# Stable components in the parameter plane of meromorphic functions of finite
type

Núria Fagella
Dept. de Matemàtiques i Informàtica
Univ. de Barcelona, Gran Via 585
08007 Barcelona, Spain
<EMAIL_ADDRESS>Partially supported by spanish grants
MTM2014-52209-C2-2-P, MTM2017-86795-C3-3-P and MDM-2014-0445 María de Maeztu.
Linda Keen
CUNY Graduate Center
365 Fifth Avenue, New York
NY, NY 10016
<EMAIL_ADDRESS><EMAIL_ADDRESS>Partially supported by PSC-CUNY
grants 67260-00 45, 80209-05 20

###### Abstract

We study the parameter planes of certain one-dimensional, dynamically-defined
slices of holomorphic families of meromorphic transcendental maps of finite
type for which infinity is not an asymptotic value. Our planes are defined by
constraining the orbits of all but one of the asymptotic values. We study the
structure of the regions in the parameter plane where the dynamics have a free
asymptotic value that is attracted to an attracting periodic orbit. The
tangent family is an example that has been studied in detail, [KK97], and our
goal is to understand in the general context of meromorphic functions to what
extent the structure one sees in parameter plane of the tangent family is
generic.

For the analogous dynamically defined slices of parameter spaces for rational
maps, there are Mandelbrot-like components that have a unique center that can
be used to give a combinatorial descripton of the components. No such center
exists for our slices, no matter how carefully they are chosen. We can show,
however, that there is an analogous concept, a “virtual” center, that plays
essentially the same role as a center would have. It lies on the boundary of
the component and corresponds to a map for which there is a ”virtual cycle”,
namely, an iterate of the free asymptotic value that lands at a pole. These
virtual centers are dense in the bifurcation locus of our one-dimensional
families.

##  1 Introduction

Our starting point is a family
$\\{f_{\mathbf{\lambda}}:{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}\\}$ of
transcendental meromorphic functions that depends holomorphically on a
parameter ${\mathbf{\lambda}}=(\lambda_{1},\lambda_{2},\ldots)$. Fix a value
for $\lambda$. Iterating the corresponding $f_{\lambda}$ gives rise to a
dynamical system. A fundamental problem is to understand what bifurcations
occur in the dynamics of these systems as one varies $\lambda$. In this
generality, the problem does not have a satisfying solution. Because the
dynamics of these systems are controlled by the orbits of the singular values
of the $f_{\lambda}$s, one effective way to approach the problem is to define
one-dimensional slices through the $\lambda$ space that constrain all but one
of these singular values and analyze the structure within these slices.

For simple holomorphic families like quadratics and exponentials, this has
been carried out more or less completely. (See for example, [BR75, DFJ02,
KK97, DB00, RG03, Sch03], as well as references quoted there). For rational
maps, it has also been carried out and shows that behavior for quadratics is
not special, but is a paradigm for rational dynamics as well.

The tangent family was the first example of transcendental meromorphic
functions to be studied this way. The essential new feature for transcendental
functions is that they have asymptotic values, which are, in effect, images of
infinity along certain paths. This introduces new phenomena into the structure
of the parameter spaces that are not seen for rational maps. In the same way
that the Mandelbrot-like sets code the influence of the critical values of the
dynamics of rational maps, these new phenomena code the influence of the
asymptotic values on the dynamics of our families of transcendental maps.

The essential characteristic of the tangent that we want to carry forward is
that it has finitely many singular values and no infinite asymptotic value.
The goal is to show for holomorphic families of such functions, there are
natural one-dimensional slices in the parameter space for which the tangent
family’s behavior is a paradigm.

We get down to one-dimension by allowing only one “free” asymptotic value to
vary. Our main results concern properties of the parameter regions for which
this free asymptotic value is attracted to an attracting cycle. These regions,
which we call “shell components”, are the analogues in this setting of the
hyperbolic components of the interior of the Mandelbrot-like sets for rational
maps, and like the Mandelbrot-like sets, they are ubiquitous in the parameter
spaces of these families.

To make these ideas precise, consider the dynamical system formed by iterating
a meromorphic transcendental map of the complex plane

$f:{\mathbb{C}}\to{\mathbb{C}}\cup\\{\infty\\}.$

Because of the essential singularity at infinity, the global degree of $f$ is
infinite and the dynamics is richer than for rational maps. The tangent map is
in this class. Other examples are generated by applying Newton’s method to an
entire function, which almost always yields a transcendental meromorphic map
as the root finder function to be iterated.

The dynamical plane of $f$ splits into two disjoint completely invariant
subsets: the Fatou set, or points for which the family of iterates is well
defined and form a normal family in some neighborhood; and its complement, the
Julia set. The Fatou set is open and contains, among other types of
components, all basins of attraction of attracting periodic orbits. By
contrast, the Julia set is closed and it is either the whole sphere or has no
interior. It can be characterized as the closure of the repelling periodic
points or, if the map is meromorphic, as the closure of the set of poles and
prepoles. For general background on meromorphic dynamics we refer to the
survey [Ber93] and the references there.

A key role in the dynamics of a meromorphic map $f$ is played by points in the
set $S(f)$ of singular values of $f$; these are either critical values (images
of zeroes of $f^{\prime}$, the critical points) or asymptotic values (points
$v=\lim_{t\to\infty}f(\gamma(t))$ where $\gamma(t)\to\infty$ as $t\to\infty$).
Indeed every periodic connected component of the Fatou set is, in some sense,
associated to the orbit of a point in the closure of $S(f)$. For example, any
basin of attraction must contain a singular value in its interior. Note that
$\overline{S(f)}$ coincides with the set of singularities of the inverse map.

Transcendental meromorphic functions with a finite number of singular values
have finite dimensional parameter spaces [EL92, GK86]. These are known as
finite type maps and denoted by

${\cal S}:=\\{f:{\mathbb{C}}\to{\mathbb{C}}\cup\\{\infty\\}{\rm\
meromorphic}\mid\\#{S(f)}<\infty\\}.$

Maps in the class ${\cal S}$ have several special properties such as the
absence of wandering domains [Sul85, GK86, EL92, BKY92] or the existence of at
most a finite number of attracting cycles.

Entire transcendental functions, whose dynamics have attracted a great deal of
attention in the last few years, are special cases of transcendental
meromorphic maps because they have no poles. The simplest example is the
exponential family $E_{\lambda}(z)=e^{z}+\lambda$. By Iversen’s theorem
[Ive14] entire functions (and meromorphic functions with finitely many poles),
all have infinity as an asymptotic value. By contrast, meromorphic maps for
which infinity is not an asymptotic value are at the other end of the
spectrum: they necessarily have infinitely many poles and this difference in
behavior at infinity has consequences for the dynamical and parameter spaces
(see e.g. [BK12]). The tangent family of maps $T_{\lambda}(z)=\lambda\tan(z)$
[DK88, KK97, GK08] is the simplest example of these. This class, which we
denote as

${\cal M}_{\infty}:=\\{f\in{\cal S}\mid\text{$\infty$ is not an asymptotic
value}\\},$

is in some sense more general than entire or meromorphic maps with a finite
number of poles since the condition of infinity not being an asymptotic value
is open. In this paper we will give special emphasis to functions in ${\cal
M}_{\infty}$. Inspired by the detailed description of the parameter space of
the tangent family [KK97], our goal is to initiate a similar study for
reasonably general one-parameter slices in ${\cal M}_{\infty}$ rather than for
particular sets of maps and to prove that, just as the Mandelbrot set is a
paradigm for one dimensional slices of rational maps, the structures in the
parameter space of the tangent family are a paradigm for one dimensional
slices of ${\cal M}_{\infty}$.

To center the discussion, we consider dynamically natural slices of these
finite type maps. Roughly speaking, we define a dynamically natural slice as a
one-parameter holomorphic family of meromorphic maps for which only one
singular value is allowed to bifurcate at any given parameter value (see
Definition 4.5 for details). We concentrate our attention on the behavior of a
marked asymptotic value called the free asymptotic value, and in particular,
on the connected components of parameters for which this marked point is
attracted to an attracting cycle whose basin contains no other singular value.
We call these components shell components and they are the transcendental
version of the hyperbolic components one finds in parameter planes of rational
maps.

A dynamically natural slice is represented by a one-dimensional manifold
isomorphic to the complex plane with perhaps a discrete set of punctures
called parameter singularities, where the function may for example reduce to a
constant. For normalization purposes, we require the parametrization to trace
the position of the marked asymptotic value in an affine fashion. Note that
because some normalization is necessary if one wishes to work in parameter
space, this is not a real restriction. Most of the best known one-parameter
slices such as quadratic polynomials, cubic polynomials with a super
attracting fixed point, quadratic rational maps with a super attracting cycle,
exponential or tangent functions, and so on, are dynamically natural slices of
maps in different classes (see Section 7).

If $\Lambda$ is a dynamically natural slice for the family ${\cal
F}=\\{f_{\lambda}\\}_{\lambda\in\Lambda}$ of meromorphic maps, it contains
different types of “distinguished” parameter values which are solutions of
algebraic or transcendental equations. Examples are: Misiurewicz parameters,
where a singular value lands on a repelling cycle; or centers, where a
critical point is periodic. The powerful tool of “bifurcation currents” has
been used successfully in recent times to study the existence and distribution
of these distinguished parameters in spaces of rational maps (see for example
[Ber13, DF08, Duj14, Gau12]). This tool is not available for meromorphic maps.
Meromorphic transcendental maps, however, may have a type of distinguished
parameter which does not occur for rational maps. This is a parameter that we
call a virtual cycle parameter; it is defined by the property that some
iterate of the asymptotic value is a pole. This name is motivated by the
existence of a virtual cycle, morally a “cycle” that contains the point at
infinity and the asymptotic value (see Definition 6.7). Virtual cycle
parameters are important in our discussion and they are generally quite
abundant in the following sense (see Proposition 5.6).

######  Proposition A.

Suppose $\Lambda$ is a dynamically natural slice for the family ${\cal
F}=\\{f_{\lambda}\\}_{\lambda\in\Lambda}$ of maps in ${\cal M}_{\infty}$. Let
$v=v(\lambda)$ denote the free asymptotic value. Then, Misiurewicz parameters
and virtual cycle parameters are dense in the set

${\cal B}(v):=\\{\lambda_{0}\in\Lambda\mid\\{\lambda\mapsto
f_{\lambda}^{n}(v_{\lambda})\\}\text{ is not a normal family in any
neighborhood of $\lambda_{0}$}\\}.$

As a consequence, ${\cal B}(v)$ has no isolated points in $\Lambda$.

The set ${\cal B}(v)$ is known as the activity locus of the free asymptotic
value and is part of the bifurcation locus ${\cal B}$, or the set of
parameters for which at least one of the singular values bifurcates. It
follows from our definition of a dynamically natural slice, that the boundary
of any shell component is a subset of ${\cal B}(v)$.

By definition, functions in a shell component have no critical points in the
basin of the attracting cycle, so the multiplier of the cycle is never zero,
and therefore the component has no center. We see however, that virtual cycle
parameters play the role of centers in our setting (see Theorem 6.12).

######  Definition 1.1 (Virtual center).

Let $\Omega$ be a shell component. We say that
$\lambda^{*}\in\partial\Omega\cap\Lambda$ is a virtual center of $\Omega$, if
for every sequence of $\lambda_{n}\in\Omega$ such that
$\lambda_{n}\to\lambda^{*}$, the multiplier $\rho_{\Omega}(\lambda_{n})$ tends
to $0$.

######  Theorem B.

Suppose $\Lambda$ is a dynamically natural slice for the family ${\cal
F}=\\{f_{\lambda}\\}_{\lambda\in\Lambda}$ of maps in ${\cal M}_{\infty}$ and
let $\Omega$ be a shell component. Then
$\lambda^{*}\in\partial\Omega\cap\Lambda$ is a virtual center if and only if
$\lambda^{*}$ is a virtual cycle parameter.

Although maps in a shell component might be non-hyperbolic, a shell component
$\Omega$ shares many properties with hyperbolic components of known families
like the exponential or tangent families, [RG03, KK97]. Indeed, shell
components can be parametrized by the multiplier map

$\rho=\rho_{\Omega}:\Omega\longrightarrow{\mathbb{D}}^{*}$

where ${\mathbb{D}}^{*}:={\mathbb{D}}\setminus\\{0\\}$, and $\rho(\lambda)$ is
the the multiplier of the attracting cycle to which $v(\lambda)$ is attracted.
Since the multiplier is never zero, $\rho$ is a covering (see Theorem 6.4) and
hence shell components are either simply connected and $\rho$ has infinite
degree or else they are homeomorphic to ${\mathbb{D}}^{*}$, where the puncture
is a parameter singularity (see Corollary 6.5).

Our last result describes the internal structure of a shell component and
establishes the existence and uniqueness of the virtual center (see Theorem
6.16, and corollaries 6.17, 6.18 and 6.20).

######  Theorem C.

Suppose $\Lambda$ is a dynamically natural slice for the family ${\cal
F}=\\{f_{\lambda}\\}_{\lambda\in\Lambda}$ of maps in ${\cal M}_{\infty}$ and
let $\Omega$ be a simply connected shell component. Let $k\geq 1$ be the
period of the attracting cycle throughout $\Omega$. Then:

  * (a)

$\partial\Omega$ is piecewise analytic and locally connected.

  * (b)

$\partial\Omega$ has a unique virtual center.

  * (c)

If $k=1$, the virtual center is at infinity and therefore $\Omega$ is
unbounded.

  * (d)

Define the internal ray in $\Omega$ of angle $\theta$ as

$R_{\Omega}(\theta):=\\{\rho^{-1}_{\Omega}(\exp(t+2\pi
i\theta)),\,t\in(-\infty,0)\\}.$

Then, all internal rays have one end at the virtual center and the other end
at a point in $\partial\Omega$ (which a priori could be infinity). If the
virtual center $\lambda^{*}$ is finite no internal ray has both ends at
$\lambda^{*}$.

When the shell component is not simply connected then the puncture, which is a
parameter singularity, plays the role of the virtual center.

Figure 1: Sketch of two shell components and their internal rays. On the left
(resp. right) the virtual center is at infinity (resp. at a finite point) and
therefore the component is unbounded (resp. bounded).

Numerical experiments show that shell components of period larger than one are
always bounded, that is, they have their virtual center at a finite parameter
value, so that it seems reasonable to conjecture that this is so in general.
At present, we can only prove this fact for particular families like the
tangent family.

