2.1 Basic dynamics

Here we give the basic definitions, concepts, and notations we will use. When we say a function is meromorphic, we mean that it is transcendental meromorphic. We refer the reader to standard sources on meromorphic dynamics for details and proofs. See e.g. [Reference BergweilerBerg, Reference Branner and FagellaBF, Reference Baker, Kotus and LüBKL1, Reference Baker, Kotus and LüBKL2, Reference Baker, Kotus and LüBKL3, Reference Baker, Kotus and LüBKL4, Reference Devaney and KeenDK, Reference Keen and KotusKK].

We denote the complex plane by ${\mathbb C}$ , the Riemann sphere by $\widehat {\mathbb C}$ , and the unit disk by ${\mathbb D}$ . We denote the punctured plane by ${\mathbb C}^* = {\mathbb C} \setminus \{ 0 \}$ and the punctured disk by ${\mathbb D}^* = {\mathbb D} \setminus \{ 0 \}$ .

Given a family of meromorphic functions, $\{ f_{\lambda }(z) \}$ , we look at the orbits of points formed by iterating the function $f(z)=f_{\lambda }(z)$ . If $f^k(z)=\infty $ for some $k>0$ , z is called a pre-pole of order k—a pole is a pre-pole of order one. For meromorphic functions, the poles and pre-poles have finite orbits that end at infinity. The Fatou set or stable set, $F_f$ , consists of those points at which the iterates $\{f_{\lambda }^{n}\}_{n=0}^{\infty }$ are well defined and form a normal family in a neighborhood of each of them. The Julia set $J_f$ is the complement of the Fatou set and contains infinity as well as all the poles and pre-poles.

A point z such that $f^n(z)=z$ is called periodic. The minimal such $n>0$ is called the period. Periodic points are classified by their multipliers, $\nu (z)=(f^n)'(z)$ , where n is the period: they are repelling if $|\nu (z)|>1$ , attracting if $0< |\nu (z)| < 1$ , superattracting if $\nu =0$ , and neutral otherwise. A neutral periodic point is parabolic if $\nu (z)=e^{2\pi i p/q}$ for some rational $p/q$ . The Julia set is the closure of the repelling periodic points. For meromorphic f, it is also the closure of the pre-poles (see e.g. [Reference Baker, Kotus and LüBKL1]).

If D is a component of the Fatou set, either $f^n(D) \subseteq f^m(D)$ for some integers $n,m$ or $f^n(D) \cap f^m(D) = \emptyset $ for all pairs of integers $m \neq n$ . In the first case, D is called eventually periodic and in the latter case, it is called wandering. The periodic cycles of stable domains are classified as follows.

• • Attracting or superattracting if the periodic cycle of domains contains an attracting or superattracting cycle in its interior.

• • Parabolic if there is a parabolic periodic cycle on its boundary.

• • Rotation if $f^n: D \rightarrow D$ is holomorphically conjugate to a rotation map. Rotation domains are either simply connected or topological annuli, which are called Siegel disks or Herman rings, respectively.

• • Essentially parabolic, or Baker, if there is a point $z_{\infty } \in \partial D$ such that $f^{n} (z_{\infty })$ is not well defined and for every $z \in D$ , $\lim _{k \to \infty } f^{nk}(z) = z_{\infty }.$

A point a is a singular value of f if f is not a regular covering map over a.

• • a is a critical value if for some z, $f'(z)=0$ and $f(z)=a$ .

• • a is an asymptotic value if there is a path $\gamma (t)$ , called an asymptotic path, such that $\lim _{t \to \infty } \gamma (t) = \infty $ and $\lim _{t \to \infty } f(\gamma (t))=a$ .

• • The set of singular values $S_f$ consists of the closure of the critical values and the asymptotic values. The post-singular set is

  $$ \begin{align*} P_f= \overline{\bigcup_{a \in S_f} \bigcup_{n=0}^{\infty} f^n(a) \cup \{\infty\}}. \end{align*} $$
  For notational simplicity, if a pre-pole s of order p is a singular value, $\bigcup _{n=0}^{p} f^n(s)$ is a finite set with $f^{p} (s)=\infty $ .

Another important concept is the following.

In rational dynamics, a map is hyperbolic if it satisfies an expansion property on its Julia set; that is, there exist constants $c>0$ and $K>1$ such that for all z in a neighborhood $V \supset J(f)$ , $|(f^{n})'(z)|> c K^n$ (see [Reference MilnorMil1]). For such maps, this property is equivalent to the condition that $J_f \cap {P_f} = \emptyset $ .

Because the Julia set of a meromorphic function is unbounded and its iterates have singularities at the pre-poles, we need a version of this condition tailored to transcendental maps. We use the following one proved in [Reference Rippon and StallardRS], which applies to hyperbolic functions in ${\mathcal F}_2$ .

Proposition 2.3. (Rippon–Stallard)

If $S(f)$ is bounded and $\overline {PS(f)} \cap J(f) = \emptyset $ , then there exist two constants $c>0$ and $K>1$ satisfying

(2) $$ \begin{align} |(f^n)'(z) |> c K^n(|f^n(z)|+1)/(|z|+1), \end{align} $$

for all $z \in J(f) \setminus A_n(f)$ and all n, where $A_n(f)$ is the set of points where $f^n$ is not analytic (pre-poles of lower order).

A standard result in dynamics is that each attracting, superattracting, parabolic, or Baker cycle of domains contains a singular value. Moreover, unless the cycle is superattracting, the orbit of the singular value is infinite and accumulates on the cycle or the orbit of $z_{\infty }$ associated with the Baker domain. The boundary of each rotation domain is contained in the post-singular set. (See e.g. [Reference MilnorMil1, Chs. 8–11] or [Reference BergweilerBerg, §4.3].)

2.1.1 Nevanlinna’s theorem

Recall that the Schwarzian derivative is defined by

$$ \begin{align*} S(f) = \bigg(\frac{f''}{f'} \bigg)' -\frac{1}{2} \bigg(\frac{f''}{f'} \bigg)^2. \end{align*} $$

It satisfies the cocycle relation

$$ \begin{align*} S(f \circ g) = S(f)( g')^2 + S(g). \end{align*} $$

Because the Schwarzian derivative of a Möbius transformations is zero, solutions to the Schwarzian differential equation $S(f)(z)=P(z)$ are unique up to post-composition by a Möbius transformation. See [Reference HilleHil, Reference NevanlinnaNev] for proofs.

Nevanlinna’s theorem characterizes transcendental functions with finitely many singular values and finitely many critical values in terms of their Schwarzian derivatives.

Theorem 2.4. (Nevanlinna [Reference NevanlinnaNev, Ch. XI], [Reference HilleHil])

Every meromorphic function g with $p < \infty $ asymptotic values and no critical values has the property that its Schwarzian derivative is a polynomial function of degree $p -2$ . Conversely, for every polynomial function $P(z)$ of degree $ p-2$ , the solution to the Schwarzian differential equation $S(g)=P(z)$ is a meromorphic function with exactly p asymptotic values and no critical points. The only essential singularity is at infinity.

A summary of the proof is given in [Reference Devaney and KeenDK1], where the behavior of the function in a neighborhood of infinity is described. There are p equally spaced asymptotic tracts separated by Julia directions along which the poles tend asymptotically to infinity. An immediate corollary of the theorem is the following.

Corollary 2.5. If f is a meromorphic function with p finite simple asymptotic values and no critical values, and h is a homeomorphism of the complex plane ${\mathbb C}$ such that $g=h^{-1} \circ f \circ h$ is holomorphic (meromorphic), then $S(g)$ is a polynomial of degree p.

In [Reference Devaney and KeenDK1], this corollary is used to prove that if f has polynomial Schwarzian derivative and all its asymptotic values are finite, then f cannot have a Baker domain.

