperbolic if and only if the Julia set $J(f)$ contains no critical points of $f$ but contains at least one parabolic point.

We recall that if $T:X\rightarrow X$ is a continuous map of a topological space $X$, then for every point $x\in X$, the $\omega$-limit set of $x$ denoted by $\omega(x)$ is defined to be the set of all limit points of the sequence $\{T^{n}(x)\}_{n\geq 0}$. We call a point $x$ recurrent if $x\in \omega(x)$; otherwise $x$ is called non-recurrent.

A rational function $f:\bf{\overline{C}}\rightarrow \bf{\overline{C}}$ is called an NCP map if all critical points contained in the Julia set $J(f)$ are non-recurrent.

The class of NCP maps obviously contains all expanding and parabolic maps. It also comprises the important class of so called subexpanding maps which are defined by the requirement that $f|_{\omega({\rm Crit}(f))\cap J(f)}$ is expanding and the class of geometrically finite maps defined by the property that the forward trajectory of each critical point contained in the Julia set is finite and disjoint from $\omega$-limit set.

Let $f(z)$ be a map of degree $\geq 2$. A component $D$ of the Fatou set $F(f)$ is said to be completely invariant, if

$ f^{-1}(D)=D=f(D) . $

A Jordan arc $\gamma$ in $\mathbb{\overline{C}}$ is defined to be the image of the real interval $[0, 1]$ under a homeomorphism $\varphi$. If the interval $[0, 1]$ is replaced by the unit circle in the above definition then $\gamma$ is said to be a Jordan curve.

In this paper, we establish the following main theorem.

Main Theorem Let $f(z)$ be an NCP map of degree $\geq 2$, and suppose that $F(f)$ is the union of exactly two completely invariant components. Then $J(f)$ is their common boundary and is a Jordan curve.

Let $f$ be an NCP map. Denote by $\Lambda(f)$ the set of all parabolic periodic points of $f$ (these points belong to the Julia set and have an essential influence on its fractal structure), and ${\rm Crit}(f)$ of all critical points of $f$. We put

$ {\rm Crit}(J(f))={\rm Crit}(f)\cap J(f). $

Set

$ {\rm Sing}(f)=\bigcup\limits_{n\geq0}f^{-n}(\Lambda(f)\cup {\rm Crit}(J(f))). $

Definition 2.1 We define the conical set $J_c(f)$ of $f$ as follow. First, say $x$ belongs to $J_c(f, r)$ if for any $\epsilon>0$, there is a neighborhood $U$ of $x$ and $n>0$ such that ${\rm diam}(U)<\varepsilon$ and $f^n: U\rightarrow B(f^n(x), r)$ is a homeomorphism. Then set $J_c(f)=\bigcup\limits_{r>0}J_c(f, r).$ We have $x\in J_c(f)$ if and only if arbitrary small neighborhood of $x$ can be blow up univalently by the dynamics to balls of definite size centered at $f^n(x)$.

Lemma 2.1 (see [2]) If $f: J(f)\rightarrow J(f)$ is an NCP map, then

$ J_c(f)=J(f)\setminus {\rm Sing}(f) . $

Note that Curtis T. McMullen used the term radial Julia set $J_{rad}(f)$ instead of conical set $J_c(f)$ in analogy with Kleinian groups, see ref. [3]. By paper [3], we have the set ${\rm Sing}(f)$ is countable.

Let $0<\lambda<1$. Then there exist an integer $m\geq1$, $C>0$, an open topological disk $U$ containing no critical values of $f$ up to order $m$ and analytic inverse branches $f^{-mn}_i:U\rightarrow \mathbb{\overline{C}}$ of $f^{mn}(i=1, \cdots, k_n\leq d^{nm}, n\geq0)$, satisfying

(1) $\forall n\geq0, \forall 1\leq i\leq k_{n+1}, \exists 1\leq j\leq k_n, f^m\circ f^{-m(n+1)}_i=f^{-mn}_j$,

(2) ${\rm diam}(f^{-mn}_i(U))\leq c\lambda^n$ for $n=0, 1, \cdots$ and $i=1, \cdots, k_n$,

(3) for each fixed $n\geq1$, for all $i=1, \cdots, k_n$ the sets $\overline{f^{-mn}_i(U)}$ are pairwise disjoint and $\overline{f^{-mn}_i(U)}\subset U$.