The paper is structured as follows. In Section 2 we recall the definition of
singular values of transcendental functions and we state some standard
theorems on the covering and connectivity properties of their Fatou
components. In Section 3 we discuss the properties of the dynamical planes of
these functions. In Section 4 we define the dynamically natural slices of
parameter spaces that are the main subject of the paper and in Section 7 we
give a number of examples. In Section 5 we define various types of
distinguished parameter values that lie in the bifurcation locus and prove
Proposition A. Section 6 is the heart of the paper. We define shell components
and prove the multiplier map is a holomorphic covering map. We prove some
important technical propositions and give the proofs of Theorems B and C. We
conclude with an Appendix where we discuss a theorem of Nevanlinna that allows
us to characterize certain functions of finite type. We also extend this
theorem to a somewhat larger class.

##  2 Preliminaries and Setup

###  2.1 Singularities of the inverse function

Let $f$ be a transcendental entire or meromorphic map. A point
$v\in{\mathbb{C}}$ is a singular value of $f$ if some branch of $f^{-1}$ fails
to be well defined in every small enough neighborhood of $v$. Singular values
may be critical, asymptotic or accumulations thereof. If $c$ is a critical
point, that is, a zero of $f^{\prime}$ then its image $v=f(c)$ is a critical
value. If there is a path $\gamma(t)$ such that $\lim_{t\to
1}\gamma(t)=\infty$ and $\lim_{t\to 1}f(\gamma(t))=v$ then the limit $v$ is an
asymptotic value of $f$.

Singular values are classified in terms of the branches of $f^{-1}$ as follows
(see [BE95],[Ive06]):

######  Proposition 2.1 (Classification of singularities).

Let $f$ be an entire or meromorphic transcendental map. For any
$z\in{\mathbb{C}}$ and $r>0$, let $D(z,r)$ be the disk of radius $r$ centered
at $z$. Let $U_{r}$ be a connected component of $f^{-1}(D(z,r))$, chosen such
that $U_{r}\subset U_{r^{\prime}}$ if $r<r^{\prime}$. Then there are only two
possibities:

  * (a)

$\bigcap_{r}U_{r}=\\{p\\},p\in{\mathbb{C}}$

  * (b)

$\bigcap_{r}U_{r}=\emptyset$.

In case $(a)$, $f(p)=z$, and either $f^{\prime}(p)\neq 0$ and $z$ is a regular
point, or $f^{\prime}(p)=0$ and $z$ is a critical value. In case $(b)$, the
chosen inverse branch with image $U_{r}$ defines a transcendental singularity
over $z$, and it can be shown that $z$ is an asymptotic value for $f$.

The map $f$ is said to be of finite type if it has a finite number of singular
values. Note that this implies all the singular values are isolated.

If an asymptotic value is isolated, the radius $r$ in the above proposition
can be chosen small enough so that $f:U_{r}\rightarrow D(v,r)\setminus\\{v\\}$
is universal covering map. In this case $U_{r}$ is called an asymptotic tract
for the asymptotic value $v$ and $v$ is called a logarithmic singular value.
The number of distinct asymptotic tracts of a given asymptotic value is called
its multiplicity.

By Iversen’s Theorem [Ive14] an entire transcendental function $f$ always has
an asymptotic value at infinity.

###  2.2 Mapping properties of meromorphic maps

In this paper we deal with the class ${\cal S}$ of transcendental functions of
finite type. Notice that because ${\cal S}$ is not closed under composition,
it is often useful to work with a larger class of maps which are meromorphic
outside of a small set. More precisely, set

$\widehat{\cal S}=\\{f:{\mathbb{C}}\setminus
E(f)\to{\widehat{\mathbb{C}}}\mid\text{$f$ is nonconstant
meromorphic}\\}\supset{\cal S}$

where $E(f)\subseteq{\widehat{\mathbb{C}}}$ a closed countable set of
essential singularities.

Functions in the class $\widehat{\cal S}$ were originally studied by M. E.
Herring [Her98] and A. Bolsch [Bol99] who extended many properties known for
rational, entire or meromorphic maps to maps of this class. The following
statements are relevant for our setting. Given $U\in{\widehat{\mathbb{C}}}$ we
denote by $\chi(U)$ its Euler characteristic (i.e.
$\chi({\widehat{\mathbb{C}}})=-2$ and $\chi(U)=c(U)-2$ when $U$ is of
connectivity $c(U)\in{\mathbb{N}}\cup\\{\infty\\}$).

######  Theorem 2.2 ([Hei57, Thm.4], [Bol99, Thm.1,3]).

Let $f\in\widehat{\cal S}$. Let $V\subset{\widehat{\mathbb{C}}}$ be an
arbitrary domain and let $U$ be a connected component of $f^{-1}(V)$
containing $k$ critical points counted with multiplicity. Then either

  1. (a)

there exists an integer $n\in{\mathbb{N}}$, with $k\leq 2n-2$, such that every
value of $V$ is assumed exactly $n$ times in $U$ and

$\chi(U)=n\chi(V)+k;$

or

  2. (b)

with at most two exceptions, every value of $V$ is assumed infinitely often in
$U$. In this case, if $\chi(V)>0$ then $\chi(U)=+\infty$

Observe that the first case occurs if and only if $f|_{U}:U\to V$ is a proper
map.

The following characterization was given in [BKY91b] for invariant Fatou
components of maps in ${\cal S}$. The same arguments applied to maps in
$\widehat{\cal S}$ extend it to Fatou components of any period.

######  Theorem 2.3 ([Bol99, Cor.2 and Remark following it]).

If $f\in\widehat{\cal S}$, a periodic Fatou component of arbitrary period has
connectivity 1,2 or infinity. If it is doubly connected then it is a Herman
ring or it is a doubly connected invariant component of infinite degree. In
this case $f$ must have infinitely many singular points and so is not in the
class ${\cal S}$.

The following classical and well known theorem on the classification of
holomorphic coverings of the punctured disk ${\mathbb{D}}^{*}=\\{z\mid
0<|z|<1\\}$ will be useful throughout the paper. The proof can be found, for
example, in [Zhe10, Thm. 6.1.1.].

######  Theorem 2.4.

Let $X$ be a Riemann surface and let $h:X\to{\mathbb{D}}^{*}$ be an unbranched
holomorphic covering map. Then one of the following statements holds.

  1. (a)

There exists a conformal mapping $\psi$ of $X$ onto the left half plane
${\mathbb{H}}_{l}=\\{z:{\rm Re}z<0\\}$ such that $f=\exp\circ\psi$.

  2. (b)

There exists a conformal mapping $\psi$ of $X$ onto ${\mathbb{D}}^{*}$ such
that for some $n\in{\mathbb{N}}$, $f=(\psi)^{n}$.

##  3 Dynamical plane

In this section we describe some properties of the dynamical plane for either
entire or meromorphic finite type maps. In particular, we are interested in
basins of attraction of attracting cycles containing a unique asymptotic value
$v$. Note that because there are only finitely many singularities, $v$ is
isolated and is thus a logarithmic singularity (see Section 2.1). It follows
that if $V^{*}$ is a punctured neighborhood of $v$ it has at least one
asymptotic tract among the components of $f^{-1}(V^{*})$.

We start with a simple observation about the connectivity of attracting
basins.

######  Lemma 3.1.

Let $f\in{\cal S}$ and let ${\cal A}$ denote the immediate basin of attraction
of an attracting or parabolic cycle. Then either every component of ${\cal A}$
is simply connected or they are all infinitely connected.

###### Proof.

Theorem 2.3 establishes that the connectivity of each component is $1,2$ or
$\infty$. Since $f$ is of finite type, and since the cycle is attracting or
parabolic, and therefore not a cycle of Herman rings, it follows that the
connectivity is either $1$ or $\infty$. Now by Theorem 2.2, the connectivity
of a component cannot be strictly larger than that of its preimage. Therefore,
since the components form a cycle, either they are all simply connected or
they are all infinitely connected. ∎

The next two lemmas describe dynamical properties of the subclass of
meromorphic functions of finite type for which infinity is not an asymptotic
value, namely ${\cal M}_{\infty}$. Recall from the general theory that the
immediate attracting basin of a periodic cycle must contain a singular value.
In the next lemma we describe properties of this basin when the singular value
is an asymptotic value.

######  Lemma 3.2.

Let $f\in{\cal M}_{\infty}$ and $p\geq 2$. Suppose $A_{0},\ldots,A_{p-1}$ are
the components of the immediate basin of attraction ${\cal A}$ of an
attracting $p-$cycle, indexed so that $A_{i+1}=f(A_{i})$ for all $i$ (with
indices taken mod $p$). Assume also that the basin contains only one singular
value $v$, which is an asymptotic value and that $v$ belongs, say, to $A_{1}$.
Then,

  1. (a)

$A_{0}$ is unbounded and maps infinite to one onto $A_{1}\setminus\\{v\\}$.
Moreover infinity is accessible from $A_{0}$ and $\partial A_{p-1}$ contains a
pole.

  2. (b)

If the components of ${\cal A}$ are simply connected then $f:A_{j}\to A_{j+1}$
is one to one for all $j\neq 0\,{\rm mod}\,p$.

###### Proof.

Since the asymptotic value $v\in A_{1}$ is the only singular value in the
basin, it follows that $A_{0}$ must contain an asymptotic tract; this implies
$f:A_{0}\setminus{f^{-1}(v)}\to A_{1}\setminus\\{v\\}$ is an infinite degree
unbranched covering. Hence $A_{0}$ is unbounded and in fact, by Theorem 2.4,
$f^{-1}(v)\cap A_{0}=\emptyset$. If $\delta:[0,1)\to A_{1}\setminus{v}$ is a
path such that $\lim_{t\to 1}\delta(t)=v$, then any path $\gamma$ in $A_{0}$
such that $\delta=f\circ\gamma$ is an asymptotic path; that is, $\lim_{t\to
1}\gamma(t)=\infty$, which proves $\infty$ is accessible from $A_{0}$.

Let $\sigma:[0,1)\to A_{p-1}$ be any path satisfying $\gamma=f\circ\sigma$. By
continuity, any accumulation point of $\sigma(t)$ when $t\to 1$ must be
infinity or a pole. Since the set of poles is discrete, it follows that
$\sigma$ must actually land at infinity or at a pole of $f$. It cannot land on
infinity because then infinity would be an asymptotic value. Therefore
$\sigma$ lands at a pole which must then be in $\partial A_{p-1}$. This proves
(a).

To see (b) assume that the basin is simply connected. From the Riemann-Hurwitz
formula (see Theorem 2.2 (a)) we conclude that for all $i=1,\cdots,p-1$, the
restrictions $f|_{A_{i}}$ must be univalent. ∎

The next lemma characterizes the mapping properties near infinity of functions
in ${\cal M}_{\infty}$. Its proof is contained in [RS99, Lemma 2.1], but we
sketch it here for completeness.

######  Lemma 3.3.

Let $f\in{\cal M}_{\infty}$. Then there exists a disk $D$ containing all the
singular values of $f$ and an infinite sequence of bounded disjoint
topological disks $\widehat{B}_{i}$, $i\in{\mathbb{Z}}$, such that if
$U={\widehat{\mathbb{C}}}\setminus D$ then

$f^{-1}(U)=\cup_{i}\widehat{B}_{i}.$

Each $\widehat{B}_{i}$ contains a pole $p_{i}$ and the restriction
$f|_{\widehat{B_{i}}}:\widehat{B_{i}}\to U$ is a covering branched over
$p_{i}$, whose order is the same as the order of the pole.

###### Proof.

Since infinity is not an asymptotic value, all the singular values are finite
and we can find a closed disk $D$ containing all of them. Set
$U_{\infty}=U\setminus\\{\infty\\}\sim{\mathbb{D}}^{*}$. Then if $B_{i}$ is a
component of $f^{-1}(U_{\infty})$, the restriction $f_{i}$ of $f$ to $B_{i}$
is a covering map

$f_{i}:B_{i}\mapsto U_{\infty}.$

Because infinity is not an asymptotic value, by Proposition 2.1 we deduce that
each $B_{i}$ is homeomorphic to a punctured disk, and, enlarging $D$ if
necessary, that these disks are all disjoint. The restricted maps $f_{i}$
extend to the puncture by continuity and send it to infinity; thus the
puncture is a pole $p_{i}$. Let $\widehat{B}_{i}=B_{i}\cup\\{p_{i}\\}$. The
extended maps are conjugate to a map of the form $\zeta\mapsto\zeta^{k}$ where
$k$ is the order of the pole. Thus if $p_{i}$ is a simple pole, there is a
single valued branch of the inverse, $g_{i}=f^{-1}_{i}$ that maps $U$ onto
$\widehat{B}_{i}$. If $k>1$ then there are $k$ such inverse branches. The
disks $\widehat{B}_{i}$ accumulate at infinity. ∎

##  4 Dynamically natural slices of parameter space

Every transcendental function $f$ with $N<\infty$ singular values belongs to
an $N+2$ dimensional complex analytic manifold consisting ofall transcendental
maps $g$ that are equivalent to $f$ in the following sense: there exist
quasiconformal homeomorphisms $\phi$ and $\varphi$ such that $g=\phi\circ
f\circ\varphi$, [BKY92, EL92, GK86]. This manifold includes the quasiconformal
deformation space of $f$. If the maps $\phi$ and $\varphi$ are affine, $f$ and
$g$ are affine conjugate and have the same dynamics. Therefore, modding out by
affine conjugation, we obtain a manifold of dimension $N$; it is known as the
natural parameter space of $f$. To define parameters for this $N$ dimensional
space, we choose a convenient normalization. Loosely speaking, each singular
value controls a different dynamical phenomenon of the dynamics of the
function. Below we consider one-dimensional slices characterized by dynamical
properties.

The theory of holomorphic motions is a good tool for studying these parameter
spaces. For rational maps, the theory is explained in [McM94] and it is
adapted to meromophic functions in [KK97]. We state the results from the
latter reference that we use here.

######  Definition 4.1 (Holomorphic family).

A holomorphic family of meromorphic maps over a complex manifold $X$ is a map
${\cal F}:X\times{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}$, such that
${\cal F}(x,z)=:f_{x}(z)$ is meromorphic for all $x\in X$ and $x\mapsto
f_{x}(z)$ is holomorphic for all $z\in{\mathbb{C}}$.