2.2 Functions with two simple asymptotic values

Our focus in this paper is on parameter spaces of meromorphic functions with two finite simple asymptotic values and no critical values. By Theorem 2.4, such functions are characterized by the property that they have a constant Schwarzian derivative. Each asymptotic value is simple and there are exactly two distinct non-equivalent asymptotic tracts. We denote this family by ${\mathbb F}_{2}$ .

It is easy to compute that $S(e^{2kz})= -2k^2$ and therefore that the most general solution to the equation $S(f)=-2k^2$ is

(3) $$ \begin{align} f(z)= \frac{a e^{kz} + be^{-kz}}{ce^{kz}+de^{-kz}}, \quad ad - bc \neq 0, \end{align} $$

and its asymptotic values are $\{ a/c, b/d \}$ . Note that both of them are omitted values. According to Theorem 2.4, the converse is true too. Moreover, by Corollary 2.5, the solution is unique up to post-composition by an affine map. Pre-composition by an affine map multiplies the constant k by the scaling factor.

Functions of the form (3) have a single essential singularity at infinity and their dynamics are invariant under affine conjugation. If one of the asymptotic values is equal to infinity, c or $d=0$ and the family is the well-studied exponential family. See e.g. [Reference Devaney, Fagella and JarqueDFJ, Reference Rempe-GillenRG]. The dynamics are quite different if both asymptotic values are finite and here we restrict ourselves to this situation.

Because we assume the asymptotic values are finite, neither c nor d can be zero. We choose a representative of an equivalence class, where the equivalence relation is defined by affine conjugation, such that $k=1$ and $f(0)=0$ ; this implies $b=-a$ . If the asymptotic values are $\lambda $ and $\mu $ , we have

$$ \begin{align*} f_{\lambda , \mu}(z) = \frac{e^{z}-e^{-z}}{({e^{z}}/{\lambda })-({e^{-z}})/{\mu}}, \end{align*} $$

where $\lambda , \mu \in {\mathbb C}^*$ . If $f'(0)= \rho \in {\mathbb C}^{*}$ , we have the relation

(4) $$ \begin{align} \frac{1}{\lambda } - \frac{1}{\mu} = \frac{2}{\rho}. \end{align} $$

We still have the freedom to conjugate by the affine map $z \to -z$ so we see that the maps $f_{\lambda , \mu }(z)$ and $f_{\lambda ', \mu '}(z)=-f_{\lambda , \mu }(-z)$ have the same dynamics. That is,

$$ \begin{align*}f_{\lambda ', \mu'}(z)=\frac{e^{z}-e^{-z}}{({e^{z}}/{\lambda '})-({e^{-z}})/{\mu'}}=\frac{e^z-e^{-z}}{({e^z}/({-\mu}))-({e^{-z}}/({-\lambda}))},\end{align*} $$

where

(5) $$ \begin{align} \frac{1}{\lambda '} - \frac{1}{\mu'} = \frac{2}{\rho}=-\frac{1}{\mu} + \frac{1}{\lambda }. \end{align} $$

Set $f_{\lambda ,\mu } \sim f_{\lambda ',\mu '}$ if $\lambda '=-\mu , \mu '=-\lambda $ and use this equivalence relation to define the space of pairs of functions:

$$ \begin{align*} \widehat{\mathbb{F}}_{2} &=\bigg\{ f_{\lambda, \rho} (z)= \frac{e^{z}-e^{-z}}{({e^{z}}/{\lambda })-({e^{-z}})/{\mu}} \; \;\bigg| \;\; \rho \in {\mathbb C}^{*}, \lambda \in {\mathbb C}^*\setminus \{\rho/2\}, \;\; \frac{1}{\lambda } - \frac{1}{\mu} = \frac{2}{\rho} \bigg \} \bigg/\sim. \end{align*} $$

Note that each pair of complex numbers $(\lambda , \rho )$ uniquely determines a pair of functions so that we also use $\widehat {\mathbb F}_{2}$ to denote the moduli space of ${\mathbb F}_{2}$ .

Because of the ambiguity left by the normalization, it is difficult to study $\widehat {\mathbb {F}}_2$ directly. This situation is similar to the space $Rat_2$ of rational functions of degree two with a fixed point at infinity discussed in §1. The affine conjugation $z \to -z$ identifies maps with the same dynamics and sends the parameter b to $-b$ . Thus the $(b, \rho )$ space is a 2-fold covering map of the space of functions. To understand the role of the parameters, however, it is easier to work in this covering space. This can be done by marking the singular points and choosing a ‘preferred’ point. In [Reference Goldberg and KeenGK], the preferred point was taken as $R(+1)$ . The conjugation $z \to -z$ interchanged the marking and corresponded to the involution ${b \to -b}$ in the lifted parameter space of functions with marked critical values. See e.g. [Reference Goldberg and KeenGK, Reference MilnorMil1] for more details.

We proceed in the same way here. To mark the asymptotic values, we choose $\lambda $ as the ‘preferred’ value and $\mu $ as the ‘non-preferred’ value. That is, $\lambda = \lim _{t \to \infty }f_{\lambda ,\rho }(\gamma (t))$ , where $\Re \gamma (t) \to +\infty $ . We call the space with marked asymptotic values $\mathbb {F}_2$ . Again the marked space is a 2-fold cover of the space of functions. Note that if $\lambda =\infty , \mu =-\rho /2$ and if $\mu =\infty , \lambda =\rho /2$ .

Because the stable dynamics of functions in $\mathbb {F}_2$ is controlled by the behavior of the orbits of the asymptotic values, it will be convenient to choose a one-dimensional ‘slice’ in this covering space of $\mathbb {F}_2$ in such a way that at each point in the slice, the orbit of one asymptotic value has fixed dynamics. One way to do this is to require that both asymptotic values have similar behavior. For example, if $\mu =-\lambda $ , so that $\lambda =\rho $ , the slice obtained is the tangent family $ f_{\rho }(z)=\rho \tanh z = i \rho \tan (iz)$ . The properties of this slice have been investigated in [Reference Chen, Jiang and KeenCJK, Reference Keen and KotusKK].

3.1 Fatou components for $f_{\lambda } \in {\mathcal F}_{2}$

For any $f_{\lambda } \in {\mathcal F}_{2}$ , the origin is an attracting fixed point with multiplier $\rho $ . Denote its attracting basin (which is non-empty) by $A_{\lambda }$ .

3.2 The parameter space trichotomy

Proposition 3.1 implies the following trichotomy for ${\mathcal F}_{2}$ .

• • $A_{\lambda }$ contains both asymptotic values; this is called the shift locus and denoted by ${\mathcal S}$ .

• • $A_{\lambda }$ contains only the preferred asymptotic value $\lambda $ ; in this case, the other asymptotic value $\mu $ is not attracted to the origin and we call the set of such $\lambda $ as ${\mathcal M}_{\mu }$ . We denote the subset where $\mu $ is attracted to an attracting periodic cycle by ${\mathcal M}_{\mu }^0$ .

• • $A_{\lambda }$ contains only the non-preferred asymptotic value $\mu $ ; in this case, the other asymptotic value $\lambda $ is not attracted to the origin and we call the set of these $\lambda $ as ${\mathcal M}_{\lambda }$ . We denote the subset where $\lambda $ is attracted to an attracting periodic cycle by ${\mathcal M}_{\lambda }^0$ .

The maps in ${\mathcal S}$ , ${\mathcal M}_{\lambda }^0$ , and ${\mathcal M}_{\mu }^0$ are hyperbolic because the orbits of their asymptotic values accumulate on attracting cycles. The connected components of these three subsets of parameter space are thus called hyperbolic components.