Now we state as a lemma the following consequence of (1)-(3).

Lemma 2.2 For each $n$, let $\mathcal {N}_n=\bigcup\{f^{-n}_j(U):j=1, \cdots, k_n\}$ and let $\mathcal {N}=\bigcup \mathcal {N}_n$. Then $\mathcal {N}$ is a net of $J_c(f)$, i.e., any two sets in $\mathcal {N}$ are either disjoint or one is a subset of the other.

In paper [4], Urbanski and Zdunik provided the framework to study infinite conformal iterated function systems. Now we recall this notion and some of its basic properties. Let $I$ be a countable index set with at least two elements and let $S=\{\phi_i: X\rightarrow X: i\in I\}$ be a collection of injective contractions from a compact metric space $X$ (equipped with a metric $\rho$) into $X$ for which there exists $0<s<1$ such that $\rho(\phi_i(x), \phi_i(y))\leq s\rho(x, y)$ for every $i\in I$ and for every pair of points $x, y\in X$. Thus system $S$ is uniformly contractive. Any such collection $S$ of contractions is called an iterated function system. We are particularly interested in the properties of the limit set defined by such a system. We can define this set as the image of the coding space under a coding map as follows. Let $I^{\ast}=\bigcup\limits_{n\geq 1}I^n$, the space of finite words, and for $\tau\in I^{\ast}$, $n\geq1$, let $\phi_{\tau}=\phi_{\tau_1}\circ\phi_{\tau_2}\circ \cdots \circ\phi_{\tau_n}$. Let $I^{\infty}=\{\{\tau_n\}_{n=1}^{\infty}\}$ be the set of all infinite sequences of elements of $I$. If $\tau\in I^{\ast}\cup I^{\infty}$ and $n\geq 1$ does not exceed the length of $\tau$, we denote by $\tau|_n$ the word $\tau_1\tau_2\cdots\tau_n$. Since given $\tau\in I^{\infty}$, the diameters of the compact sets $\phi_{\tau|_n}(X)$, $n\geq1$, converge to zero and since they form a descending family, the set

$ \bigcap\limits_{n=0}^{\infty}\phi_{\tau|_n}(X) $

is a singleton therefor, denoting its only element by $\pi(\tau)$, defines the coding map

$ \pi: I^{\infty}\rightarrow X . $

The main object in the theory of iterated function systems is the limit set defined as follows.

$ J=\pi(I^{\infty})=\bigcup\limits_{\tau\in I^{\infty}}\bigcap\limits_{n=1}^{\infty}\phi_{\tau|_n}(X)=\bigcap\limits_{n\geq 1}\bigcup\limits_{|\tau|=n}\phi_{\tau}(X) . $

Observe that $J$ satisfied the natural invariance equality, $J=\bigcup\limits_{i\in I}\phi_i(J)$.

Notice (1) If $I$ is finite, then $J$ is compact and this property fails for infinite systems by paper [4].

(2) In Lemma 3.3, we shall build recursively our iterated function system $S_t=\{S_t^1, S_t^2, $ $ \cdots, S_t^{n}\}$, and $n(=I)$ is finite.

Let $X(\infty)$ be the set of limit points of all sequences $x_i\in\phi_i(X)$, $i\in I^{'}$, where $I^{'}$ ranges over all infinite subsets of $I$, see ref. [4].

Lemma 3.1 (see [4]) If $\lim\limits_{i\in I}{\rm diam}(\phi_i (X))=0$, then $\overline{J}=J\cup\bigcup\limits_{\omega\in I^{\ast}}\phi_{\omega}(X(\infty))$.

An iterated function system $S=\{\phi_i: X\rightarrow X: i\in I\}$ is said to satisfy the open set condition if there exists a nonempty open set $U\subset X$ (in the topology of $X$) such that $\phi_i(U)\subset U$ for every $i\in I$ and $\phi_i(U)\cap \phi_j(U)=\emptyset$ for every pair $i, j\in I$, $i\neq j$ (we do not exclude $cl\phi_i(U)\cap cl\phi_j(U)\neq\emptyset$).

An iterated function system $S=\{\phi_i: X\rightarrow X: i\in I\}$ is said to be conformal if $X\subset \mathbb{R}^d$ for some $d\geq1$ and the following conditions are satisfied.