######  Definition 4.2 (Holomorphic motion).

A holomorphic motion of a set $V\subset\widehat{\mathbb{C}}$ over a connected
complex manifold with basepoint $(X,x_{0})$ is a map $\phi:X\times
V\rightarrow\widehat{\mathbb{C}}$ given by $(x,v)\mapsto\phi_{x}(v)$ such that

  1. 1.

for each $v\in V$ , $\phi_{x}(v)$ is holomorphic in $x$,

  2. 2.

for each $x\in X$, $\phi_{x}(v)$ is an injective function of $v\in V$, and,

  3. 3.

at $x_{0}$, $\phi_{x_{0}}\equiv{\rm Id}$.

A holomorphic motion of a set $V$ respects the dynamics of the holomorphic
family ${\cal F}$ if $\phi_{x}(f_{x_{0}}(v))=f_{x}(\phi_{x}(v))$ whenever both
$v$ and $f_{x_{0}}(v)$ belong to $V$ .

The following equivalencies are proved for rational maps in [McM94] and
extended to the transcendental setting in [KK97].

######  Theorem 4.3.

Let ${\cal F}$ be a holomorphic family of meromorphic maps with finitely many
singularities, over a complex manifold $X$, with base point $x_{0}$. Then the
following are equivalent.

  1. (a)

The number of attracting cycles of $f_{x}$ is locally constant in a
neighborhood of $x_{0}$.

  2. (b)

There is a holomorphic motion of the Julia set of $f_{x_{0}}$ over a
neighborhood of $x_{0}$ which respects the dynamics of ${\cal F}$.

  3. (c)

If in addition, for $i=1,\ldots,N$, $s_{i}(x)$ are holomorphic maps
parameterizing the singular values of $f_{x}$, then the functions $x\mapsto
f_{x}^{n}(s_{i}(x))$ form a normal family on a neighborhood of $x_{0}$.

######  Definition 4.4 ($J-$stability).

A parameter $x_{0}\in X$ is a $J$-stable parameter for the family ${\cal F}$
if it satisfies any of the above conditions. We denote by $X^{stab}$, the set
of $J$-stable parameters for the family ${\cal F}$.

The set of non $J$-stable parameters is precisely the set where bifurcations
occur, and it is often called the bifurcation locus of the family ${\cal F}$,
and denoted by ${\cal B}_{X}$. In families of maps with more than one singular
value, however, it makes sense to consider subsets of the bifurcation locus
where only some of the singular values are bifurcating, in the sense that the
families $\\{g_{n}^{i}(x):=f_{x}^{n}(s_{i}(x))\\}$ are normal in a
neighborhood of $x_{0}$ for some values of $i$, but not for all. We define

${\cal B}_{X}^{i}:={\cal B}_{X}(s_{i})=\\{x_{0}\in
X\mid\\{g_{n}^{i}(x)\\}\text{\ is not normal in any neighborhood of $x_{0}$ in
$X$}.\\}$

This is also known as the activity locus (in $X$) of the singular value
$s_{i}$ (see [DF08, Gau12]).

In this paper we investigate one dimensional slices of holomorphic families in
which the activity loci of the different singular values are disjoint. This
means that at any given parameter on our slice only one of the singular values
is allowed to bifurcate. We will require our slices to be dynamically defined
in a sense we describe below. Such slices can be found, for example, by fixing
the dynamical behavior of all but one of the singular values.

######  Definition 4.5 (Dynamically natural slice).

Let ${\cal F}$ be a holomorphic family of entire or meromorphic maps over $X$
(and by abuse of notation also denote the set of maps $\\{f_{x}\\}_{x\in X}$
by ${\cal F}$). Assume $S(f_{x})$, the set of finite singular values of
$f_{x}$, counted with multiplicity has cardinality $N<\infty$ for all $x\in
X$. A one dimensional subset $\Lambda\subset X$ is a dynamically natural slice
with respect to ${\cal F}$ if the following conditions are satisfied.

  1. (a)

$\Lambda$ is holomorphically isomorphic to the complex plane with perhaps a
discrete set of points removed. The removed points are called parameter
singularities. By abuse of notation we denote its image in ${\mathbb{C}}$ by
$\Lambda$ again, and denote the variable in ${\mathbb{C}}$ by $\lambda$.

  2. (b)

The singular values are given by distinct holomorphic functions
$s_{i}(\lambda)$, $i=1,\ldots,N-1$, and an asymptotic value $v_{\lambda}$ that
is an affine function of $\lambda$. We call $v_{\lambda}$ the free asymptotic
value; we require that ${\cal B}_{\Lambda}(v_{\lambda})\neq\emptyset,\Lambda$.

  3. (c)

The poles (if any) are given by distinct holomorphic functions
$p_{i}(\lambda)$, $\lambda\in\Lambda$, $i\in{\mathbb{Z}}$.

  4. (d)

There are $N-1$ distinct attracting or parabolic cycles whose period and
multiplier are constant for all $\lambda\in\Lambda$.

  5. (e)

Suppose $v_{\lambda_{0}}$ is the only singular value in ${\cal
A}_{\lambda_{0}}$, the basin of attraction of an attracting cycle whose
multiplier is not a constant function of $\lambda$. Then the slice $\Lambda$
contains, up to affine conjugacy, all meromorphic maps
$g:{\mathbb{C}}\to{\widehat{\mathbb{C}}}$ that are quasiconformally conjugate
to $f_{\lambda_{0}}$ in ${\mathbb{C}}$ and conformally conjugate to
$f_{\lambda_{0}}$ on ${\mathbb{C}}\setminus{\cal A}_{\lambda_{0}}$.

  6. (f)

$\Lambda$ is maximal in the sense that if
$\Lambda^{\prime}=\Lambda\cup\\{\lambda_{0}\\}$ where $\lambda_{0}$ is a
parameter singularity, then $\Lambda^{\prime}$ does not satisfy at least one
of the conditions above.

From now on, as long as it is understood from the context, we will drop the
subindex $\Lambda$. In other words ${\cal B}:={\cal B}_{\Lambda}$.

Let us make some remarks on the conditions above.

######  Remark 4.6.

  * •

In Condition (b), asking for the asymptotic value to be an affine function is
not a real restriction, since it can usually be arranged by a change of
coordinate. In any event, because the properties we are interested are
dynamical, some normalization is required to make statements about the
parameter plane.

  * •

Condition (d) could actually be weakened for our purposes. Morally what we
really need is that $N-1$ dynamical phenomena are kept constant throughout the
slice so that only one singular value is free at any given parameter value.
For example, if two singular orbits were eventually equal (i.e. shared the
same “tail”) throughout the slice, that would count as one dynamical
phenomenon. Another possibility is that throughout the slice some singular
value lands on an attracting cycle after a fixed number of iterations. A third
possibility is that there is an asymptotic value whose orbit is tied to the
orbit of the free asymptotic value by a relation which persists throughout the
slice. An example of this last possibility is the tangent map, for which the
two singular orbits are symmetric. Although most of our results hold in these
more general situations, for simplicity of exposition we choose to require
condition (d) as stated. We will, however, also consider dynamical slices in
the more relaxed sense at some points and some examples in the paper.

  * •

Condition (e) is generally easy to verify in concrete families. If the initial
holomorphic family is in some sense complete, and the slice is determined by
the required dynamical conditions, deformations can be controlled. This will
become clearer in the examples in Section 7. If, however, we relax condition
(d) as in the above remark, condition (e) has may have to be modified
accordingly. For example, if there is a relation tying the orbit of a non-free
asymptotic value to that of the free asymptotic value, condition (e) would say
that up to affine conjugacy, all meromorphic maps
$g:{\mathbb{C}}\to{\widehat{\mathbb{C}}}$ that are quasiconformally conjugate
to $f_{\lambda_{0}}$ in ${\mathbb{C}}$, conformally conjugate to
$f_{\lambda_{0}}$ on ${\mathbb{C}}\setminus{\cal A}_{\lambda_{0}}$ and whose
asymptotic values satisfy the given relation must lie in the slice. The
tangent family is an example.

  * •

Condition (f) is imposed to avoid artificial parameter singularities. In
general, these singularities occur because the functions become constant or
the number of singular values drops, or some poles escape to infinity when
approaching these values.

  * •

Suppose $\Lambda$ is a dynamically natural slice. For a given parameter value
$\lambda_{0}\in\Lambda$, if the free asymptotic value $v_{\lambda_{0}}$ is
attracted to an attracting cycle, it follows that $\lambda_{0}\in X^{stab}$
(indeed, if any of the cycles is parabolic, it must be persistent and
therefore the map is $J-$stable). Because there are $N-1$ attracting or
parabolic basins which need to attract $N-1$ different singular values, only
one of the singular values can be active at any given parameter. That is, if
we set ${\cal B}^{0}:={\cal B}(v_{\lambda})$, then for $i,j=0,1,\ldots,N-1$,

${\cal B}^{i}\cap{\cal B}^{j}=\emptyset$

whenever $i\neq j$. If we relax the definition of a dynamically natural slice
as indicated in Remark 4.6 above, then we must also allow the possibility that
${\cal B}^{i}={\cal B}^{j}$ for some $i\neq j$. This is the case, for example,
in the tangent family.

#### Examples

Most of the families which have been systematically studied in the literature
like the exponential family $e^{z}+\lambda$ or the tangent family $\lambda\tan
z$ are dynamically natural slices. Many others can be constructed by pre- or
post- composing functions in these families with rational maps. In Section 7
we discuss dynamically natural slices for a number of families formed in this
way. In the Appendix we give methods to create them.

##  5 Distinguished parameter values

Let $\Lambda$ be a dynamically natural slice for the family ${\cal
F}=\\{f_{\lambda},\lambda\in\Lambda\\}$ and let $v_{\lambda}$ denote the free
asymptotic value of $f_{\lambda}$

In this section we discuss special parameter values for which the forward
orbit of some singular value is finite.

######  Definition 5.1 (Misiurewicz parameter).

A parameter $\lambda\in\Lambda$ (or the map $f_{\lambda}$) is called
Misiurewicz  if there is a singular value $s\in S(f_{\lambda})$ such that
$f_{\lambda}^{n}(s)$ is a repelling periodic point for some $n\geq 0$.

######  Remark.

It might happen that for $\lambda$ in a dynamically natural slice, some
iterate of the free asymptotic value $v_{\lambda}$ lands on an attracting or
neutral periodic cycle $\mathbf{a}$. If $\mathbf{a}$ is attracting or
parabolic, there must be a singular value $v^{\prime}\neq v_{\lambda}$ of
$f_{\lambda}$ in the immediate basin of attraction of $\mathbf{a}$. Because we
are in a dynamically natural slice $\mathbf{a}$ must be one of the $N-1$
persistent attracting cycles. Since $v_{\lambda}$ belongs to a preperiodic
component of the full basin of attraction of $\mathbf{a}$, $\lambda$ is a
$J$-stable parameter in $\Lambda$. We therefore choose not to call these
parameters Misiurewicz parameters.

By definition, Misiurewicz parameters are solutions of

$f_{\lambda}^{m}(s(\lambda))=f_{\lambda}^{n}(s(\lambda))$ (5.1)

for some $m>n\in{\mathbb{N}}$ and some singular value function $s(\lambda)$.
If $f_{\lambda}$ is conjugate to a Misiurewicz map $f_{\lambda^{\prime}}$
which satisfies Equation (5.1) for certain values of $m,n\in{\mathbb{N}}$,
then $f_{\lambda}$ satisfies the same equation for the same $m,n$.

A meromorphic function that is not entire must have at least one pole that is
not an omitted value. The pre-images of the poles have a finite forward orbit
and play an important role in the dynamics.

######  Definition 5.2 (Order of a prepole).

A point $p\in{\mathbb{C}}$ is a prepole of order $n>0$ for $f_{\lambda}$ if
$f_{\lambda}^{k}(p)$ is defined for $k<n$ and $f_{\lambda}^{n}(p)=\infty$.

With this definition a pole is a prepole of order 1.

For every $\lambda\in\Lambda$, the poles form a discrete set so they can only
accumulate at infinity. Unless there are at most two poles, and both are
omitted values (and this is never the case for functions of finite type),
Picard’s theorem implies that the prepoles of order 2 form an infinite set.
Their accumulation set is the set of poles and the point at infinity. Prepoles
of order $n\geq 3$ accumulate both at prepoles of order $n-1$ and at infinity.
It is well known that prepoles are dense in the Julia set of $f_{\lambda}$
[Ber93, BKY91a].

We now look at the parameter plane and consider parameters for which some
iterate of the free asymptotic value is a pole.

######  Definition 5.3 (Virtual cycle).

Let $f$ be a meromorphic map. A virtual cycle for $f$ of period $p\geq 2$ is a
set of points $\\{a_{1},a_{2},\ldots,a_{p-1},\infty\\}$,
$a_{i}\in{\mathbb{C}}$ for $i=1,\ldots,p-1$, where $a_{1}$ is an asymptotic
value, $a_{p-1}$ is a pole and $f(a_{i})=a_{i+1}$ for $1\leq i\leq p-2$. In
other words a virtual cycle is the forward orbit of an asymptotic value which
is a prepole.

Note that it does not make sense to define virtual cycles of period one.

######  Definition 5.4 (Virtual cycle parameter).

A parameter $\lambda\in\Lambda$ is called a virtual cycle parameter of order
$p\geq 2$ if $f_{\lambda}$ has a virtual cycle of period $p$.

For $\lambda\in\Lambda$, let $\\{p_{i}(\lambda)\\}_{0\leq i\leq M}$ denote the
set of poles of $f_{\lambda}$, for some $M\leq\infty$. Recall that since we
are working in a dynamically natural slice, $p_{i}(\lambda)$ are distinct
holomorphic functions of $\lambda$ throughout $\Lambda$. By definition,
virtual cycle parameters of order $p\geq 2$ are $\lambda-$values which are
solutions of

$f_{\lambda}^{p-2}(s(\lambda))=p_{i}(\lambda)$ (5.2)

for some $0\leq i\leq M$, and some asymptotic value $s(\lambda)\in
S(f_{\lambda})$.

######  Remark.

Virtual cycle parameters are parameter values for which an orbit relation
exists. Hence if $\lambda$ is a virtual cycle parameter and $f_{\lambda}$ and
$f_{\lambda^{\prime}}$ are topologically conjugate, then $\lambda^{\prime}$
must be a virtual cycle parameter of the same order.

Observe that in a dynamically natural slice, the sets of Misiurewicz
parameters and virtual cycle parameters are disjoint.

######  Proposition 5.5.

Misiurewicz parameters and virtual cycle parameters belong to the bifurcation
locus ${\cal B}$.