As with the space $Rat_2$ , there is an inversion of the space ${\mathcal F}_{2}$ that interchanges the regions ${\mathcal M}_{\lambda }$ and ${\mathcal M}_{\mu }$ , and leaves ${\mathcal S}$ invariant.

Let ${\mathcal C}_0$ be the circle in the $\lambda $ plane centered at the parameter singularity $\rho /2$ with radius $|\rho /2|$ and let D be the disk it bounds, punctured at the singularity $\rho /2$ . (See Figure 11 later.) The inversion

$$ \begin{align*} I(\lambda )=-\mu = \frac{\lambda }{2\lambda /\rho -1} \end{align*} $$

leaves ${\mathcal C}_0$ invariant and interchanges $\lambda $ and $-\mu $ .

Proof. Suppose $f_{\lambda }^n(\lambda ) \not \rightarrow 0$ so that $f_{\lambda }^n(\mu ) \rightarrow 0$ . Because $ I(\lambda )=-\mu $ and $I(\mu )=-\lambda $ , we can write

$$ \begin{align*} f_{-\mu}(-\mu)=\frac{e^{-2\mu}-1}{({e^{-2\mu}})/({-\mu}) -({1}/({-\lambda }))} = \frac{e^{2\mu}-1}{({e^{2\mu}}/{\lambda }) -({1}/{\mu})}= f_{\lambda }(\mu) \end{align*} $$

and

$$ \begin{align*} f_{-\mu}(-\lambda )=\frac{e^{-2\lambda }-1}{({e^{-2\lambda }}/({-\mu})) -({1}/({-\lambda }))} = \frac{e^{2\lambda }-1}{({e^{2\lambda }}/{\lambda }) -({1}/{\mu})}= f_{\lambda }(\lambda ).\\[-42pt] \end{align*} $$

It follows that the inversion also preserves the region ${\mathcal S}$ where both asymptotic values go to zero.

When $\rho $ is real, we can say more.

Proof. Set $\lambda _1=I(\lambda )$ . Then $-\mu = I(\lambda ) = \bar {\lambda } =\lambda _1$ . Thus,

$$ \begin{align*}\overline{f_{\lambda } (\lambda )} = f_{\bar{\lambda }}(\bar{\lambda })=f_{\lambda _1}(\lambda _1); \end{align*} $$

so if $f_{\lambda }^n(\lambda ) \rightarrow 0$ , $f_{\lambda _1}^n(\lambda _1) \rightarrow 0$ .

We can rewrite this as

$$ \begin{align*} f_{\lambda }(\mu)=f_{\lambda }(-\bar{\lambda }) = - \overline{f_{\lambda }(\lambda )}.\end{align*} $$

Therefore, either both asymptotic values iterate to zero or neither does. Because the origin is an attracting fixed point with multiplier $\rho $ , at least one must iterate to zero and so they both do.

This proposition says when $\rho $ is real, the region where both asymptotic values are attracted to zero contains the invariant circle of the inversion.

Notice that the point $\lambda = \rho $ is on the circle ${\mathcal C}_0$ . At that point, we have $\mu = -\lambda $ , $f_{\lambda } = \lambda \tanh z$ , and $I(\lambda )=\lambda $ so that it is a branch point of the double covering defined by the marking. Moreover, because of the symmetry, both asymptotic values are attracted to zero.

If $\lambda \in {\mathcal M}_{\mu }^0 \text { or } {\mathcal M}_{\lambda }^0$ , $f_{\lambda }$ has an attracting periodic cycle different from the origin. This cycle has an attractive basin, which we denote by $K_{\lambda }$ , and $A_{\lambda } =\widehat {\mathbb C} \setminus \overline {K_{\lambda }}$ . Thus $\partial {K_{\lambda }}$ is the Julia set and $\overline {K_{\lambda }}$ is the ‘filled Julia set’. Both of them are unbounded sets in ${\mathbb C}$ .

3.3 The set $\overline {K_{\lambda }}$ and the Julian set dichotomy

In [Reference Keen and KotusKK], it is proved that for $\rho \in {\mathbb D}^*$ , the Julia set $J_{\lambda }$ of the function $T_{\rho }(z)=\rho \tanh (z)$ is a Cantor set. Moreover, it is homeomorphic to a space consisting of finite and infinite sequences on an alphabet isomorphic to the natural numbers and infinity. The finite sequences end with infinity. The homeomorphism conjugates $T_{\rho }$ to the shift map on this alphabet. See [Reference Devaney and KeenDK1, Reference MoserMo] for details.

At this point in this paper, we can prove the following.

Note that the argument above adapts easily to show that if $f_{\lambda }$ has a non-zero attracting or parabolic fixed point, the attracting basin of this fixed point is unbounded and completely invariant. Other standard arguments [Reference MilnorMil1] show that if $f_{\lambda }$ has a neutral fixed point with a Siegel multiplier, its boundary must be contained in the post-singular set. Thus there are two completely invariant domains in the Fatou set separated by the Julia set. An example of this is shown in Figure 1, where $\rho =2/3$ and $\lambda =2+2i$ . The yellow is the basin of $0$ and the blue is the basin of the fixed point $2.25818 + 2.12632i$ . The proof of the main structure theorem uses another example of a function with two attractive fixed points and its dynamic space is shown in Figure 7 (see later).

Figure 1 The dynamic plane of $f_{\lambda }$ with $\rho =2/3$ and $\lambda =2+2i$ . The stars are fixed points and the black dot is a pole.

4.1 Virtual cycle parameters and virtual centers

Let $\Omega $ be a hyperbolic component in ${\mathcal M}_{\lambda }$ and let $\lambda \in \Omega $ . Both $\lambda $ and $\mu $ are attracted by attracting cycles of $f_{\lambda }$ , and because $\lambda \in {\mathcal M}_{\lambda }$ , $\mu $ is attracted to the origin and $\lambda $ is attracted to a different cycle of order $n \geq 1$ . Because all the $f_{\lambda }$ , $\lambda \in \Omega $ are quasiconformally conjugate, all the functions in $\Omega $ have non-zero attracting cycles of the same period, say n. We say $\Omega $ has period n and, where appropriate, denote it by $\Omega _n$ .

We need the following definitions.

This definition is justified by the following. Assume for argument’s sake that we are in the first case. Let $\gamma (t)$ be an asymptotic path for $\lambda =a_1$ , that is, $\lim _{t\to \infty } \gamma (t) =\infty $ and $\lim _{t\to \infty } f(\gamma (t))=a_1$ .

Then $\widehat \gamma =f^{-n}(\gamma (t))$ is again an asymptotic path where the inverse branches are chosen so that $f^{-1}(a_i)=a_{i-1}$ ; that is,

$$ \begin{align*} \lim_{t \to \infty} f(\gamma(t))=\lim_{t \to \infty} f^{n+1}(\widehat\gamma(t))=\lambda =a_1, \end{align*} $$

so that in this limiting sense, the points form a cycle.

Definition 4.2. Let $\Omega _n$ be a shell component of period n and let

$$ \begin{align*} {\mathbf a}_{\lambda} =\{a_0, a_1, \ldots, a_{n-2}, a_{n-1} \} \end{align*} $$

be the attracting cycle of period n that attracts $\lambda $ or $\mu $ . Suppose that as $k \to \infty $ , $\lambda _k \to \lambda ^* \in \partial {\Omega _n}$ , and the multiplier $\nu _{\lambda _k}=\nu ({\mathbf a}_{\lambda _k})=\Pi _{i=0}^{n-1} f'(a_{i}(\lambda _k)) \to 0.$ Then $\lambda ^*$ is called a virtual center of $\Omega _n$ .

Because the attracting basin of the cycle ${\mathbf a}_{\lambda }$ must contain an asymptotic value, we will assume throughout the paper that the points in the cycle are labeled so that $\lambda $ or $\mu $ and $a_1$ are in the same component of the immediate basin.