(a) Open set condition (OSC). $\phi_i({\rm Int} X)\cap \phi_j({\rm Int} X)=\emptyset$ for every pair $i, j\in I$, $i\neq j$.

(b) $\overline{\bigcup\limits_{i\in I}\phi_i(X)}\subset {\rm Int} X$.

(c) There exists an open connected set $V$ such that $X\subset V\subset \mathbb{R}^d$ such that all maps $\phi_i$, $i\in I$, extend to $\mathcal{C}^1$ conformal diffeomorphisms of $V$ into $V$ (note that for $d=1$ this just means that all the maps $\phi_i$, $i\in I$, are monotone diffeomorphism, for $d=2$ the words conformal mean holomorphic and antiholomorphic, and for $d=3$, the maps $\phi_i$, $i\in I$ are Möbius transformations).

(d) (Cone condition) There exist $\alpha, l>0$ such that for every $x\in \partial X$ and there exists an open cone ${\rm Con}(x, u, \alpha)\subset {\rm Int}(V)$ with vertex $x$, the symmetry axis determined by vector $u$ of length $l$ and a central angle of Lebesgue measure $\alpha$, here ${\rm Con}(x, u, \alpha)= \{y: 0<(y-x, u)\leq\cos\alpha||y-x||\leq l\}$.

(e) Bounded distortion property (BDP). There exists $K\geq 1$ such that

$ |\phi^{'}_{\omega}(y)|\leq K|\phi^{'}_{\omega}(x)| $

for every $\omega\in I^{\ast}$ and every pair of points $x, y\in V$, where $|\phi^{'}_{\omega}(x)|$ means the norm of the derivative, see ref. [9, 10].

Definition 3.1 A bounded subset $X$ of a Euclidean space (or Reimann sphere) is said to be porous if there exists a positive constant $c>0$ such that each open ball $B$ centered at a point of $X$ and of an arbitrary radius $0<r\leq 1$ contains an open ball of radius $cr$ disjoint from $X$. If only balls $B$ centered at a fixed point $x\in X$ are discussed above, $X$ is called porous at $x$, see ref. [5].

Lemma 3.2 (see [5]) The Julia set of each NCP map, if different from $\mathbb{\overline{C}}$, is porous.

Lemma 3.3 If $f$ is an NCP map, then $J_c(f)$ admits a conformal iterated function system satisfying conditions (a)-(e).

Proof Let $f$ be an NCP map. By Lemma 2.2, $J_c(f)$ admits a net such that $B_i\cap B_j=\emptyset$, $i\neq j$. Moreover, we may require the existence of an integer $q\geq1$ and $\sigma>0$ such that the following holds:

If $x\in J_c(f)$, say $x\in B_i$, and $f^{qn}(x)\in B_t$, then there exists a unique holomorphic inverse branch $f^{-qn}_x: U(B_t, 2\sigma)\rightarrow \mathbb{\overline{C}}$ of $f^{qn}$ sending $f^{qn}(x)$ to $x$. Moreover $f^{-qn}_x(B_t)\subset B_i$ and taking $q$ sufficiently large, we have

$ f_x^{-qn}(U(B_t, \sigma/2))\subset {\rm Int}(B_i) $

for sufficiently small $\sigma$, then

$ \begin{eqnarray}\overline{f_x^{-qn}(B_t)}\subset B_i={\rm Int}(B_i).\end{eqnarray} $

(3.1)

Let $n>1$ be finite. For every $t=1, 2, \cdots, n$, we now build recursively our iterated function system $S_t$ as a disjoint union of the families $S_t^j$, $j\geq 1$, as follows. $S_t^1$ consists of all the maps $f_x^{-q}$, where $x, f^q(x)\in J_c(f)\cap B_t$. $S_t^2$ consists of all the maps $f_x^{-2q}$, where $x, f^{2q}(x)\in J_c(f)\cap B_t$ and $f^{q}(x)\notin B_t$. Suppose that the families $S_t^1, S_t^2, \cdots, S_t^{n-1}$ have been already constructed. Then $S_t^n$ is composed of all the maps $f_y^{-qn}$ such that $y, f^{qn}(y)\in J_c(f)\cap B_t$ and $f^{qj}(y)\notin B_t$ for every $1\leq j\leq n-1$.