###### Proof.

Virtual cycle parameters are solutions of (5.2) while Misiurewicz parameters
are solutions of (5.1). Since the zeroes of holomorphic functions are
discrete, for fixed $m,n$ and $p$ both equations have a discrete set of
solutions. Hence, since both conditions are invariant under topological
conjugacy, the maps must be structurally unstable. ∎

######  Proposition 5.6.

For every asymptotic value $s(\lambda)\in S(f_{\lambda})$, virtual cycle
parameters corresponding to $s(\lambda)$ are dense in ${\cal B}(s(\lambda)$.
Misiurewicz parameters are dense in ${\cal B}$. As a consequence of both these
statements, ${\cal B}(v_{\lambda})$ has no isolated points in $\Lambda$.

###### Proof.

The proof is a standard application of Montel’s Theorem. We first show the
first statement. Let ${\lambda_{0}}\in{\cal B}(s(\lambda))$ and let $U$ be a
neighborhood of ${\lambda_{0}}$ in $\Lambda$. Let $p_{i}(\lambda)$, $i=1,2,3$
be three prepoles of order 3 which are distinct for every $\lambda\in U$.
These exist since the set of prepoles of order 3 or larger is infinite. Now
observe that, for some $\lambda\in U$,
$g_{n}(\lambda)=f_{\lambda}^{n}(s(\lambda))$ must take one of the values
$p_{i}(\lambda)$, for some $i=1,2$ or $3$ and some $n$, because otherwise
$\\{g_{n}\\}$ would be normal in $U$, contradicting the assumption that
${\lambda_{0}}\in{\cal B}(s(\lambda))$.

To show that Misiurewicz parameters are dense we use a repelling periodic
point $a(\lambda)$ of period larger than or equal to 3 for all $\lambda\in U$,
and apply the above arguments to the functions
$\\{a(\lambda),f_{\lambda}(a(\lambda)),f_{\lambda}^{2}(a(\lambda))\\}$, and to
all functions $g_{n}^{i}(\lambda):=f_{\lambda}^{n}(s_{i}(\lambda))$ at once. ∎

##  6 Shell components

Let $\Lambda$ be a dynamically natural slice for the family ${\cal
F}=\\{f_{\lambda}\\}$ and let $v_{\lambda}$ denote the free asymptotic value
of $f_{\lambda}$ which depends affinely on $\lambda$.

We are interested in studying those components in the slice $\Lambda$ that
reflect the behavior of the free asymptotic value. We first define the set

${\cal H}:=\\{\lambda\in\Lambda\mid v_{\lambda}\text{\ is (strictly) attracted
to an attracting cycle with nonconstant multiplier}\\}.$

Observe that if $\lambda\in{\cal H}$, then all remaining singular values must
belong to the $N-1$ attracting or parabolic basins of cycles with constant
multiplier. As a consequence,

${\cal H}\subset\Lambda\setminus{\cal B}=\Lambda^{stab}.$

######  Definition 6.1 (Shell component).

A set $\Omega\subset\Lambda$ is a shell component of the family $f_{\lambda}$
if it is a connected component of ${\cal H}$.

Since $\partial\Omega$ is in ${\cal B}$, it follows that $\Omega$ is also a
connected component of $\Lambda\setminus{\cal B}$.

Shell components may be hyperbolic components in the classical sense or may
not be; for example, if some of the cycles with constant multipliers are
parabolic. Conversely, capture components, where one of the persistent cycles
attracts $v_{\lambda}$ and another singular value, may be hyperbolic
components but are not shell components.

The following property follows directly from the definition.

######  Lemma 6.2 (The period is constant).

If $\Omega$ is a shell component, the period of the attracting cycle to which
$v_{\lambda}$ is attracted is constant throughout $\Omega$. It is called the
period of $\Omega$.

Since the dynamics of all but one of the attracting cycles of $f_{\lambda}$
are fixed as $\lambda$ varies in a shell component, the attracting cycle for
this shell component is the one whose multipler varies with $\lambda$. Below,
when we talk about an attracting cycle or its multiplier, in relation to a
shell component, it is these that we mean.

Every hyperbolic component of the parameter plane of the well known
exponential family $E_{\lambda}=z\mapsto e^{z}+\lambda$ is an example of a
shell component. In this case, the multiplier of the attracting cycle of
$E_{\lambda}$ is never zero. This also happens in more general settings.

######  Lemma 6.3 (Nonzero multipliers).

If $\Omega$ is a shell component, the multiplier of the attracting cycle is
nonzero for all $\lambda\in\Omega$.

###### Proof.

If the multiplier of the attracting cycle is zero, the cycle contains a
critical point. Hence $v_{\lambda}$ is not the only singular value in the
basin of attraction of this cycle, contradicting the definition of a shell
component. ∎

The following theorem explains how shell components are parameterized by the
multiplier of the attracting cycle. Let $\Omega$ be a shell component of
period $k>0$. We define the multiplier map

$\rho=\rho_{\Omega}:\Omega\to{\mathbb{D}}^{*}$

where $\rho(\lambda)$ is the multiplier of the attracting cycle which attracts
$v_{\lambda}$.

######  Theorem 6.4 (Multiplier map is a covering).

The multiplier map $\rho:\Omega\to{\mathbb{D}}^{*}$ is a holomorphic covering.

###### Proof.

The proof uses the method of surgery on the basin of attraction of the
attracting cycle that attracts $v_{\lambda}$. For details see [BF14]. Observe
that because $f_{\lambda}$ is a holomorphic family both $\rho$ and its
derivative are holomorphic functions of $\lambda$.

Choose a parameter ${\lambda_{0}}\in\Omega$ such that
$\omega_{0}:=\rho({\lambda_{0}})\in{\mathbb{D}}^{*}$. Let $k>0$ be the period
of $\Omega$ and ${\bf a^{0}}=\\{a_{0}^{0},\cdots,a_{k-1}^{0}\\}$ the
attracting periodic orbit to which the free asymptotic value $v_{\lambda_{0}}$
is attracted. Let ${\cal A}^{0}$ denote its basin of attraction and let
$A_{i}^{0}$ be the connected component of ${\cal A}^{0}$ containing
$a_{i}^{0}$ for $i=0,\cdots,k-1$. Assume without loss of generality that
$v_{\lambda_{0}}\in A_{1}^{0}$.

Let $W\subset{\mathbb{D}}^{*}$ be a (simply connected) neighborhood of
$\omega_{0}$. For any $\omega\in W$ we define a Beltrami form $\mu_{\omega}$
in $A_{1}^{0}$ invariant under $f_{\lambda_{0}}^{k}$ as follows. Since ${\bf
a^{0}}$ is attracting, there is a holomorphic homeomorphism $\psi$ from a
neighborhood $U$ of $a_{1}^{0}$ to the unit disk $\mathbb{D}$ such that
$\psi(a_{1}^{0})=0$ and $\psi(f_{\lambda_{0}}^{k}(z))=\omega_{0}\psi(z)$ for
every $z\in U$. In $\mathbb{D}$ choose an annular fundamental domain
$R_{0}=\\{|\omega_{0}|<|z|<1\\}$ for the map
$\psi(z)\mapsto\omega_{0}\psi(z)$. Now we define the surgery using standard
arguments: Construct a quasiconformal map
$\tilde{\phi}_{\omega}:\mathbb{D}\to\mathbb{D}$ such that it fixes the origin
and the unit circle and it maps $R_{0}$ to $R_{\omega}=\\{|\omega|<|w|<1\\}$
conjugating multiplication by $\omega_{0}$ to multiplication by $\omega$.
Additionally, we require that $\tilde{\phi}_{\omega_{0}}=Id$ and that
$\tilde{\phi}_{\omega}$ depends holomorphically on $\omega$. To do this, first
define the map on $R_{0}$ and then extend it by the dynamics to all of
$\mathbb{D}$. The Beltrami coefficient $\tilde{\mu}_{\omega}$ of
$\tilde{\phi}_{\omega}$ lifts to a Beltrami coefficient $\mu_{\omega}$ on $U$
invariant under the map $\psi$ that satisfies
$\mu_{\omega}(f^{k}_{\lambda_{0}}(z))\frac{\overline{(f^{k}_{\lambda_{0}})^{\prime}(z)}}{(f^{k}_{\lambda_{0}})^{\prime}(z)}=\mu_{\omega}(z)$.
Extend $\mu_{\omega}$ to the full backward orbit of $U$ so that for any
inverse branch $g_{n}$ of $f^{-n}_{\lambda_{0}}$ defined on a component of
${\cal A}^{0}$,
$\mu_{\omega}({g_{n}}(z))\frac{\overline{{g_{n}}^{\prime}(z)}}{{g_{n}}^{\prime}(z)}=\mu_{\omega}(z)$.
Extend it to be identically zero on the complement of the full basin of ${\bf
a^{0}}$ in ${\mathbb{C}}$. A Beltrami coefficient with this property is said
to be compatible with $f_{\lambda_{0}}$. Notice that the Beltrami coefficient
$\mu_{\omega}$ depends holomorphically on $\omega$.

By the Measurable Riemann Mapping Theorem [AB60], there is a quasiconformal
homeomorphism $\phi_{\omega}:{\mathbb{C}}\to{\mathbb{C}}$ with Beltrami
coefficient $\mu_{\omega}$. It is unique up to post-composition by an affine
map and depends holomorphically on $\omega$. Moreover, $\phi_{\omega_{0}}$ is
conformal.

At this point we need to assume that the functions in the family $f_{\lambda}$
are normalized in the same way. For example, that all the functions fix points
$\alpha$ and $\beta$ (although many other normalizations are possible). Then
we can chose a uniform normalization for $\phi_{\omega}$ fixing these two
points and define the new meromorphic map

$F_{\omega}:=\phi_{\omega}\circ f_{\lambda_{0}}\circ\phi^{-1}_{\omega}.$

This map respects the dynamics: it has an attracting periodic cycle ${\bf
a}(\omega)=\phi_{\omega}(\bf a^{0})$ of multiplier $\omega$. Moreover,
$F_{\omega}$ is quasiconformally conjugate to $f_{\lambda_{0}}$ in the
respective basins of attraction and conformally conjugate to $f_{\lambda_{0}}$
everywhere else. Since $\phi_{\omega_{0}}$ is conformal and fixes two points
it follows that it is the identity map. Hence
$F_{\omega_{0}}=f_{\lambda_{0}}$.

Since $f_{\lambda}$ is a dynamically natural slice, property (d) ensures that
$F_{\omega}=f_{\lambda(\omega)}$ for some $\lambda(\omega)\in\Lambda$. Observe
that the asymptotic value of $f_{\lambda(\omega)}$ is
$v_{\omega}=\phi_{\omega}(v_{\lambda_{0}})$ and it depends holomorphically on
$\omega$. Since $v_{\lambda}=v(\lambda)$ is affine, $\lambda=\lambda(v)$ is
also affine and hence $\lambda(\omega)=\lambda(v_{\omega})$ is holomorphic.
Since the multiplier changes with $\omega$, the map is nonconstant and thus
open. Finally, $\lambda(\omega_{0})=\lambda_{0}$ and therefore
$\lambda(W)\subset\Omega$.

This construction defines a holomorphic local inverse of the multiplier map
$\rho$ in a neighborhood of any point
$\omega_{0}=\rho({\lambda_{0}})\in{\mathbb{D}}^{*}$. That is, for every
$\omega\in{\mathbb{D}}^{*}$, there exists $\lambda\in\Omega$ with
$\rho(\lambda)=\omega$; it follows that $\rho$ is a covering map from $\Omega$
to ${\mathbb{D}}^{*}$. ∎

######  Corollary 6.5 (Connectivity of shell components).

Let $\Omega$ be a shell component of $f_{\lambda}$. Then one of the two
following possibilities occurs:

  1. (a)

$\Omega$ is simply connected and $\rho:\Omega\to{\mathbb{D}}^{*}$ is a
universal covering (hence of infinite degree), or

  2. (b)

$\Omega$ is homeomorphic to ${\mathbb{D}}^{*}$ and
$\lim_{|\omega|->0}\rho^{-1}(\omega)=\lambda^{*}$ where
$\lambda^{*}\notin\Lambda$ is a parameter singularity.

###### Proof.

Since $\rho:\Omega\to{\mathbb{D}}^{*}$ is a covering, it follows from Theorem
2.4 that either $\rho$ is a universal covering and hence has infinite degree
and $\Omega$ is simply connected, or $\rho$ has finite degree and $\Omega$ is
isomorphic to ${\mathbb{D}}^{*}$. In this case, the point
$\lambda^{*}:=\lim_{|\omega|->0}\rho^{-1}(\omega)$ cannot belong to ${\cal B}$
because, by Proposition 5.6, ${\cal B}$ has no isolated points. It follows
from Lemma 6.3 that $\rho$ cannot extend continuously to $\lambda^{*}$ because
$f_{\lambda}$ has no superattracting cycles. Therefore $\lambda^{*}$ is a
parameter singularity. ∎

######  Corollary 6.6.

Let $\Omega$ be a shell component. Then, any two maps $f_{\lambda}$ and
$f_{\lambda^{\prime}}$ with $\lambda,\lambda^{\prime}\in\Omega$ are (globally)
quasiconformally conjugate.

###### Proof.

This follows from the proof of Theorem 6.4. Indeed, let
$\gamma:[0,1]\to\Omega$ be a continuous path with endpoints $\lambda$ and
$\lambda^{\prime}$. Then $\sigma(t):=\rho(\gamma(t))$ is a continuous path in
${\mathbb{D}}^{*}$ joining $\omega=\rho(\lambda)$ and
$w^{\prime}=\rho(\lambda^{\prime})$. The image of $\sigma$ is a compact set so
it can be covered by a finite number of open disks $\\{D_{i}\\}_{1\leq i\leq
N}$ in ${\mathbb{D}}^{*}$ each centered at a point $\sigma(t_{i})$. Using the
construction in Theorem 6.4, we can construct local inverses of $\rho$ mapping
$D_{i}$ to an open neighborhood $D^{\prime}_{i}$ of $\gamma(t_{i})$. By
compactness, $\gamma[0,1]\subset\cup_{i=i}^{N}D_{i}^{\prime}$. Since all pairs
of parameters in each $D_{i}^{\prime}$ correspond to quasiconformally
conjugate maps, it follows that $f_{\lambda}$ is quasiconformally conjugate to
$f_{\lambda^{\prime}}$. ∎

###  6.1 Virtual centers

In standard rational dynamics hyperbolic components have a distinguished point
called the center of the component at which the attracting cycle is
superattracting. This is not the case for the shell components we are
studying, where the multiplier of a finite cycle cannot vanish. Nevertheless
we shall see that there are points in the boundary of the shell component that
play the role of the center.