In the next theorem, we collect the results reported in [Reference Fagella and KeenFK] about shell components for fairly general families of functions. The proof of parts (b) and (c) are based on an estimate of the growth of the orbits of the singular values given in lemma 2.2 of [Reference Rippon and StallardRS], and on proposition 6.8 of [Reference Fagella and KeenFK]. Part (d) is theorem 6.10 of [Reference Fagella and KeenFK]. Part (e) combines the accessibility in part (d) and theorem A of [Reference Chen and KeenCK], whose proof contains a construction that shows that every virtual cycle parameter is on the boundary of a shell component.

As a corollary, we have the following.

Proof. The poles of $f_{\lambda }$ are given by

(6) $$ \begin{align} p_k(\lambda)=\frac{1}{2}{\text{Log}} \bigg(\frac{\rho-2\lambda}{\rho}\bigg)+ik\pi, \end{align} $$

where ${\text {Log}}$ is the branch of the logarithm with an imaginary part in $[-\pi ,\pi )$ . They and all their pre-images are holomorphic functions of $\lambda $ .

Let V be a neighborhood of $\lambda ^*$ in the parameter plane that does not contain any poles of $f_{\lambda }^{k}$ for all $1\leq k<n-1$ . Such a neighborhood exists because the poles of $f_{\lambda }^{k}$ form a discrete set. The holomorphic function $h(\lambda )=f_{\lambda }^{n-1}(\lambda )$ maps V to a neighborhood W of infinity and $h(\lambda ^*)=\infty $ . Because infinity is an essential singularity, for $\lambda \in V$ and large enough $|k|$ , W contains infinitely many poles $p_k(\lambda )$ of the functions $f_{\lambda }(z)$ ; moreover, for each $\lambda $ , as $k \to \infty $ , $p_k(\lambda )$ converge to infinity. The zeroes of the functions $h_k(\lambda )=h(\lambda )-p_k(\lambda )$ are virtual centers of components of period $n+1$ . We want to show there is a sequence of these zeroes in V converging to $\lambda ^*$ .

The functions $\widehat {h}(\lambda )=1/h(\lambda )$ and $\widehat {p}_k(\lambda )=1/p_k(\lambda )$ take values in a neighborhood of the origin. Because the $p_k(\lambda )$ converge to infinity as $|k| \to \infty $ uniformly on $\overline {V}$ as long as V is small enough, we can find N large enough so that if $k>N$ and $\lambda \in \partial V$ , then $|\widehat {p}_k(\lambda )| < |\widehat {h}(\lambda )|$ . By Rouché’s theorem, we conclude $\widehat {h}$ and $\widehat {h} -\widehat {p}_k$ have the same number of zeroes in V; $\widehat {h}$ has a zero at $\lambda ^*$ and thus each $\widehat {h} -\widehat {p}_k$ has a zero $\lambda _k \in V$ . It then follows that $h(\lambda _k) =p_k(\lambda _k)$ so that $\lambda _k$ is a virtual center of period $n+1$ .

4.2 Combinatorics

Theorem 4.4 allows us to assign a label to each of the shell components of ${\mathcal M}_{\lambda }$ in terms of its virtual center. To label the virtual centers, we need to know that the indices of the poles are well defined. In §6.3.1, we will prove Lemma 6.18 that says that we can find a simply connected domain $\Sigma $ , containing ${\mathcal M}_{\lambda }$ and not containing ${\mathcal M}_{\mu }$ , in which, after an initial choice, as above, of a basepoint and a branch of the logarithm, the poles and inverse branches of $f_{\lambda }$ can be labeled consistently. The discussion here will assume that lemma.

Pick a basepoint that is not in ${\mathcal M}_{\lambda }$ , for example, the symmetric point $\lambda _0 =-\mu _0=\rho $ . It has poles $p_k(\lambda _0)$ defined by the principal branch of the logarithm. With the poles $p_k(\lambda _0)$ defined by equation (6), denote the branch of $f_{\lambda _0}^{-1}$ that maps $\infty $ to $p_k(\lambda _0)$ by $g_{\lambda ,k}(z)=g_{\lambda _0,k}(z)$ . With the choice of a fixed base point and logarithm branch, the inverse branches are well defined because the set $\Sigma $ (to be defined in §6.3.1) is simply connected.

We use these branches to define labels for the pre-poles of all orders, and thus for labels of the virtual cycle parameters. Because of part (d) of Theorem 4.4, each virtual cycle parameter is a virtual center of a shell component so the label of the virtual center defines a label for the shell component.

The formula for $p_{k}(\lambda )$ shows that the poles are injective functions of $\lambda $ in $\Sigma $ . Let

$$ \begin{align*} {\mathcal V}_1=\{ \lambda ^*_k \in \Sigma \, | \, g_{\lambda ^*,k}(\infty)= \lambda ^* \}. \end{align*} $$

That is, ${\mathcal V}_1$ is the set of $\lambda ^*_k$ such that $f_{\lambda ^*_k}(\lambda ^*_k)=\infty $ . It is the set of virtual cycle parameters of order one and hence virtual centers of shell components $\Omega _2$ of period two. We assign the label k to each point in ${\mathcal V}_1$ and the same label to the component for which it is the virtual center.

The pre-poles $p_{k_1k_2}(\lambda )=g_{\lambda ,k_2}(p_{k_1}(\lambda ))$ are defined for all $\lambda \in \Sigma \setminus {\mathcal V}_1$ . Because they are holomorphic functions of $\lambda $ with non-vanishing derivative, each $g_{\lambda , k}$ is an injective function of both $\lambda $ and z. Next, we inductively define the sets of virtual cycle parameters of order $n-1$ with labeled points by

$$ \begin{align*} {\mathcal V}_{n-1}=\bigg\{\lambda ^*_{k_{n-1} \ldots k_1} \in \Sigma \setminus \bigcup_{i=1}^{n-2} {\mathcal V}_i \, | \, g_{\lambda ^*_{k_{n-1} \ldots k_1},k_{n-1} \ldots k_1}(\infty)= \lambda ^*_{k_{n-1} \ldots k_1}\bigg\}. \end{align*} $$

The pre-poles $p_{k_{n-1} \ldots k_1}(\lambda )$ are defined for all $\lambda \not \in {\mathcal V}_{n-1}$ and, as above, move injectively.

We now assign the label $k_{n-1} \ldots k_1$ to the shell component of order n for which $\lambda ^*_{k_{n-1} \ldots k_1}$ is the virtual center.

We can also use the labeling of the inverse branches to assign an itinerary to each attractive cycle.

Definition 4.7. For simplicity, we suppress the dependence on $\lambda $ and assume the shell component is in ${\mathcal M}_{\lambda }$ . Suppose $f^n(a_0)=a_0$ for $n \geq 1$ , where, by our numbering convention in part (c) of Theorem 4.4, $a_0$ is in the asymptotic tract of $\lambda $ and $a_j=f(a_{j-1})$ , $j=1, \ldots n$ . Then for some $k_j$ , $a_{j-1}= g_{k_j}(a_{j})$ . In fact, there is a unique sequence $\{k_1,\ldots , k_{n}\}$ such that

$$ \begin{align*} a_0= g_{k_{n}} \circ \cdots \circ g_{k_2} \circ g_{k_{1}}( a_0). \end{align*} $$

We say the cycle $\mathbf a$ has itinerary ${\mathbf k}_n= k_{n}k_{n-1}\ldots k_2 k_1$ .

Above we assigned a label, or itinerary, to the virtual center of each shell component. We now address the questions of the uniqueness of these labels and their relation to the itineraries of their attracting cycles. As we stated above, this is based on Lemma 6.18, which will be proved later, where the inverse branches are defined as single valued functions of $\lambda $ .