Let $V\supset J_c(f)$ be an open set constructed by the net such that it disjoints from the parabolic and critical points and their inverse orbits of $f$. For any $x\in V$ and finite $n<\infty$, we have

$ 0<|(f^{-n}(x))^{'}|\leq M<\infty , $

then

$ |(f^{-n}(y))^{'}|\leq K|(f^{-n}(x))^{'}| , $

where $x$, $y\in V$ and $1\leq K<\infty$ is a constant. So condition (e) bounded distortion property (BDP) holds. It is evident that $f^n$ is holomorphic and antiholomorphic of $V$ into $V$ for all $n\geq1$, then condition (c) holds. Since $J(f)$ is porous, and condition (d) is satisfied. Condition (b) follows immediately from (3.1). In order to prove condition (a), take two distinct maps $f_x^{-qm}$ and $f_y^{-qn}$ belong to $S_t$. Without loss of generality we may assume that $m\leq n$. Suppose on the contrary that

$ f^{-qm}_x(B_t)\cap f^{-qn}_y(B_t)\neq \emptyset . $

Then

$ \emptyset\neq f^{qm}(f^{-qm}_x(B_t)\cap f^{-qn}_y(B_t))\subset B_t\cap f^{qm}(f^{-qn}_y(B_t))=B_t\cap f^{-q(n-m)}_{f^{qm}(y)}(B_t) . $

Hence $f^{-q(n-m)}_{f^{qm}(y)}(B_t)\subset B_t$, and therefor $f^{qm}(y)\in B_t$. Due to our construction of the system $S_t$, we have $f^{qn}(y)\in B_t$, and this implies that $m=n$. But then $f_x^{-qn}(B_t)\cap f_y^{-qn}(B_t)=\emptyset$ since $f_x^{-qn}$ and $f_y^{-qn}$ are distinct inverse branches of the same map $f^{qn}$. This contradiction finishes the proof of Lemma 3.3.

Given $x\in \mathbb{C}$, $\theta, r>0$, we put

$ {\rm Con}(x, \theta, r)={\rm Con}(x, \eta, r)\cup {\rm Con}(x, -\eta, r) , $

where $\eta$ is a representative of $\theta$. We recall that a set $Y$ has a tangent in the direction $\theta$ at a point $x\in Y$ if for every $r>0, $

$ \lim\limits_{r\rightarrow0}\frac{\mathcal H{^1}(Y\cap(B(x, r)\setminus {\rm Con}(x, \theta, r)))}{r}=0 , $

where $\mathcal{H}^1$ denotes the 1-dimensional Hausdorff measure (see refs. [6, 7]). Following [6], we say that a set $Y$ has a strong tangent in the direction $\theta$ at a point $x$ provided for each $0<\beta\leq1$, there is a some $r>0$ such that $Y\cap B(x, r)\subset {\rm Con}(x, \theta, \beta)$.

Lemma 4.1 (see [7]) If $Y$ is locally arcwise connected at a point $x$ and $Y$ has a tangent $\theta$ at $x$, then $Y$ has strong tangent $\theta$ at $x$.

We call a point $\tau\in I^{\infty}$ transitive if $\omega(\tau)=I^{\infty}$, where $\omega(\tau)$ is the $\omega$-limit set of $\tau$ under the shift transformation $\sigma: I^{\infty}\rightarrow I^{\infty}$. We denote the set of these points by $I^{\infty}_t$ and put $\Gamma_t=\pi(I^{\infty}_t)$. We call the $\Gamma_t$ the set of transitive points of $\Gamma_{S_t}$ and notice that for every $\tau\in I^{\infty}_t$, the set $\{\pi(\sigma^n\tau): n\geq0\}$ is dense in $\Gamma_{S_t}$ or $\overline{\Gamma}_{S_t}$.

Lemma 4.2 (see [7]) If $\overline{\Gamma}_{S_t}$ has a strong tangent at a point $x=\pi(\tau)$, $\tau\in I^{\infty}$, then $\overline{\Gamma}_{S_t}$ has a strong tangent at every point $\overline{\pi(\omega(\tau))}$.