######  Definition 6.7 (Virtual center).

Let $\Omega$ be a shell component and $\rho:\Omega\to{\mathbb{D}}^{*}$ its
multiplier map. A point $\lambda\in\partial\Omega\cap\Lambda$ is a virtual
center of $\Omega$ if whenever $\lambda_{n}\to\lambda$, with
$\lambda_{n}\in\Omega$, the multipliers $\rho(\lambda_{n})$ tend to 0.

In this section we will restrict our attention to dynamically natural slices
$\Lambda$ of maps in ${\cal M}_{\infty}$. Other cases like families of entire
maps, have been studied previously in a more direct way. More precisely, let
$\\{f_{\lambda}\\}_{\lambda\in\Lambda}$ be a dynamically natural slice of a
holomorphic family of finite type meromorphic maps for which infinity is not
an asymptotic value. Let $v(\lambda)$ denote the free asymptotic value.

For maps in this class, we show that every shell component has a unique
virtual center which may be finite or infinite. In addition, we show that if
the virtual center of a shell component is finite then it is a virtual cycle
parameter. To this end, we need some preliminary results.

The following two technical propositions describe the possibilities for what
happens in the dynamical plane when we take a sequence of parameters in a
shell component approaching a finite boundary point. The first one deals with
the case where the cycle stays bounded, and applies to both entire and
meromorphic functions.

######  Proposition 6.8 (Approaching the boundary I)).

Let $\Omega$ be a shell component of period $k\geq 1$ with
$\lambda_{n}\in\Omega$, and suppose
$\lambda_{n}\to\lambda^{*}\in\partial\Omega\cap\Lambda$. Let ${\bf
a}^{n}=\\{a_{0}^{n},\ldots,a_{k-1}^{n}\\}$ be the attracting cycle of period
$k$ for $f_{\lambda_{n}}$, labeled so that $a_{1}^{n}$ belongs to the
component of the basin containing the asymptotic value
$v_{n}:=v(\lambda_{n})$, and so that $a_{i+1}^{n}=f_{\lambda_{n}}(a_{i}^{n})$
(with indices taken mod $k$). Suppose the cycle ${\bf a}^{n}$ stays bounded as
$n\to\infty$. Then, $f_{\lambda^{*}}$ has an indifferent cycle ${\bf a}$ whose
period divides $k$ and ${\bf a}^{n}\to{\bf a}$ as $n\to\infty$.

###### Proof.

Since the cycle stays bounded and $\lambda^{*}\in\Lambda$, it follows that for
all $0\leq i\leq k-1$, there exists a subsequence of the sequence $a_{i}^{n}$
that has a limit $a_{i}$ and that this limit point is a fixed point of
$f_{\lambda^{*}}^{k}$. This cycle must be non-repelling. Since
$\lambda^{*}\in\partial\Omega$, it cannot be attracting. Hence it is an
indifferent cycle. ∎

######  Proposition 6.9 (Approaching the boundary II)).

Let $\Omega$ be a shell component of period $k\geq 2$ with
$\lambda_{n}\in\Omega$, and suppose
$\lambda_{n}\to\lambda^{*}\in\partial\Omega\cap\Lambda$. Let ${\bf
a}^{n}=\\{a_{0}^{n},\ldots,a_{k-1}^{n}\\}$ be the attracting cycle of period
$k$ for $f_{\lambda_{n}}$, labeled so that $a_{1}^{n}$ belongs to the
component of the basin containing the asymptotic value
$v_{n}:=v(\lambda_{n})$, and so that $a_{i+1}^{n}=f_{\lambda_{n}}(a_{i}^{n})$
(with indices taken mod $k$). Suppose $|a_{j}^{n}|\to\infty$ as $n\to\infty$,
for some $0\leq j\leq k-1$. Then, $j=0$ and as $n\to\infty$,

  1. (a)

$a_{1}^{n}\to v(\lambda^{*})$, the free asymptotic value of $f_{\lambda^{*}}$;

  2. (b)

$a_{k-1}^{n}$ tends to a pole of $f_{\lambda^{*}}$;

  3. (c)

$a_{k-i}^{n}$ tends to a prepole of order $i$ for all $1\leq i\leq k-1$;

  4. (d)

The multiplier of the cycle tends to 0.

Consequently, $\lambda^{*}$ is a virtual cycle parameter. Moreover, it is also
a virtual center.

###### Proof.

Observe that $f_{\lambda^{*}}$ is a well defined map in our family because
$\lambda^{*}\in\Lambda$. Let $D$ be a closed disk containing all the singular
values of $f_{\lambda_{n}}$ for all $n$ sufficiently large. Since infinity is
not an asymptotic value, by Lemma 3.3 the preimage of ${\mathbb{C}}\setminus
D\simeq{\mathbb{D}}^{*}$ under $f_{\lambda_{n}}$ consists of countably many
bounded topological punctured disks $\\{B_{n}^{i}\\}_{i}$. The punctures
correspond to distinct poles $p_{n}^{i}$ and
$f_{\lambda_{n}}:B_{n}^{i}\cup\\{p_{n}^{i}\\}\to\widehat{\mathbb{C}}\setminus
D$ is a finite degree branched cover. Infinitely many of these punctured disks
must be entirely contained in ${\mathbb{C}}\setminus D$, and provided that $D$
is large enough, they are disjoint. If $p_{n}^{i}$ is a simple pole, there is
a single valued branch of the inverse of $f_{\lambda_{n}}$ that maps
neighborhood of infinity in ${\mathbb{C}}$ onto $B_{n}^{i}$. If the pole is
multiple then there are finitely many such inverse branches.

Note that since we are in a dynamically natural slice, the poles
$p_{n}^{i}=p^{i}(\lambda_{n})$ of $f_{\lambda_{n}}$ are holomorphic functions
of $\lambda$. It follows that because $\lambda_{n}$ converges to a finite
point $\lambda^{*}\in\Lambda$, the functions $p^{i}(\lambda_{n})$ converge to
the poles $p^{i}(\lambda^{*})$ of $f_{\lambda^{*}}$. Therefore if we take $n$
sufficiently large (and assume we have indexed the disks consistently), the
disks $\\{B_{n}^{i}\\}_{i}$ and the poles $p_{n}^{i}$ are arbitrarily close
(in the spherical metric) to $B^{i}_{*}$ and $p^{i}(\lambda^{*})$, the
analogous sets and points for $f_{\lambda^{*}}$. Let
$U:={\mathbb{C}}\setminus\cup_{i}\overline{B^{i}_{*}}$.

By hypothesis, for some $0\leq j\leq k-1$, $|a_{j}^{n}|\to\infty$ as
$n\to\infty$. We first observe that $a_{j+1}^{n}=f_{\lambda_{n}}(a_{j}^{n})$
stays bounded for all $n\geq 0$. Indeed if it tends to infinity, for every $n$
large enough, $a_{j}^{n}$ belongs to a disk $B_{n}^{i}$ for some $i$. But
since $a_{j}^{n}$ tends to infinity with $n$ it follows that $i$ must change
with $n$. This is equivalent to saying that the inverse branch of
$f_{\lambda_{n}}^{-1}$ mapping $a_{j+1}^{n}$ to $a_{j}^{n}$ changes as $n$
increases. But inside the shell component $\Omega$ all parameters can be
joined by a path where the inverse branches are well defined and
quasiconformally conjugate and thus cannot change.

Hence $a_{j+1}^{n}$ converges to a finite point $a_{j+1}$. We now prove that
$a_{j+1}$ is an asymptotic value of $f_{\lambda^{*}}$. Suppose it is not.
Since $f_{\lambda}$ is of finite type for all $\lambda$, there exists a small
neighborhood $V$ of $a_{j+1}$ such that $f_{\lambda^{*}}^{-1}(a_{j+1})$
consists of infinitely many bounded disjoint topological disks
$\\{D_{i}\\}_{i}$. Let $N$ be such that $a_{j+1}^{n}\in V$ for $n>N$. Then the
points $f_{\lambda^{*}}^{-1}(a_{j+1}^{n})$ are contained in the disks $D_{i}$.
Since the inverse branch taking $a_{j+1}^{n}$ to $a_{j}^{n}$ cannot change
within $\Omega$, it follows that the points in the attracting cycle must
actually lie in a single one of these disks, which contradicts the hypotheses
that $a_{j}^{n}\to\infty$ when $n\to\infty$.

We now know that $a_{j+1}$ is an asymptotic value of $f_{\lambda^{*}}$. Since
the singular values of $f_{\lambda}$ are holomorphic functions of $\lambda$,
there is a holomorphic map determining an asymptotic value $u(\lambda)$ of
$f_{\lambda}$ such that $u(\lambda^{*})=a_{j+1}$. Let $T^{*}=T(\lambda^{*})$
be the asymptotic tract of $u(\lambda^{*})$ which contains $a_{j}^{n}$ for $n$
large enough (which exists because $a_{j+1}^{n}\to u(\lambda^{*})$). Let
$T_{n}$ be the corresponding asymptotic tracts for $u_{\lambda_{n}}$. By the
same convergence argument, for $n$ large enough, $T^{*}$ and $T_{n}$ contain a
non-empty domain $T^{\prime}$ such that $a_{j}^{n}\in T^{\prime}$. If
$V^{\prime}=f_{\lambda^{*}}(T^{\prime})$, it follows that $V^{\prime}$
contains $a_{j+1}^{n}$ and $u(\lambda^{*})$; moreover, since $T^{\prime}$ is
contained in the asymptotic tracts $T_{n}$ it follows that $u(\lambda_{n})$
also lie in $V^{\prime}$.

We claim that $u(\lambda_{n})$ lies in the component of the immediate basin
containing $a_{j+1}^{n}$ when $n$ is large enough. By the definition of a
dynamically natural slice, $v(\lambda_{n})$ is the only asymptotic value in
the basin of $a_{j+1}^{n}$ and there are $N-1$ other attracting or parabolic
cycles with fixed multiplier. Each of these attracts a singular value. Thus if
$u(\lambda_{n})$ were outside the basin, it would be attracted to one of the
other $N-1$ attracting or parabolic cycles that persist throughout the slice
and thus in a different basin. Taking a pre-image of a neighborhood of
$u(\lambda_{n})$ inside this basin, we see that any asymptotic tract
corresponding to $u(\lambda_{n})$ is mapped within the basin itself and so the
basin could not contain $a_{j}^{n}$ for any $n$. Therefore since
$v(\lambda_{n})$ is the only asymptotic value in the basin of $a_{j+1}^{n}$,
$u(\lambda_{n})=v(\lambda_{n})$ and the claim follows. Hence $j+1=1$ and $j=0$
and the limit of $a_{j+1}^{n}$ as $n\to\infty$ is $v(\lambda^{*})$. This
proves (a).

To show (b), observe that by continuity $a_{k-1}^{n}$ is a sequence which must
tend either to infinity or to a pole of $f_{\lambda^{*}}$, given that its
image $a_{0}^{n}$ is converging to infinity. As above, if it converges to a
pole, this is a finite point and we are done. By the same arguments as above
if it were to converge to infinity then $a_{0}^{n}$ would converge to the
asymptotic value and infinity is not an asymptotic value. Hence
$a_{k-1}:=\lim_{n\to\infty}a_{k-1}^{n}$ is a pole of $f_{\lambda^{*}}$ and (b)
is proved.

If $k=2$ there is nothing to show for (c). If $k>2$, using the same arguments
as in (a) we can see that no other point of the cycle can tend to infinity.
Indeed, if $a_{i}^{n}$ were to tend to infinity, we have shown that
$v(\lambda_{n})$ would have to be the component containing $a_{i+1}^{n}$ and
hence $i=0$. Thus all $a_{i}^{n}$ must converge to finite prepoles of the
appropriate order. This shows that the cycle is virtual and hence
$\lambda^{*}$ is a virtual cycle parameter.

Finally we prove (d). The multiplier of the cycle for a given $n$ is
$\rho_{n}=\prod_{i=0}^{k-1}f_{\lambda_{n}}^{\prime}(a_{i}^{n})$. When
$n\to\infty$ all factors tend to a finite number except for $i=0$ or $i=k-1$.
For $n$ large enough, $a^{n}_{k-1}$ belongs to a neighborhood $W$ of the pole
$p$ from statement (b), where the maps behave like $f_{\lambda_{n}}(z)\simeq
c/(z-p)^{m}$ for some $m\geq 1$ and $c\in{\mathbb{C}}$. Hence,

$a_{0}^{n}\simeq c/(a_{k-1}^{n}-p)^{m}$

and

$f_{\lambda_{n}}^{\prime}(a_{k-1}^{n})\simeq
c^{\prime}(a_{0}^{n})^{1+\frac{1}{m}}.$

Likewise, the points $a_{0}^{n}$ belong to $T^{*}$, the asymptotic tract of
$v(\lambda^{*})$. Let $f^{*}:=f_{\lambda^{*}}$ and
$\psi:T^{*}\to{\mathbb{H}}_{l}$ be a conformal map from the tract to the left
half plane so that, restricted to $T^{*}$, we have
$f^{*}=e^{\psi(z)}+v(\lambda^{*})$. Now, since $a_{1}^{n}\to v(\lambda^{*})$
we know that ${\rm Re}(\psi(a_{0}^{n}))\to-\infty$. Hence, for $n$ large
enough

$|f_{\lambda_{n}}^{\prime}(a_{0}^{n})|\simeq|e^{\psi(a_{0}^{n})}||\psi^{\prime}(a_{0}^{n})|=e^{{\rm
Re}(\psi(a_{0}^{n}))}|\psi^{\prime}(a_{0}^{n})|$

and

$|f_{\lambda_{n}}^{\prime}(a_{k-1}^{n})||f_{\lambda_{n}}^{\prime}(a_{0}^{n})|\simeq|a_{0}^{n}|^{1+\frac{1}{m}}e^{{\rm
Re}(\psi(a_{0}^{n}))}|\psi^{\prime}(a_{0}^{n})|.$

We will suppose that the limit of this this product does not go to zero and
obtain a contradiction. This assumption means that although ${\rm
Re}(\psi(a_{0}^{n}))$ goes to $-\infty$, it must go slowly enough to counter
the exponential contraction, but since all the multipliers are bounded by $1$,
it cannot go to infinity. To make this more precise, we first prove a general
lemma that quantifies how the exponential contraction counters polynomial
expansion well inside the asymptotic tract of an arbitrary transcendental
function.