Because the shell components of period one have virtual centers that do not belong to the parameter space, we cannot label them in this way. There are only two such points, $\rho /2$ and $\infty $ , and hence only two such components with no label. To have a label for every component, we arbitrarily assign the label $\infty $ to these components.

The proof of Corollary 4.5 also implies the following.

Thus the combinatorics of the pre-poles enable us to label each shell component $\Omega _n \in {\mathcal M}_{\lambda }$ and $\Omega _n' \in {\mathcal M}_{\mu }$ by the itinerary of its virtual center. If $ {\mathbf k}_{n-1}= k_{n-1}k_{n-2}\ldots k_1 $ is the itinerary of the virtual center of a shell component of period n, and we want to emphasize it, we write $\Omega _{{\mathbf k}_{n-1}}$ or $\Omega _{{\mathbf k}_{n-1}}'$ .

The above discussion, modulo the proof of Lemma 6.18, gives us a proof of the combinatorial structure theorem.

4.3 Parameter space pictures

Figure 2 shows a picture of the $\lambda $ parameter plane for $\rho =2/3.$ The large region on the left (green in the color version) is ${\mathcal S}$ where both $\lambda $ and $\mu $ are attracted to the origin. The unbounded multishaded (multicolored region) on the right in Figure 2 is ${\mathcal M}_{\lambda }$ and the small bounded multishaded (multicolored) region inside ${\mathcal S}$ is ${\mathcal M}_{\mu }$ . Since this figure is drawn to scale, in ${\mathcal M}_{\mu }$ the different shades (colors other than yellow) are not visible. To make the structure of this region visible and show it is similar to ${\mathcal M}_{\lambda }$ ’s, in Figure 3 we place a blown up neighborhood of ${\mathcal M}_{\mu }$ near ${\mathcal M}_{\lambda }$ .

Figure 2 The $\lambda $ plane divided into the shift locus and shell components. The large region on the left (green in the color version) is the shift locus ${\mathcal S}$ . The region ${\mathcal M}_{\mu }$ is the small disk inside it. The region ${\mathcal M}_{\lambda }$ is on the right and shell components of different periods are shaded: the large region to the right is period 1; the biggest of the regions attached are period 2; the next smallest period 3, etc. Because ${\mathcal M}_{\mu }$ is so small, only the period-1 region is visible.

Figure 3 Blow-up of ${\mathcal M}_{\mu }$ placed near ${\mathcal M}_{\lambda }$ for comparison. The coloring scheme is the same as in Figure 2 and is now visible in the blown-up ${\mathcal M}_{\mu }$ .

Figure 4 Blow-up of the $\lambda $ plane near ${\mathcal M}_{\lambda }$ with the periods labeled.

The periods of the shell components decrease with size, with components of the same period shaded similarly. The period one component is on the right. In the color version, the shell components are colored according to their period: yellow is period $1$ , cyan is period $2$ , red is period $3$ and so on. Periods higher than $10$ are colored black. Note that there is only one unbounded domain, the yellow period- $\kern-1pt 1$ domain on the right, $\Omega _1$ ; its virtual center is the point at infinity. The virtual center of the period- $\kern-1pt 1$ component of ${\mathcal M}_{\mu }$ is the leftmost point. It is the singular point $\rho /2$ of the parameter space. There is a cusp boundary point of $\Omega _1$ on the real axis where the multiplier of the cycle attracted to $\lambda $ is $+1$ . There are cyan period- $2$ components appearing as ‘buds’ off of the yellow component $\Omega _1$ where the multiplier of the cycle attracted to $\lambda $ is $e^{(2m+1) \pi i}$ ; each of these has a virtual center with itinerary $\mathbf k_1=m$ .

In Figure 4, we see a period- $2$ component $\Omega _2$ budding off $\Omega _1$ . Although $M_{\lambda }$ and $M_{\mu }$ look disconnected in the figure, as we will prove, they are not. Here we have only computed shell components for periods up to $10$ . To make a figure where ${\mathcal S}$ and ${\mathcal M}_{\lambda }$ look connected would require much more computation and many more colors to show components with much higher periods. What we do see, however, is a period $3$ , red component that is not a bud component of the period- $\kern-1pt 1$ component. In fact, there are infinitely many such converging to the virtual center of $\Omega _2$ marked as v. We postpone a full discussion of the finer structure of the shell components to future work.

5.1 J-stability and the bifurcation locus

Denote the set of virtual cycle parameters by $\mathcal {B}_{cv}$ . By Theorem 4.4, each such parameter is on the boundary of a unique shell component and in §4.2, we used these parameters to enumerate the shell components. Here we will prove that these parameters are dense in the boundary of the shift locus. To do so, we need two definitions.

Theorem B of [Reference Mané, Sad and SullivanMSS] in our context states the following.

As a corollary (see [Reference Keen and KotusKK, Corollary 3.2]), we have the following.

Consider a component U of ${\mathcal J}$ . Suppose $\lambda $ and $\lambda '$ are two points in U and $\gamma (t): [0, 1]\to U$ is an analytic curve connecting $\lambda $ and $\lambda '$ with $\gamma (0)=\lambda $ and $\gamma (1)=\lambda '$ . In a neighborhood V of $\gamma $ with basepoint $\lambda $ , the set $E_{c}=\{p_{c}\}$ , $c\in V$ , of all repelling periodic points of $f_{c}$ defines a holomorphic motion,

$$ \begin{align*} h(z, c) =p_{c}: E_{\lambda } \times V\to {\widehat{\mathbb C}}. \end{align*} $$

The $\lambda $ -lemma (see [Reference Gardiner, Jiang and WangGJW, Reference Mané, Sad and SullivanMSS]) implies that the holomorphic motion h can be extended to the closure $\overline {E}_{\lambda }$ of $E_{\lambda }$ ; that is, there is a holomorphic motion

$$ \begin{align*} H(z,c): \overline{E}_{\lambda } \times V\to {\widehat{\mathbb C}} \end{align*} $$

such that $H|E\times V=h$ and $H_{\lambda , \lambda '} (z)=H(z, \lambda '): \overline {E}_{\lambda } \to \overline {E}_{\lambda '}$ is a quasiconformal homeomorphism. Because the repelling periodic points of $f_{\lambda }$ are dense in the Julia set $J_{\lambda }$ , it follows that $\overline {E}_{\lambda } = J_{\lambda }$ and

$$ \begin{align*} H_{\lambda , \lambda '} \circ f_{\lambda } =f_{\lambda '}\circ H_{\lambda , \lambda '} \quad \text{on } J_{\lambda}. \end{align*} $$

Note that the construction of $H_{\lambda , \lambda '}$ depends on the choice of the curve $\gamma $ and it may not be unique.

We also need the following generation of Montel’s theorem.

The set of virtual center parameters $\mathcal {B}_{cv}$ is clearly not contained in $\mathcal {J}$ . By Theorem 4.4 and Theorem 5.2, the points in $\mathcal {B}_{cv}$ are on the boundaries of both a shell component and the shift locus. In addition, we have the following.

Proof. Because $0$ is an attracting fixed point, at least one of the families $\{f_{\lambda }^n(\lambda )\}$ and $\{f_{\lambda }^n(\mu )\}$ converges to $0$ ; that is, for each $\lambda _0$ , one of them is always normal and, by Proposition 5.8, in a neighborhood of $\lambda _0$ , it is the same family that is normal. Suppose $\lambda _0\in \partial \mathcal {J}$ ; without loss of generality, we may assume that $\{f_{\lambda }^n(\lambda )\}$ is not normal at $\lambda _0$ .