Remark 4.1 If $f$ is an NCP map, by Lemma 3.3, $J_{c}(f)$ admits a conformal iterated function system $S_t$. It is obvious that the Julia set $J(f)$ coincides with the limit set $\overline{\Gamma}_{S_t}$ by Lemma 3.1. By Lemma 3.1, 3.3 and 4.2 we have

Lemma 4.3 If $f$ is an NCP map, then $J(f)$ has a strong tangent at every point of $J(f)$.

Proof of Main Theorem Let $f$ be an NCP map and denoted by $F_{\infty}$ the unbounded component of the Fatou set $F(f)$. As $F_{\infty}$ is completely invariant, applying Riemann-Hurwitz formula (see $\S5.4$ in [8]) to $f:F_{\infty}\rightarrow F_{\infty}$, we find that $F_{\infty}$ has exactly $d-1$ critical points of $f$, and all of these lie at $\infty$. Now take any disk $D$ centered at $\infty$, which is such that

$ f(\overline{D})\subset D\subset F_{\infty} . $

For each $n$, let $D_n=f^{-n}(D)$: then $D_n$ is open and connected,

$ D=D_0\subset D_1\subset D_2\subset \cdots, $

and as

$ \chi(D_{n+1})+(d-1)=d\chi(D_n), $

where $\chi(D_{n+1})$ and $\chi(D_n)$ denote the Euler characteristics of domains $D_{n+1}$ and $D_n$ as above, we see that each $D_n$ is simply connected. Let $\gamma_n$ be the boundary of $D_n$; then $\gamma_n$ is a Jordan curve and $f^{n}$ is a $d^{n}$-fold map of $\gamma_n$ onto $\gamma_0$. Set $\lim\limits_{n\rightarrow\infty}\gamma_n=\Gamma .$ Roughly speaking, we shall show that $\gamma_n$ converges to $\partial F_{\infty}(=J(f))$, i.e. $\Gamma=J(f)$.

If $\xi\in\Gamma$ then there are points $\xi_n$ on $\gamma_n$ which converge to $\xi$, so, in particular, $\xi$ is in the closure of $F_{\infty}$. However, $\xi$ cannot lie in $F_{\infty}$ else it has a compact neighbourhood $K$ lying in some $D_n$ (for the $D_j$ are an open cover of $K$), and hence not meeting $\gamma_n$, $\gamma_{n+1}$, $\cdots$ for sufficiently large $n$. We deduce that $\Gamma\subset J(f)$.

$J(f)$ is porous, then $J_c(f)$ admits a conformal iterated function system $S_t=\{f_t^{-i}: t\in s\}$ for finite $s$ satisfying conditions (a)-(e) by Lemma 3.3.

To prove that $J(f)\subset\Gamma$, let $w\in J(f)$ be a repelling fixed point (or an image of a repelling fixed point) and $l$ be the straight line determined by the strongly tangent direction of $J(f)$ at $w$ as in Lemma 4.3. Then $w$ is an attracting fixed point of $f^{-1}$. Moreover,

$ f^{-1}: U(w)\rightarrow U(w) $

is a conformal map, where $U(w)$ is a disk centered at $w$. Now suppose that $J(f)$ is not contained in $\Gamma$. Consider $x\in J(f)\setminus \Gamma$ such that $x\in U(w)$, then $\lim\limits_{n\rightarrow\infty}f^{-n}(x)=w$ and for every $n\geq0$, we have $f^{-n}(x)\in J(f).$ Since the map $f^{-1}: U(w)\rightarrow U(w)$ is conformal, we get

$ \angle(w-f^{-n}(x), l)=\angle(f^{-n}(w)-f^{-n}(x), l)=\angle(f^{-n}(w-x), f^{-n}(l)) =\angle(x-w, l) . $

It follows that $w$ and $f^{-n}(x) (n\geq 0)$ are contained in the same line $\widetilde{l}\neq l$ and this implies that $\widetilde{l}$ is the strongly tangent straight line of $J(f)$ at $w$. Therefore, we conclude that $l$ is not a strongly tangent straight line of $J(f)$ at $w$. This contradiction proves that $J(f)\subset \Gamma$.

Remark If Main Theorem only with the hypothesis: the Fatou set $F(f)$ has a completely invariant component, $J(f)$ need not be a Jordan curve; for example, the map $z\mapsto z^2-1$ is expanding on its Julia set (certainly NCP map), see Theorem 9.7.5 and Figure 1.5.1 in [8].