######  Lemma 6.10.

Let $f$ be a meromorphic map and let $U$ be a logarithmic tract such that $f$
is a universal covering from $U$ onto a punctured neighborhood of an
asymptotic value. Let $\psi:U\to{\mathbb{H}}_{l}$ be a conformal map to the
left half plane so that $f$ factors as $f(z)=\exp(\psi(z))$. Suppose that
there is an unbounded sequence $z_{n}\in U$ such that ${\rm
Re}(\psi(z_{n}))\to-\infty$ as $n\to\infty$. If $|{\rm Im}(\psi(z_{n}))|$ is
bounded, then, for any $k\in[0,\infty)$,

$\lim_{n\to\infty}|z_{n}|^{k}|f^{\prime}(z_{n})|=0$

We shall use the two following Koebe estimates (see e.g. [Pom92]).

######  Lemma 6.11.

Let $\phi:D(a,r)\in{\mathbb{C}}$ be a conformal map. Then,

  * (a)

$|f(z)-f(a)|\leq\frac{r^{2}|z-a||f^{\prime}(a)|}{(r-|z-a|)^{2}}$.

  * (b)

$|f^{\prime}(a)|\leq\frac{(r+|z-a|)^{3}}{(r-|z-a|)r^{2}}|f^{\prime}(z)|$.

###### Proof of Lemma 6.10.

Let $z_{0}=\psi^{-1}(-1)$ and set

$r_{n}:=|{\rm Re}(\psi(z_{n}))|\text{\ and \
}d_{n}:=|\psi(z_{n})-\psi(z_{0})|=|\psi(z_{n})+1|.$

Using our assumption that the imaginary parts of $\psi(z_{n})$ are bounded,
there is an $\epsilon>0$ such that

$1-\epsilon\leq r_{n}-d_{n}\leq 1;\quad\text{and\ \ }\frac{r_{n}}{2}\leq
d_{n}\leq r_{n}.$

For all $n>0$, the map $\psi^{-1}$ restricted to the disc
$D(\psi(z_{n}),r_{n})$ is conformal, hence the Koebe estimates in Lemma 6.11
and the assumption tell us

  * (a)

$|z_{0}-z_{n}|\leq\frac{r_{n}^{2}d_{n}}{|\psi^{\prime}(z_{n})|(r_{n}-d_{n})^{2}}\leq\frac{k_{1}r_{n}^{3}}{|\psi^{\prime}(z_{n})|}$;

  * (b)

$\frac{1}{|\psi^{\prime}(z_{n})|}\leq\frac{(r_{n}+d_{n})^{3}}{(r_{n}-d_{n})r_{n}^{2}}\frac{1}{|\psi^{\prime}(z_{0})|}\leq
k_{2}r_{n}$.

Observe also that

$|z_{n}|\leq|z_{0}|+|z_{0}-z_{n}|\leq 2|z_{0}-z_{n}|.$

Since $|z_{n}|\to\infty$, the claim is obvious for $k=0$. For $k\geq 1$,
combining the two estimates above we obtain

$\displaystyle|z_{n}|^{k}|f^{\prime}(z_{n})|$
$\displaystyle=|z_{n}|^{k}e^{-r_{n}}|\psi^{\prime}(z_{n})|\leq
k_{3}|z_{0}-z_{n}|^{k}e^{-r_{n}}|\psi^{\prime}(z_{n})|$ $\displaystyle\leq
k_{4}\,\frac{r_{n}^{3k}}{|\psi^{\prime}(z_{n})|^{k-1}}e^{-r_{n}}\leq
k_{5}\,r_{n}^{4k-1}e^{-r_{n}},$

which tends to 0 when $r_{n}\to\infty$, i.e. when $n\to\infty$. ∎

We next prove that we can define an inverse branch of $f^{*}$ in a punctured
neighborhood of $v_{\lambda^{*}}$. Since the points $a_{1}^{n}$ belong to
periodic cycles for $\lambda_{n}$, there is a branch
$g_{n}=f_{\lambda_{n}}^{-1}$ that takes $a_{1}^{n}$ to $a_{0}^{n}$ in the
cycle. Let $V$ be the open set obtained by removing from the punctured disk
$U:=f^{*}(T^{*})$ a closed segment $\beta$, joining $v_{\lambda^{*}}$ and
$\partial U$, and closed segments $\beta_{n}$ joining $v_{\lambda^{*}}$ to
$v_{\lambda_{n}}$. Notice that $V$ is simply connected, and locally connected
since the diameter of $\beta_{n}$ tends to 0 as $n\to\infty$. On $V$, the
inverses $g_{n}$ are well defined and continuous and form a normal family.
After taking a subsequence if necessary, let $g^{*}$ be the limit of $g_{n}$
which is by construction an inverse of $f^{*}$ on $V$.

Now we use $g^{*}$ to define a curves in $T^{*}$ and
${\mathbb{H}}_{l}=\psi(T^{*})$ to which we can apply Lemma 6.10. The group of
deck transformations of the universal covering map $f^{*}$ is isomorphic to
${\mathbb{Z}}$. Let $S^{*}=g^{*}(D^{*})\subset T^{*}$ be a fundamental domain
for the map $f^{*}$ that contains $a_{0}^{n}$ for all large $n$. Its boundary
in $T^{*}$ consists of two unbounded curves $\gamma$ and $\gamma^{\prime}$
identified by a deck transformation. We can assume without loss of generality
that $\gamma,\gamma^{\prime}$ do not meet any of the points $a_{1}^{n}$ and
that they are smooth curves. The left half plane
${\mathbb{H}}_{l}=\psi(T^{*})$ is also a universal covering of $D^{*}$ by the
exponential and the group of deck transformations is translation along the
imaginary axis. The domain $\psi(S^{*})\subset{\mathbb{H}}_{l}$ is a
fundamental domain for this group. Clearly $\psi(\gamma)$ and
$\psi(\gamma^{\prime})$ are unbounded and have unbounded length. Applying
Lemma 6.10, we see that if the multiplier of the cycle does not go to zero,
the imaginary parts of these curves in ${\mathbb{H}}_{l}$ are unbounded. This
in turn implies that $f^{*}(\gamma)\subset D^{*}$ is a curve that spirals
infinitely often around $v_{\lambda^{*}}$ as it approaches it.

Consider ${f^{*}}^{-1}(T^{*})$ where the inverse is chosen so it lies in the
neighborhood $W$ of the pole $p$. Because $f^{*}$ is a finite type map, its
asymptotic tracts remain in sectors bounded by the Julia directions, and hence
$T^{*}$ cannot spiral around infinity; ${f^{*}}^{-1}(T^{*})$ at $p$ is a pre-
asymptotic tract, and it stays between the pre-images of the Julia rays, two
rays at $p$ that make an angle of less than $2\pi$.

Now apply ${f^{*}}^{k-2}$ to $f^{*}(\gamma)$; because we are in a dynamically
natural slice, this is a holomorphic homeomorphism onto its image which lies
in the neighborhood $W$ of the pole $p$. Since $f^{*}(\gamma)$ spirals around
$v_{\lambda}^{*}$ its image in $W$ spirals around the pole and remains inside
a Fatou component. Because it spirals, this Fatou component must intersect
${f^{*}}^{-1}(T^{*})$ and so contain it, which implies that
${f^{*}}^{-1}(T^{*})$ spirals. This contradiction proves (d).

Since ${\cal F}$ is a holomorphic family, the functions depend continuously on
$\lambda$. Therefore if we approach $\lambda^{*}$ with a different sequence
$\lambda^{\prime}_{n}\in\Omega$, the attracting cycle of
$f_{\lambda^{\prime}_{n}}$ must be arbitrarily close to the virtual cycle of
$f_{\lambda}^{*}$ as $n$ tends to infinity. It follows that the multiplier
also tends to 0 and we have proved that $\lambda^{*}$ is a virtual center. ∎

We are now ready to prove the main theorems in this section.

######  Theorem 6.12 (Virtual centers and virtual cycles).

Let $\Lambda$ be a dynamically natural slice for a family of meromorphic maps
$\\{f_{\lambda},\lambda\in\Lambda\\}$ in ${\cal M}_{\infty}$. Let
$\Omega\subset\Lambda$ be a shell component of period $k\geq 2$ and
$\lambda^{*}$ a parameter in $\partial\Omega\cap\Lambda$. Then $\lambda^{*}$
is a virtual center if and only if $\lambda^{*}$ is a virtual cycle parameter.

###### Proof.

Assume $\lambda^{*}$ is a virtual center. Let $\lambda_{n}\to\lambda^{*}$,
with $\lambda_{n}\in\Omega$ and let ${\bf
a}_{n}=\\{a_{0}^{n},\cdots,a_{k-1}^{n}\\}$ be the cycle to which
$v_{n}=v(\lambda_{n})$ is attracted for $f_{\lambda_{n}}$. We claim that one
of the points in the cycle tends to infinity as $n\to\infty$. Indeed,
otherwise, the cycle would converge to a finite cycle ${\bf a}^{*}$ of the
same period and multiplier $0$. Hence ${\bf a}^{*}$ is superattracting and
some point in the limit cycle is a critical point. But this implies that for
every $\lambda$ in a neighborhood of $\lambda^{*}$ the analytic continuation
of this cycle is attracting and its basin contains a critical point. This
contradicts the assumption that $\lambda^{*}$ is in $\partial\Omega$ and that
some part of this neighborhood belongs to the shell component. It now follows
from Proposition 6.9 that $\lambda^{*}$ is a virtual cycle parameter.

Conversely, suppose that $\lambda^{*}$ is a virtual cycle parameter, and let
$\lambda_{n}\to\lambda^{*}$ as above. Then, by the definition of a virtual
cycle parameter, at least one of the points in the cycle is tending to
infinity when $n\to\infty$ . By Proposition 6.9, the multiplier is then
tending to zero. Since the sequence was arbitrary, $\lambda^{*}$ is a virtual
center.

∎

The next proposition deals with the special case of shell components of period
one.

######  Proposition 6.13.

Let $\Lambda$ be a dynamically natural slice for a family of meromorphic maps
$\\{f_{\lambda},\lambda\in\Lambda\\}$ of finite type. Let $\Omega$ be a simply
connected shell component of period one. Then, for any sequence
$\lambda_{n}\to\lambda^{*}\in\partial\Omega\cap\Lambda$, the attracting fixed
point $a_{n}$ converges to an indifferent finite fixed point $a^{*}$ of
$f_{\lambda_{*}}$. It follows that $\Omega$ has no finite virtual centers.

######  Remark 6.14.

Note that if $\Omega$ is isomorphic to a punctured disk, the puncture is a
parameter singularity and does not belong to $\Lambda$ and the proposition
does not apply.

######  Remark 6.15.

Note that this proposition also holds if functions in the family have infinity
as an asymptotic value; they may even be entire.

###### Proof.

Suppose that the attracting fixed point $a_{n}$ for $f_{\lambda_{n}}$ tends to
infinity as $n\to\infty$. If $f_{\lambda}$ is meromorphic and infinity is not
an asymptotic value of any of the functions in our family we argue as in the
proof of Proposition 6.9. Let $D$ be a large disk eventually containing all
the singular values of $f_{\lambda_{n}}$ and let $B_{n}^{i}$ be the components
of $f_{\lambda_{n}}^{-1}({\mathbb{C}}\setminus D)$. Since the $a_{n}$ are
fixed points and form an unbounded sequence, they must belong to $B_{n}^{i}$
for $n$ large enough. Because the maps $f_{\lambda_{n}}$ are polynomially
expanding on $B_{n}^{i}$, the derivative in these disks is larger than one, so
there cannot be any attracting fixed points inside them. Therefore the $a_{n}$
must converge to a finite fixed point which, by Proposition 6.8, is an
indifferent fixed point. This shows the first part of the Proposition.

If infinity is an asymptotic value for functions in the family, we choose the
disk $D$ so that it contains all the finite singular values of
$f_{\lambda_{n}}$ for large $n$. Then
$f_{\lambda_{n}}^{-1}({\mathbb{C}}\setminus D)$ has at least one simply
connected unbounded component that is an asymptotic tract for the asymptotic
value infinity. In this tract the function is expanding exponentially so again
it cannot contain an attracting fixed point.

Now suppose $\lambda^{*}\in\partial\Omega\cap\Lambda$ is a virtual center, and
let $\lambda_{n}\to\lambda^{*}$ with $\lambda_{n}\in\Omega$. Since the
attracting fixed point $a_{n}$ of $f_{\lambda_{n}}$ stays bounded, it must
converge to a finite fixed point of multiplier 0. But this is impossible by
Proposition 6.8. ∎

We are now ready to study the properties of the boundary of the simply
connected shell components in more detail. Let $\Omega$ be a simply connected
shell component and let ${\mathbb{H}}_{l}$ denote the left half plane. Recall
from Corollary 6.5 that the multiplier map $\rho:\Omega\to{\mathbb{D}}^{*}$ is
a universal covering. Hence, there exists a conformal homeomorphism
$\varphi:{\mathbb{H}}_{l}\to\Omega$, unique up to precomposition by a Möbius
transformation, such that

$(\rho\circ\varphi)(w)=e^{w}.$

######  Theorem 6.16.

Let $\Lambda$ be a dynamically natural slice for a family of mermorphic maps
$\\{f_{\lambda},\lambda\in\Lambda\\}$ in ${\cal M}_{\infty}$. Let $\Omega$ be
a simply connected shell component in $\Lambda$ and let $\rho,\varphi$ be as
above. Then, $\varphi$ extends continuously to a map
$\varphi:\overline{{\mathbb{H}}_{l}}\cup\infty\to\overline{\Omega}$, where the
closure is taken in ${\widehat{\mathbb{C}}}$.

###### Proof.

Let $\gamma(t)$, $t\in(0,1)$ be an arbitrary arc in ${\mathbb{H}}_{l}$ such
that $\lim_{t\to 1}\gamma(t)=i\theta^{*}$ for some
$\theta^{*}\in{\mathbb{R}}$. We want to show that
$\sigma(t)=\varphi(\gamma(t))$ lands at a point in
$\overline{\Omega}\setminus\Omega$ when $t\to 1$.