Let U be any neighborhood of $\lambda _0$ . The poles of $f_{\lambda ,}$

$$ \begin{align*} p_k(\lambda)=\frac{1}{2}\log \bigg(\frac{\rho-2\lambda}{\rho}\bigg)+ik\pi, \quad k\in \mathbb{Z}, \end{align*} $$

form a holomorphic family in U. If $f^n_{\lambda ,\mu }(\lambda ) \neq p_k(\lambda )$ for any k or $\lambda \in U$ , then Theorem 5.9 implies $f^n_{\lambda }(\lambda )$ is normal in U. This contradicts the hypothesis that $\lambda _0\in \partial \mathcal {J}$ .

The parabolic cusps and Misiurewicz points are contained in the bifurcation locus.

6.1 The model space

Every point $\lambda \in \Omega _1$ corresponds to a function $f_{\lambda }$ with a non-zero attracting fixed point denoted by $q_{\lambda }$ ; its attractive basin is denoted by $K_{\lambda }$ . By Propositions 3.1 and 3.6, it is simply connected and completely invariant. In fact, $f_{\lambda }$ is quasiconformally conjugate to $ t \tan z$ for some real $t\geq 1$ whose Julia set is the real line (see [Reference Devaney and KeenDK]), so its Julia set is the quasiconformal image of a line (see Figure 7 where $\rho =2/3$ ). In Figure 7, the cyan colored region is $K_{\lambda }$ and the yellow region is the basin of $0$ , $A_{\lambda }$ . The black dots are poles on the boundary of $K_{\lambda }$ and are in the Julia set. Denote the closure of $K_{\lambda }$ by $\overline {K_{\lambda }}$ . It is the analogue of the filled Julia set for a quadratic map.

Because the multiplier map $\nu $ is a universal covering from $\Omega _1$ to ${\mathbb D}^*$ , we can find a sequence of points $\lambda _j \in \Omega _1$ , $j \in {\mathbb Z}$ such that $\nu (\lambda _j)=f_{\lambda _j}'(q(\lambda _j))= \rho $ . We choose one, denote it by $\lambda _0$ , and set $q_{0}=q_{\lambda _0}$ . We set $Q(z)=f_{\lambda _0}(z)$ and let $K_0$ denote the attracting basin of $q_0$ .

In Figure 8, the set $\overline {K}_{0}$ is depicted for $\rho $ real and $\lambda _0$ taken as the real solution to $\nu (\lambda _0)=f_{\lambda _0}'(q(\lambda _0))= \rho $ . Because the multipliers of both attracting fixed points, $0$ and $q_0$ , are the same, there is a real $t=t(\rho )$ , $\sigma =it$ , such that $Q(z)$ and $it \tan iz =t \tanh z$ are not only quasiconformally conjugate but affine conjugate and the Julia set of $Q(z)$ is a vertical line.

Figure 7 The ‘filled Julia set’ of $Q(z)$ . The black dots are poles.

Figure 8 Domains and curves in the construction.

There is a local uniformizing map, which we denote by $\phi _0$ , $\phi _0: K_0 \rightarrow {\mathbb C}$ , normalized so that $\phi _0$ maps $q_0$ to $0$ , $\phi _0'(q_0)=1$ , and $\phi _0$ conjugates Q to $\zeta \to \rho \zeta $ in a neighborhood of $q_0$ . We can extend $\phi _0$ to all of $K_0$ by analytic continuation. Note that $\phi _0(z)=0$ if and only if $Q^n(z)=q_0$ for some n.

Let $r=|\phi _0(\lambda _0)|$ and let $\gamma ^*=\phi ^{-1}_0(re^{i\theta }), \, \theta \in {\mathbb R}$ . It is a simple closed curve; let $\Delta $ be the closed topological disk in $K_0$ bounded by $\gamma ^*$ . Then $\phi _0$ is injective on $\Delta $ and $\lambda _0$ is on $\partial \Delta $ .

Proof. The map E is defined as follows. Given $\lambda \in {\mathcal S}_{\lambda }^0$ , we define a conformal homeomorphism $\phi _{\lambda }$ from the neighborhood $O_{\lambda }$ in the attracting basin $A_{\lambda } $ to a disk centered at zero with $\phi _{\lambda }(0)=0$ that conjugates $f_{\lambda }$ to $ \zeta \mapsto \rho \zeta $ . The map is defined up to affine conjugation and depends holomorphically on $\lambda $ . We assume here that it is normalized so that $\phi _{\lambda }(\mu )=\phi _0(\lambda _0)$ . For each $\lambda $ , define a map

$$ \begin{align*} \xi_{\lambda}=\phi_0^{-1}\circ \phi_{\lambda}: O_{\lambda}\to \Delta. \end{align*} $$

From the definitions of $\phi _{\lambda }$ and $\phi _0$ , it follows that $\xi _{\lambda }(0)=q_0$ and $\xi _{\lambda }(\mu )=\lambda _0$ . Because $f_{\lambda _0}'(0)=Q'(q_0)=\rho $ , the map $\xi _{\lambda }$ is a conformal homeomorphism from $O_{\lambda }$ to $\Delta $ and it conjugates $f_{\lambda }$ to Q.

Now we are ready to define the map E from $\mathcal {S}_{\lambda }^0\to K_0\setminus \Delta $ as

$$ \begin{align*} E(\lambda ) = \xi_{\lambda }(\lambda ). \end{align*} $$

By construction, E satisfies properties (i) and (ii).

Suppose $\xi _{\lambda '}(\lambda ')=\xi _{\lambda }(\lambda )$ . The map $\xi _{\lambda ', \lambda }=\xi _{\lambda '}^{-1}\xi _{\lambda }=\phi _{\lambda '}^{-1}\phi _{\lambda }$ , restricted to a neighborhood of the origin in the basin $A_{\lambda }$ , defines a holomorphic conjugacy between $f_{\lambda }$ and $f_{\lambda '}$ on this neighborhood. Because $\xi _{\lambda ', \lambda } (\lambda )=\lambda '$ , we can extend this holomorphic conjugacy by the dynamics of $f_{\lambda }$ and $f_{\lambda '}$ to a holomorphic conjugacy, which we still denote by $\xi _{\lambda ', \lambda }$ , defined on the whole stable set $A_{\lambda }$ . Furthermore, by using dynamics of $f_{\lambda }$ and $f_{\lambda '}$ , we can extend this holomorphic conjugacy to to the Julia set $J_{\lambda }$ as a topological conjugacy that fixes infinity. Because ${\widehat {\mathbb C}}=A_{\lambda }\cup J_{\lambda }$ , if we denote this extension by $\xi _{\lambda , \lambda '}$ again, we have

$$ \begin{align*} \xi_{\lambda ', \lambda }\circ f_{\lambda } = f_{\lambda '}\circ \xi_{\lambda ', \lambda } \quad \text{on } {\widehat{\mathbb C}}. \end{align*} $$

From the discussion on J-stability in §5.1, we can find a holomorphic motion $H(z, c): \overline {E}_{\lambda }\times V\to {\widehat {\mathbb C}}$ such that $\xi _{\lambda , \lambda '}|\overline {E}_{\lambda } =H_{\lambda , \lambda '}=H(\cdot , \lambda ')$ is a quasiconformal homeomorphism on $\overline {E}_{\lambda }=J_{\lambda }$ . On $A_{\lambda }= {\widehat {\mathbb C}}\setminus \overline {E}_{\lambda }$ , $\xi _{\lambda , \lambda '}$ is holomorphic and injective, thus it is conformal. Now by a theorem of Rickman (see [Reference RickmanRick, Theorem 1], [Reference Douady and HubbardDH], or [Reference JiangJ, Theorem 5.1]) it follows that $\xi _{\lambda , \lambda '}$ is a global quasiconformal mapping of ${\widehat {\mathbb C}}$ . It follows from the paper of Zheng (see [Reference ZhengZJ, Theorem 3.1]), that the area of the Julia set $J_{\lambda }$ is zero and $\xi _{\lambda , \lambda '}$ is a global conformal mapping of ${\widehat {\mathbb C}}$ . Because $\xi _{\lambda ', \lambda }$ fixes zero and infinity, $\xi _{\lambda ', \lambda }(z)=az$ . Equation (5) implies that $a=1$ , $\lambda =\lambda '$ , which proves that E is injective. Note that as we saw in §2.2, there are two choices for $\lambda '$ but if we require that $\lambda '$ is the preferred asymptotic value so that $\lambda ' \in {\mathcal S}_{\lambda }$ , then $\xi _{\lambda ',\lambda }$ is the identity.