To this end, let $\lambda^{*}$ be a finite accumulation point of $\sigma(t)$
when $t\to 1$. This means that there exists a sequence $t_{n}\to 1$ as
$n\to\infty$, such that $\lambda_{n}=\sigma(t_{n})\in\Omega$ satisfies
$\lambda_{n}\to\lambda^{*}$. Let ${\bf a}^{n}$ denote the attracting cycle
corresponding to $\lambda_{n}$. If one point in this cycle goes to infinity as
$n\to\infty$, it follows from Proposition 6.9 that the multiplier tends to
zero which is impossible. Therefore the cycle stays bounded and, by
Proposition 6.8, it tends to an indifferent cycle of period dividing the
period of $\Omega$ and multiplier $e^{2\pi i\theta^{*}}$. Since the zeroes of
holomorphic functions are discrete, the parameters for which the function has
a periodic cycle with this multiplier and period either form a discrete set or
constitute the whole slice $\Lambda$. They cannot fill out the whole slice
because by the definition of a dynamically natural slice, the singular value
that can accumulate on such a neutral cycle must be the free asymptotic value
$v_{\lambda}$, and if $\lambda\in\Omega$ the cycle is attracting. Since the
accumulation set of $\sigma(t)$ must be connected it cannot be a discrete set.
It then follows that $\sigma(t)$ actually lands at
$\lambda^{*}\in\partial\Omega$.

We conclude that $\sigma$ lands either at infinity or at a finite parameter
value $\lambda^{*}$ for which $f_{\lambda^{*}}$ has an indifferent cycle. It
follows from the Lindelöf theorem [CG93, Theorem 2.2] that the image of every
other possible arc $\gamma$ approaching $i\theta^{*}$ must have the same
landing point as $\gamma$.

It remains to consider curves in ${\mathbb{H}}_{l}$ that tend to infinity.
First suppose $\gamma(t)=t$, $t\in(-\infty,0)$, the negative real axis. We
claim that $\sigma(t)=\varphi(\gamma(t))$ lands at a point in
$\partial\Omega\cap{\mathbb{C}}$ or at infinity. Indeed, let
$\lambda^{*}\in\partial\Omega$ be a finite accumulation point of $\sigma$. As
above, there exists a sequence $t_{n}\to-\infty$ as $n\to\infty$, such that
$\lambda_{n}=\sigma(t_{n})\in\Omega$ and $\lambda_{n}\to\lambda^{*}$. Let
${\bf a}^{n}$ denote the attracting cycle corresponding to $\lambda_{n}$. If
all the points in the cycle stay bounded as $n$ grows then, by Proposition
6.8, the multiplier of the cycle tends to one in modulus, which contradicts
the fact that $\exp(\gamma(t))$ is tending to 0. Hence at least one point of
the cycle must tend to infinity and, by Proposition 6.9, $\lambda^{*}$ is a
virtual cycle parameter of order equal to the period of $\Omega$, and a
virtual center. But virtual cycle parameters of a given period form a discrete
set. Thus all the accumulation points of $\sigma$ must equal $\lambda^{*}$
which implies that $\sigma$ actually lands at $\lambda^{*}$.

The considerations above imply that the radial limit of $\varphi$ at infinity
exists, and in fact, by the Lehto-Virtanen Theorem [Pom92, Sect 4.1], the
angular limit exists (i.e., the limit for any curve tending to infinity in a
Stolz angle), and equals the virtual center $\lambda^{*}$ or infinity. If we
take any other curve in ${\mathbb{H}}_{l}$ tending to infinity, Lindelöf’s
theorem implies that any accumulation point must equal the radial limit.

We conclude that unrestricted limits exist at all finite points of
$\partial{\mathbb{H}}_{l}$, and the radial limit exists at infinity. Hence
$\phi$ extends continuously to the boundary in ${\widehat{\mathbb{C}}}$. ∎

An immediate corollary of the proof is the following.

######  Corollary 6.17.

Let $\Lambda$ be a dynamically natural slice for a family of mermorphic maps
$\\{f_{\lambda},\lambda\in\Lambda\\}$. Let $\Omega$ be a simply connected
shell component in $\Lambda$. Then,
$\partial\Omega\subset{\widehat{\mathbb{C}}}$ is locally connected and has a
unique virtual center, which may be at infinity.

Combining the existence of a unique virtual center with Proposition 6.13, we
obtain the following immediate corollary of Theorem 6.16.

######  Corollary 6.18.

Under the hypotheses of the corollary above, every simply connected shell
component of period 1 has its virtual center at infinity and it is therefore
unbounded.

######  Remark 6.19.

If a shell component is not simply connected, the role of the virtual center
may be taken by a parameter singularity. An example can be found in the
tangent family $z\mapsto\lambda\tan(z)$, where ${\mathbb{D}}^{*}$ is a shell
component of period one with a parameter singularity in its center. In this
case, the multiplier map is a conformal homeomorphism.

Theorem 6.16 also allows us to define an internal structure in the simply
connected shell component $\Omega$.

######  Corollary 6.20.

Under the hypotheses of Corollary 6.17, define the internal ray in $\Omega$ of
angle $\theta$ by

$R_{\Omega}(\theta):=\\{\varphi(t+2\pi i\theta),t\in(-\infty,0)\\}.$

Then, all internal rays have one end at the virtual center and the other end
at a point in $\partial\Omega$ (which a priori could be infinity). If the
virtual center $\lambda^{*}$ is finite no internal ray has both ends at
$\lambda^{*}$.

Notice that the internal rays foliate the shell component. The rays landing at
parameters corresponding to parabolic cycles of multiplier one, that is the
preimages of $\\{0,1\\}$ under the multiplier map extended to
$\partial\Omega$, divide $\Omega$ into fundamental domains which are mapped
one to one to the punctured disk by the multiplier map $\rho$.

We have shown that simply connected shell components of period one are always
unbounded. In the next section we look at some examples of dynamically natural
slices of parameter space for specific families. The first is the exponential
family $\\{E_{\lambda}=e^{z}+\lambda,\lambda\in{\mathbb{C}}\\}$ where all
components are unbounded shell components. In the other examples, the families
of meromorphic maps $\\{f_{\lambda},\lambda\in\Lambda\\}$ are in ${\cal
M}_{\infty}$ and in all of these we see that all of the simply connected shell
components of period greater than one are bounded. This leads us to the
following conjecture.

######  Conjecture 6.21.

Let $\Lambda$ be a dynamically natural slice for a family of meromorphic maps
$\\{f_{\lambda},\lambda\in\Lambda\\}$ in ${\cal M}_{\infty}$ in the strict
sense. Let $\Omega\subset\Lambda$ be a simply connected shell component of
period $k=2$. Then $\Omega$ is bounded.

######  Remark 6.22.

If the slice is defined in the relaxed sense of Remark 4.6, there may be a
finite number of unbounded components of period greater than $1$ because of
the relations among the asymptotic values. Example (c) is a case in point.

##  7 Examples

To find examples we start with some fairly simple entire or meromorphic
functions, as described in the appendix, and pre-compose or post-compose with
rational functions. The coefficients of these rational functions determine a
natural parameter space. To define a dynamically natural family, we must cut
down the dimension and show that condition (e) holds.

######  Example 1.

The family of meromorphic functions with exactly two asymptotic values and no
critical points is given by

${\cal
F}_{2}=\left\\{\frac{ae^{z}+be^{-z}}{ce^{z}+de^{-z}},\,\,a,b,c,d\in{\mathbb{C}},ad-
bc\neq 0\right\\}.$

See the Appendix and the references there. The asymptotic values are $a/c$ and
$b/d$. If $b$ or $c$ is equal to zero, the function is the exponential
function and infinity is an asymptotic value. Otherwise both asymptotic values
are finite. This is really a two dimensional family because we can normalize
so that $ad-bc=1$ and conjugate by an appropriate affine transformation to
reduce to two parameters. We will look at several different dynamically
natural slices of ${\cal F}_{2}$ and use different affine transformations for
each.

  1. (a)

The exponential family $E_{\lambda}(z)=e^{z}+\lambda$ is one of the best
understood dynamically natural slices of ${\cal F}_{2}$. It contains, up to
affine conjugacy, every entire transcendental map with one finite asymptotic
value and no critical values; property (d) holds. Since $E_{\lambda}$ is
entire, one of the asymptotic values is at infinity. The finite asymptotic
value is $v_{\lambda}=\lambda$. In this case $\Lambda={\mathbb{C}}$. Note that
another normalization for this family in common use is one that fixes the
second asymptotic value at the origin. The family then takes the form $ae^{z}$
and the parameter is an affine function of the first iterate of the finite
asymptotic value. Standard references include [BR75, DFJ02, Sch03, RG03].

Figure 2 shows the parameter plane for $E_{\lambda}$. It was shown that all
shell components (here, all hyperbolic) in $\Lambda$ are unbounded and simply
connected and that the multiplier goes to zero as the parameter goes to
infinity inside the component; that is the components have a virtual center at
infinity. The functions in this family have no poles so there are no virtual
cycle parameters. In the figure, components are colored according to their
respective period, yellow for 1, cyan-blue for 2, red for 3, brownish green
for 4, etc.

Figure 2:  The parameter plane of the exponential family
$E_{\lambda}(z)=\exp(z)+\lambda$.

  2. (b)

Another dynamical slice that can be extracted from ${\cal F}_{2}$ is formed by
requiring the origin to be a fixed point with persistent multiplier
$\rho_{0},\,|\rho_{0}|<1$. The resulting slice is
$\Lambda={\mathbb{C}}\setminus\\{0,\rho_{0}/2\\}$ and the maps have the form

$f_{\lambda}(z)=\frac{e^{z}-e^{-z}}{(1/\lambda)e^{z}+(1/\mu)e^{-z}},$

with $\mu=\rho_{0}\lambda/(2\lambda-\rho_{0})$. The asymptotic values are
$\lambda$ and $-\mu$ and at least one of them is attracted by the origin. See
[GK08] for a discussion of properties of maps in the slice where
$\rho_{0}=1/2$. To see that condition (e) is satisfied, observe that any
conjugacy which is conformal in the basin of attraction of 0 keeps the
multiplier of this fixed point unchanged. Since the resulting conjugate map
again has two asymptotic values and no critical points, it belongs to ${\cal
F}_{2}$. Using an affine conjugacy, we may require that the new fixed point be
the origin again; thus the new map belongs to this family.

Figure 3 shows the parameter plane for for $\rho_{0}=2/3$and hence
$\mu=-\lambda/(1-3\lambda)$ and $\Lambda={\mathbb{C}}\setminus\\{0,1/3\\}$.

$0$

Figure 3:  Left: The $\lambda-$plane of the meromorphic family $f_{\lambda}$,
showing the dynamics of the free asymptotic value $\lambda$. Dimensions
$[-12,12]\times[-12,12]$. Right: Zoom in on a shell component of period two
(cyan-blue). Color coding is explained in the text. Dimensions
$[0.5,1.9]\times[-3.25,1.92]$

As in Figure 2, the color represents the period of the attracting orbit that
the free asymptotic value $\lambda$ is attracted to. It does not reflect the
behavior of the orbit of the other asymptotic value $-\mu$. In the unbounded
component on the left (green) both $\lambda$ and $-\mu$ are attracted to the
origin. Observe that this is a capture component as opposed to a shell
component; the origin is always an attracting fixed point so it attracts at
least one of the asymptotic values and in this component it attracts both. In
the yellow shell component on the right, $\lambda$ is attracted to an
attracting fixed point different from zero while $-\mu$ is attracted to zero.
It follows from Theorem C (c) that this component is indeed unbounded. In the
cyan-blue bounded shell components, $-\lambda$ is attracted to a cycle of
period $2$; in the red ones the attracting cycle has period three, etc.
Bifurcations occur at parabolic parameters as usual, giving rise to shell
components of all periods attached to the boundary of any given shell
component.

In Figure 4 we again plot the $\lambda-$plane but we show the dynamics of the
second asymptotic value $-\mu$. On the unbounded green capture component both
asymptotic values are attracted to the origin. On the complement, $\lambda$ is
attracted to the origin and $-\mu$ is free. The yellow region is a shell
component of period one which should have a virtual center at the parameter
$\lambda=1/3$, (where $\mu=\infty$) but this is a parameter singularity.

Observe that $\mu$ is not an affine function of the parameter $\lambda$; this
would explain the numerics which seem to indicate that this period one
component is bounded. In fact, if we were to reparametrize the family so that
$\mu$ depended affinely on the parameter, we would see the same picture as in
Figure 4 because of the symmetry in the equation connecting $\mu$ and
$\lambda$.

$\frac{1}{3}$

Figure 4:  The $\lambda-$ plane of the family $f_{\lambda}$, showing the
dynamics of the second asymptotic value $-\mu=\lambda/(1-3\lambda)$.
Dimensions $[0.28,0.44]\times[-0.08,0.08]$.

  3. (c)

A slice of ${\cal F}_{2}$ is given by the tangent family of maps

$T_{\lambda}(z)=\lambda\tan z$

with $\Lambda={\mathbb{C}}^{*}$ and $v_{\lambda}=\lambda i$ [DK88, KK97,
KY06]. This is a dynamically natural slice in the relaxed sense of Remark 4.6
because the second asymptotic value is $s_{1}(\lambda)=-v_{\lambda}=-\lambda
i$ and $T_{\lambda}$ is symmetric. It follows that the bifurcation locus is
the same for both singular values. Condition (e) here means that the relation
$s_{1}(\lambda)=-v_{\lambda}=-\lambda i$ must persist under deformation.
Figure 5 shows the parameter plane of $T_{\lambda}$.111 Note that the maps
$\lambda\tan{z}$ and $\lambda\tanh{z}$ are conjugate under $z\to iz$ so they
are dynamically the same. There is a lot of symmetry in this family. We have

$T_{\lambda}(-z)=T_{-\lambda}(z)\mbox{ and
}T_{\lambda}(-z)=-T_{\lambda}(z)\mbox{ so that }$
$T_{-\lambda}(-z)=T_{\lambda}(z).$

In addition

$T_{\bar{\lambda}}(z)=\overline{T_{\lambda}(\bar{z})}.$

This says, for example, that if for some $\lambda_{0}$, $z_{1}$ is a fixed
point of $T_{\lambda_{0}}$, then $T_{-\lambda_{0}}(z_{1})=-z_{1}$ so that
$z_{1}$ is a period $2$ point of $T_{-\lambda_{0}}$. It also says that complex
conjugate values of $\lambda$ have conjugate periodic cycles.

$0$

Figure 5:  The parameter plane of the family $T_{\lambda}(z)=\lambda\tan z$.
Dimensions $[-6,6]\times[-6,6]$.

The figure shows the $\lambda$ plane where we can observe these symmetries. We
only follow the orbit of ONE asymptotic value, $\lambda i$, and we color the
components based on the period of the cycle it is attracted to. The colors do
not reflect the behavior of the orbit of the other asymptotic value. Thus, we
see the same color for a $\lambda$ value for which there are two separate
attracting periodic cycles of period $2k$ as we do when there is a single
attracting periodic cycle of period $2k$ attracting both asymptotic values.