Property (iii) follows because as $\lambda $ tends to the boundary ${\mathcal S}_*$ of ${\mathcal S}_{\lambda }$ , the asymptotic value $\lambda $ tends toward the leaf of the dynamically defined level curve containing $\mu $ in the dynamic plane of $f_{\lambda }$ ; thus $E(\lambda )$ tends to a point on the corresponding level curve, $\partial \Delta \setminus \{ \lambda _0 \}$ in $K_0$ .

Note that the map $E^{-1}$ is not defined at the point $\lambda _0$ on $\partial \Delta $ because if it were, the asymptotic values of the function corresponding to the image point would be equal. Thus the point omitted by $E^{-1}$ would be a parameter singularity, and by continuity, a punctured neighborhood of it would contain points in both ${\mathcal S}_{\lambda }$ and ${\mathcal S}_{\mu }$ . There are only two parameter singularities, $0$ and $\rho /2$ ; the latter is a virtual center on the boundary of ${\mathcal M}_{\mu }$ so small neighborhoods do not contain points of ${\mathcal S}_{\lambda }$ . Therefore, the point omitted by $E^{-1}$ is $0$ . We can extend $E^{-1}$ to $\lambda _0$ by setting $E^{-1}(\lambda _0)=0$ so that $E^{-1}(\partial \Delta )$ is the closed curve ${\mathcal S}_* \cup \{0\}$ .

6.2 Construction of an inverse for E

6.2.1 Dynamic decomposition of $K_0$

To define inverse branches $R_j$ of Q on the $K_0$ , let $l^*$ be the gradient curve joining $Q(\lambda _0)$ to $\lambda _0$ in $\Delta $ and let $l \in Q^{-1}(l^*)$ be the curve joining $\lambda _0$ to infinity. Remove the line l from $K_0$ and define an inverse branch on its complement by the condition $R_0(q_0)=q_0$ . Label the other branches as $R_j(q_0)=q_0+\pi i j=q_j$ . This is equivalent to choosing a principal branch for the logarithm. Having made this choice, we can extend the $R_j$ analytically to all of $K_0$ . Denote the pre-images of $q_0$ under $Q^{-1}$ by $q_j$ , enumerated so that $q_0$ is fixed, and denote the inverse branch of Q that sends $q_0$ to $q_j$ by $R_j$ . Denote the upper and lower sides of the line l by $l^+$ and $l^-$ and let $l_j=R_j(l^-)$ , $l_{j+1}=R_j(l^+)=R_{j+1}(l^-)$ . Then $R_0$ is a homeomorphism between the open region bounded by the lines $l_0$ , $l_1$ , and l onto $K_0 \setminus l$ and $R_j$ , $j \neq 0$ is a homeomorphism from the open region between $l_j$ and $l_j+1$ onto $K_0 \setminus l$ .

If $\rho $ and $\lambda _0$ are real, this choice for the logarithm agrees with the labeling of the poles and inverse branches in §4.2 where $R_0=g_{\lambda _0,0}$ , the branch of $Q^{-1}=f_{\lambda _0}^{-1}$ that fixes the origin. This is the labeling in Figure 8. If $\rho $ and/or $\lambda _0$ is not real and a different branch of the logarithm is chosen, there could be a shift by some k in the labeling. It would be the same shift throughout the rest of the paper so would not change the essence of the argument.

Recall that $\gamma ^{*}$ is the boundary of $\Delta $ in $K_0$ and it contains $\lambda _0$ . Then, because $\lambda _0$ is an omitted value of Q, the curves $\{\gamma _{j} \}=R_j(\gamma ^*), \, j \in {\mathbb Z}$ , are a countable collection of bi-infinite disjoint curves whose infinite ends approach infinity asymptotic to the lines $l_{j}$ and $l_{j+1}$ . Thus $R_j(\Delta )$ is an unbounded domain, with boundary $\gamma _j$ , that contains $q_j$ . Note that $R_0(\Delta )$ contains the removed line l. We label the complementary components of the $\gamma _j$ as follows (see Figure 8).

• • $A_0=R_0(\Delta )$ is the component of the complement of $\gamma _0$ containing the fixed point $q_0$ and the point $\lambda _0$ .

• • $B_{j}, \, j \in {\mathbb Z} \setminus \{0 \}$ are the components of the complement of $\gamma _j$ containing the non-fixed pre-images $q_{j}$ of $q_0$ and

• • $C_0$ is the common complementary component in $K_0$ of $A_0$ and all the $B_{j}$ .

To define the second pre-images of $q_0$ and $\gamma ^*$ , we need two indices. Thus $q_{j_2 j_1}=R_{j_2}R_{j_1}(q_0)$ and $\gamma _{j_2 j_1} =R_{j_2}R_{j_1}(\gamma ^*)$ , where $j_1, j_2, \in {\mathbb Z}$ . They divide $K_0$ into domains as follows (see Figure 8).

• • Because $A_0$ is simply connected and contains one asymptotic value, $Q: A_1=Q^{-1}(A_0)\to A_0\setminus \{\lambda _0\}$ is a universal covering. Set $A_{j0}=R_j(A_0)$ ; it is bounded by $l_j$ , $l_{j+1}$ , and $\gamma _{j0}$ . Each $\{\gamma _{j0}\}$ joins the pole $R_j(\infty )$ to the pole $R_{j+1}(\infty )$ ; these two poles are different but adjacent because the infinite ends of $\gamma _0$ are on opposite sides of the line l defining the principal branch.

• • Because $B_{j_1}$ is simply connected and contains no asymptotic value for any $j_1\neq 0$ , each component of $Q^{-1}(\gamma _{j_1})$ is homeomorphic to $\gamma _{j_1}$ . The curves $\gamma _{j_2 j_1}$ bound domains containing the pre-images $q_{j_2 j_1}$ . Label these domains $B_{j_2 j_1}=R_{j_2}(B_{j_1})$ .

• • There are domains $C_{j0}=R_j(C_0)$ .

Inductively we have curves

$$ \begin{align*} \gamma_{j_n \ldots j_1} = R_{j_n}(\gamma_{j_{n-1}} \ldots j_1). \end{align*} $$

and the regions they define are as follows (see Figure 8).

• • $A_{n}=R_0(A_{n-1})$ ; it contains $q_0$ and $\lambda _0$ . It also contains all pre-images of $q_0$ up to order $n-1$ but not those of order n. It is bounded by a curve $Q^{-n}(\gamma _0)$ that is a union of open arcs with endpoints at adjacent pre-poles of order n. These are the red curves without labels closest to the vertical line in Figure 8.

• • $B_{j_n j_{n-1} \ldots j_1} =R_{j_n}(B_{j_{n-1} \ldots j_1})$ ; it contains the pre-image $q_{j_n j_{n-1} \ldots j_1}$ of $q_0$ . These are not shown in the figure. They are bounded by a single curve with a boundary point at a pre-pole of order $n-1$ .

• • $C_{j_n j_{n-1} \ldots , 0} = R_{j_n}(C_{ j_{n-1} \ldots 0} ) $ .

6.2.2 Inductive construction of the pair $(U_{\infty }, Q_{\infty })$

See Figure 9.

Figure 9 The point p is in $A_{00}$ and the set U is the region to the right of the dotted curves.

• • Pick $p \in K_0 \setminus \Delta $ . In Figure 9, p is in $A_{00}$ . Following the outline above in part (3), we construct a map with the asymptotic values p and $\lambda _0$ as follows. Let $\widehat {p}_j=R_j(p)$ ; in the figure, the $\widehat {p}_j$ are in $A_{j00}$ . Let N be the smallest integer such that p is in $A_{N} \cup B_{j_N, j_{N-1} \ldots , j_1}$ . The boundaries of the sets in this union are the level sets $\phi _{0}^{-1}(\rho ^{-N} re^{i\theta })$ where, as above, $r=|\phi _0(\lambda _0)|$ . For every small $\epsilon>0$ , one component of the level set $\phi _0^{-1}( (\rho ^{-N} r +\epsilon )e^{i\theta })$ is an analytic curve, except at the pre-poles of order $N-1$ . It bounds a simply connected domain containing $A_N$ ; it thus contains the points $p, q_0, \lambda _0$ and the curves $\gamma _{j_N, \ldots j_1}$ , but none of pre-images $\widehat {p}_j$ of p. Fix $\epsilon $ and denote the resulting domain by U. Its boundary is denoted by the dotted black curve in Figure 9. Because it is contained in the attracting basin of $q_0$ , $Q(U) \subset U$ . Moreover, because U does not contain any of the points $\widehat {p}_j$ , $p \not \in Q(U)$ . Set $\widetilde {U}= U \setminus \{\lambda _0 , p\}$ .

• • Lemma 6.1 implies there is a holomorphic unramified covering map $\Pi _1: U_1 \rightarrow \widetilde {U}$ , where $U_1$ is a Riemann surface that is topologically a disk minus a countable set of topological disks.

• • Note that $Q : U \rightarrow Q(U)$ is a holomorphic universal covering map with omitted value $\lambda _0$ . Set $U_1'=\Pi _1^{-1}(Q(U))$ ; because $\lambda _0\in Q(U)$ and $p\notin Q(U)$ , it is a topological disk so that $\Pi _1 : U_1' \rightarrow Q(U)$ is also a holomorphic unramified covering map that omits the value $\lambda _0$ .

  

• • Because both $\Pi _1$ and Q are regular coverings whose domains are simply connected, we can lift them to obtain a conformal map $i_1: U\to U_1'$ such that $\Pi _1 \circ i_1=Q.$ The choice of inverse branch will affect $i_1$ but the argument works for any choice. We now define $Q_1: U_1 \rightarrow U_1'$ by $Q_1= i_1\circ \Pi _1$ . It is an unramified regular infinite to one holomorphic endomorphism and omits the values $i_1(\lambda _0)$ and $i_1(p)$ . Moreover,

  $$ \begin{align*} Q_1\circ i_1=i_1\circ \Pi_1\circ i_1=i_1\circ Q; \end{align*} $$
  that is, $i_1$ conjugates Q and $Q_1$ , therefore $Q_1$ fixes $i_1(q_0)$ .

  We may, without loss of generality, assume $i_1(\lambda _0)=\lambda _0$ , $i_1(p)=p$ and ${i_1(q_0)=q_0}$ .

• • Now set $Q_{0}=Q$ , $U_{0}=U$ , and $U_{0}'=Q(U)$ . We proceed by induction: we assume that for $1 \leq j \leq n-1$ , we have the following.

  1. (1) Domains $U_j$ , homeomorphic to an open disk from which infinitely many open disks have been removed, and infinite to one unramified covering maps $\Pi _j: U_j \rightarrow U_{j-1}$ with two asymptotic values.

  2. (2) Holomorphic endomorphisms, $Q_j: U_j \rightarrow U^{\prime }_j\subset U_j$ , that are infinite to one, unramified, have one fixed point and two asymptotic values.

  3. (3) Conformal maps $i_{j}: U_{j-1} \to U^{\prime }_{j}$ satisfying

     $$ \begin{align*} Q_j\circ i_j=i_j\circ Q_{j-1}. \end{align*} $$

  For the inductive step, we use Remark 6.2 to obtain the holomorphic unramified covering map $\Pi _n: U_{n} \rightarrow U_{n-1}$ , where $U_{n}$ is homeomorphic to $U \setminus \{U_{j_{n-1} \ldots j_1 j}, \, \, ( j_{n-1}, \ldots , j_1,j) \in {\mathbb Z}^{n} \}$ . As in the first step, we set $U_{n}'=\Pi _{n}^{-1}(Q_{n-1}(U_{n-1}))$ . Both

  $$ \begin{align*} Q_{n-1}: U_{n}' \rightarrow Q_{n-1}(U_{n-1}) \quad\text{and}\quad \Pi_{n}:U_n \rightarrow Q_{n-1}(U_{n-1}) \end{align*} $$
  are unramified coverings with asymptotic values $\lambda _0$ and p. Lemma 6.1 implies there are infinitely many choices for a holomorphic isomorphism
  $$ \begin{align*} i_{n}: U_{n-1} \rightarrow U_{n}'\quad \text{satisfying}\quad \Pi_{n} \circ i_{n}= Q_{n-1} \text{ on } U_{n-1}. \end{align*} $$
  Making one such choice (the choice does not matter), we define $Q_{n} : U_{n} \rightarrow U_{n}$ by $ Q_{n}=i_{n} \circ \Pi _{n}$ so that
  $$ \begin{align*} Q_n\circ i_n= i_n \circ \Pi_n\circ i_n=i_n\circ Q_{n-1}. \end{align*} $$
  Therefore, $i_{n}$ conjugates $Q_n$ to $Q_{n-1}$ and the induction hypotheses are satisfied, completing the inductive step.

• • The direct limit $U_{\infty }$ of the system $(U_n, i_n)$ is the quotient

  $$ \begin{align*} \bigcup_n U_n / \sim, \end{align*} $$
  where the equivalence relation is defined by the identifications, $ z \sim i_n(z)$ , and the equivalence class is denoted by $[z]$ . The Riemann surface $U_{\infty }$ has infinite type. There is an infinite unramified holomorphic covering map $Q_{\infty }$ defined by
  $$ \begin{align*} Q_{\infty}([z])=[Q_n(z)], \quad z \in U_n \end{align*} $$
  that has two omitted values $[\lambda _0]$ and $[p]$ . It also fixes $[q_0]$ and because the maps $\Pi _n$ and $i_n$ are holomorphic, we have $Q_{\infty }'([q_0])=\rho $ .

  Topologically, $U_{\infty }$ is the complement in ${\mathbb C}$ of a Cantor set C isomorphic to the space of infinite sequences $\Sigma _{\infty } = s_1, \ldots s_{n-1}, s_n \ldots , s_j \in {\mathbb Z}$ together with the finite sequences $\Sigma _{n+1}=s_1, \ldots , s_n, \infty $ of length $n+1$ . The map $Q_{\infty }$ is conjugate to the shift map on C. See [Reference MoserMo].

The final step of the proof is to show there is a conformal embedding $e: \, U_{\infty } \rightarrow {\mathbb C}$ such that

$$ \begin{align*} e \circ Q_{\infty} = f_{\lambda } \circ e \end{align*} $$

for some $\lambda \in {\mathcal S}_{\lambda }^0$ with $\xi _{\lambda }(\lambda )=\lambda _0$ . To do this, we first give a brief informal review of the results we need from Teichmüller theory and the theory of mapping classes of tori and punctured tori. We refer the reader to [Reference BirmanBir, Reference MasseyMa] for a full discussion and to [Reference Goldberg and KeenGK] for a discussion analogous to what we need here.