There is a single non-simply connected component, the punctured unit disk, for
which both asymptotic values are attracted to the origin. There are two
unbounded components, one on the right (yellow) and one on the left (cyan-
blue). In the one on the right, $\lambda i$ is attracted to a fixed point
$a_{0}$ with multiplier $\rho_{0}$, and $-\lambda i$ is attracted to $-a_{0}$
with the same multiplier. It has the same color as the unit disk. In the
unbounded component on the left, both $\lambda i$ and $-\lambda i$ are
attracted to the attracting period two cycle $a_{0},-a_{0}$ and its multiplier
is $\rho_{0}^{2}$. Neither of these components has a finite virtual center.
Although the left one is colored for period $2$, it cannot have a finite
virtual center because if $\lambda^{*}$ were a finite virtual center,
$-\lambda^{*}$ would be a virtual center for the unbounded component on the
right of period $1$, contradicting Corollary 6.18.

Note that, except for the punctured disk, each bounded component is paired
with another bounded component; the two have a common virtual center and are
tangent there. The two unbounded components can be thought of as paired at
infinity. The relationship between these component pairs is discussed in in
[KK97] and in detail for $\lambda$ on the imaginary axis in [CJK18].

  4. (d)

Another slice is formed again by keeping the origin fixed and fixing one of
the asymptotic values at $\pi i$ so that its image is $0$. The other
asymptotic value is $\lambda$. Since the dynamical behavior of one asymptotic
value is pre-periodic, but the multiplier at the origin is a function of
$\lambda$, this is a dynamically natural slice in the relaxed sense. The
functions have the form

$f_{\lambda}(z)=\frac{e^{z}-e^{-z}}{(1/\lambda)e^{z}+(i/\pi)e^{-z}}.$ (7.1)

We take the slice defined by the parameter $v_{\lambda}=\lambda$. The
parameter singularities are at $0$ and $\pi i$ where the denominator becomes
$0$ or infinity, or the asymptotic values coincide.

We see the parameter plane in Figure 6 where we show the dynamics of the free
asymptotic value $\lambda$.

$0$$\pi i$

Figure 6:  Left: The parameter plane of the family $f_{\lambda}(z)$ in (7.1).
Dimensions $[-6,15]\times[-10,11]$. Right: Zoom showing the two parameter
singularities. Dimensions $[-4,5.5]\times[-3,6.5]$

The parameter $\lambda=0$ is a parameter singularity and belongs to the
central yellow disk-like component. This component corresponds to parameter
values for which the free asymptotic value is attracted to the origin. It is
conformally homeomorphic to ${\mathbb{D}}^{*}$ and the multiplier is
univalent. The puncture is located at $\lambda=0$.

In the unbounded component on the right (yellow), $\lambda$ is attracted to an
attracting fixed point different from the origin. The second parameter
singularity $\lambda=-\pi i$, is located on the boundary of this component.
When $\lambda$ approaches this value, the multiplier tends to $-1$. Since the
parameter is singular, the usual period doubling bifurcation does not occur
and we see a cusp instead of an attached period two component.

######  Example 2.

Functions in ${\cal F}_{2}$ can be composed with rational or polynomial
functions and, up to affine conjugation, the dynamics of the composed
functions remain invariant under quasi-conformal deformations supported on the
Fatou set (see Theorem 8.3 in the Appendix). Below we describe dynamically
natural slices formed by pre- and post- composition with a quadratic
polynomial $Q(z)=az^{2}+bz+c$.

  1. (a)

If we start with an an $f\in{\cal F}_{2}$ with finite asymptotic values and
pre-compose by a degree two polynomial, $Q(z)=az^{2}+bz+c$, the resulting
function

$g(z)=f(az^{2}+bz+c)$

has the same two asymptotic values as $f$ does. Since each asymptotic tract of
$f$ has two pre-images under $Q$, each asymptotic value of $g$ has two
asymptotic tracts and so has multiplicity two. Since $f$ has no critical
points, $g$ has a single critical point at $-b/2a$ and a single critical value
at $f(-b/2a)$. If we assume the origin is fixed, then $c=0$. If we assume it
is also a critical value then $b=0$. Now making one of the asymptotic values
land on the fixed point $0$, we can form a dynamically natural slice. This
determines the coefficient $a$. This is a slice in the strict sense because
the origin is a super-attractive fixed point. The functions in the slice can
then be written

$g_{\lambda}(z)=\frac{e^{z^{2}}-e^{-z^{2}}}{\frac{1}{\lambda}e^{z^{2}}+(\frac{1}{\sqrt{\pi
i}})e^{-z^{2}}}$

The asymptotic values are $\lambda$ and $-\sqrt{\pi i}$. The singular
parameter values are at $0$ and $-\sqrt{\pi i}$. Theorem 8.3 implies that
condition (d) holds.

Figure 7 represents the $\lambda$-plane. The green components are capture
components where the orbit of $\lambda$ falls into the basin of attraction of
the origin. In the unbounded yellow shell components, $\lambda$ is attracted
to an attracting (not super attracting) fixed point. In the cyan-blue
(bounded) components it is attracted to a cycle of period two, in the red ones
to a cycle of period three and in the brownish green ones to a cycle of period
four. The components come in pairs because the asymptotic values have
multiplicity two. This is discussed in [CK].

Figure 7:  Dynamically natural slice for functions in ${\cal F}_{2}$ pre-
composed with a quadratic polynomial.

  2. (b)

We can form a dynamically natural slice, in the relaxed sense, with the
functions $h=af^{2}+bf+c$, for $f\in{\cal F}_{2}$ by assuming the origin is a
super attracting fixed point and that the two asymptotic values coincide. In
this case the functions in the slice each have one asymptotic value of
multiplicity two. We can write these functions as

$\lambda\tanh^{2}z.$

The parameter $\lambda$ is the free (double) asymptotic value and so
determines a dynamically natural slice of the parameter space
$\Lambda={\mathbb{C}}\setminus\\{0\\}$. Condition (e) is satisfied in the
modified sense.

In the left plot of Figure 8 we see the $\lambda-$plane. In the center of the
figure we see a green capture component formed by parameters for which the
asymptotic value belongs to the immediate basin of $0$. This component is
doubly connected because of the puncture at $\lambda=0$. The remaining green
components are capture components for higher iterates.

In the two yellow shell components the period is one. They are unbounded in
accordance with Proposition 6.13. The remaining shell components are all of
higher period and numerics indicate that they are all bounded. We see that
they are grouped into quadruples that share a virtual center. This is because
the asymptotic value has two asymptotic tracts and each asymptotic tract has
two pre-asymptotic tracts.

Figure 8:  Left: Parameter plane of the family $\lambda\tanh^{2}z$. Dimensions
$[-6,6]\times[-6,6]$. Right: Parameter plane of the family $\lambda\tanh
z^{2}$. Dimensions $[-3,3]\times[-3,3]$

Compare this figure to the right plot in Figure 8 which shows the parameter
plane for the family $\lambda\tanh z^{2}$. It is essentially the “square root”
of this slice, since both maps are semiconjugate by $z^{2}$. The asymptotic
values now have multiplicity $2$. These examples are investigated further in
[CK] where it is proved that all shell components of period greater than $1$
are bounded.

##  8 Appendix

Any meromorphic function that is a branched cover of the sphere of finite
degree $d$ is a rational function and so can be expressed as a quotient of
relatively prime polynomials at least one of which has degree $d$. The $2d+1$
coefficients of these polynomials define a natural embedding of the space of
such functions into ${\mathbb{C}}^{2d+1}$. Up to affine conjugation then,
these functions are represented as a complex analytic manifold of dimension
$2d-2$. This family is denoted by $Rat_{d}$ in the literature.

Not all holomorphic families of meromorphic functions have such an obvious
representation as a complex manifold. In this appendix we consider some
transcendental families that do. These are the families that most of our
examples are drawn from. The functions have explicit expressions that involve
complex constants. We show that under quasiconformal deformation, the deformed
function has a similar expression in which only the constants are changed.
Thus the constants determine the embedding of the manifold into
${\mathbb{C}}^{n}$ for the appropriate $n$.

We begin with families of meromorphic functions with $p<\infty$ asymptotic
values, no critical points and a single essential singularity at infinity. We
denote these by ${\cal F}_{p}$. (See [DK88, KK97, EG09] for further
discussion.)

These functions have a particularly nice characterization. Recall that the
Schwarzian derivative of a function $g$ is defined by

$S(g)(z)=(g^{\prime\prime}/g^{\prime})^{\prime}-\frac{1}{2}(g^{\prime\prime}/g^{\prime})^{2}.$
(8.1)

Nevanlinna, [Nev70, Nev32], Chap X1, proved

######  Theorem 8.1.

Every meromorphic function $g$ with $p<\infty$ asymptotic values and
$q<\infty$ critical points has the property that its Schwarzian derivative is
a rational function of degree $p+q-2$. If $q=0$, the Schwarzian derivative is
a polynomial $P(z)$. In the opposite direction, 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.

We sketch the proof of the theorem here and refer the interested reader to the
literature. The proof follows from the construction of a function with $p$
asymptotic values and no critical points as a limit of rational functions with
$p$ branch points. Letting the order of the branching at some or all of these
$p$ points increase, one obtains a sequence of rational functions. In the
limit, the images of the branch points whose order goes to infinity become
logarithmic singularities and the limit function has finitely many branch
points and finitely many logarithmic singularities. The limit function is a
parabolic covering map from the plane to itself. The Schwarzian derivatives of
the rational functions in the sequence are again rational functions with
degree determined only by the number of branch points, not their order. The
limit of the Schwarzian derivatives is the Schwarzian derivative of the limit
and so must be rational.

If all the branch points become asymptotic values in the limit, the limit
function has no critical points or critical values and hence its derivative
never vanishes. It follows from the definition of the Schwarzian that in the
limit, it must be a polynomial.

It is classical, ( see e.g. [Hil76]), that solutions to the Schwarzian
equation $S(g)=P(z)$ are related to solutions of the linear second degree
ordinary differential equation

$w^{\prime\prime}+\frac{1}{2}P(z)w=0.$ (8.2)

Such an equation has a two dimensional space of holomorphic solutions. If
$w_{1},w_{2}$ are a pair of linearly independent solutions, and

$g=\frac{aw_{2}+bw_{1}}{cw_{2}+dw_{2}},\,\,a,b,c,d\in{\mathbb{C}},\,ad-bc=1,$

it is easy to check that $S(g)=P(z)$. Moreover, if $g$ is any solution of the
Schwarzian equation, $w=\sqrt{1/g^{\prime}}$ is a solution of the linear
equation.

######  Remark 8.2.

One of the basic features of the Schwarzian derivative is that it satisfies
the following cocycle relation: if $f,g$ are meromorphic functions then

$S(g(f))(z)=S(g(f))f^{\prime}(z)^{2}+S(f(z)).$ (8.3)

In particular, if $T$ is a Möbius transformation, $S(T(z))=0$ and $S(T\circ
g(z))=S((g(z))$ so that post-composing by $T$ does not change the Schwarzian.
Under pre-composition by a Möbius transformation the Schwarzian behaves like a
quadratic differential. In particular, pre-composing by an affine
transformation multiplies the Schwarzian by a constant.

This means that if we find a specific pair $(w_{1}^{*},w_{2}^{*})$ of
solutions to the second order linear equation and set
$g^{*}=w_{2}^{*}/w_{1}^{*}$, then every solution of the Schwarzian equation
$S(g)=P(z)$ has the formula

$g=\frac{ag^{*}+b}{cg^{*}+d},\,\,a,b,c,d\in{\mathbb{C}},\,ad-bc=1,$

In particular, if $p=2$, $P(z)$ is identically constant. Using one of the
constants in an affine conjugation, we may assume $P\equiv-1/2$; then a
specific pair of solutions to equation (8.2) is
$w_{1}^{*}=e^{-\frac{z}{2}},w_{2}^{*}=e^{\frac{z}{2}}$ so we have
$g^{*}=w_{2}^{*}/w_{1}^{*}=e^{z}$. The functions in the family ${\cal F}_{2}$
have the form

$g(z)=\frac{ae^{z}+b}{ce^{z}+d},\,\,a,b,c,d\in{\mathbb{C}},\,ad-bc=1.$

Since there is one more degree of freedom from the affine conjugation, we see
this is a two dimensional family. We have discussed several dynamically
natural slices of ${\cal F}_{2}$ in this paper.

For ${\cal F}_{3}$, $P(z)$ is linear and two specific solutions to the second
order linear equation

$w^{\prime\prime}+\frac{1}{2}\zeta w=0$

are given by the Airy functions222 These are named after the British
astronomer G.B.Airy (1801-92). Others have studied this equation and solutions
are expressed in terms of Bessel functions and Gamma functions. We won’t write
the formulas but will describe the properties we need. $Ai(\zeta),Bi(\zeta)$.
Setting $g^{*}(\zeta)=Ai(\zeta)/Bi(\zeta)$ we obtain a solution with three
asymptotic values, $0,i$ and $-i$. Since we are interested in the dynamics, we
may conjugate by the affine transformation $\zeta=rz+s$. We may thus transform
any function whose Schwarzian is linear, and hence any function in ${\cal
F}_{3}$, up to affine conjugation, to one given by the formula

$g(\zeta)=\frac{ag^{*}(\zeta)+b}{cg^{*}(\zeta)+d},\,\,a,b,c,d\in{\mathbb{C}},\,ad-
bc=1.$

The asymptotic values of $g$ are
$\frac{b}{d},\frac{ai+b}{ci+d},\frac{-ai+b}{-ci+d}$.

We could define a dynamically natural slice, for example, by choosing the
coefficients $a,b$ and $d$ so that

$g(1)=1,\mbox{ and }g^{\prime}(1)=1/2.$

It is not hard to compute that the remaining coefficient $c$ is an affine
function of the asymptotic value $\frac{b}{d}$.

There are standard functions that solve the second order equation when the
coefficient polynomial is of degree two or three, and they give formulas for
functions in ${\cal F}_{4}$ and ${\cal F}_{5}$.

#### Compositions

The following theorem shows us how we can find more families of functions with
singular points that are closed under quasiconformal deformation. We have

######  Theorem 8.3.

Let $f_{0}\in{\cal F}_{p}$

  1. 1.

Suppose $g_{0}=f_{0}\circ Q_{0}$ is a function such that $Q_{0}$ is a
polynomial of degree $d$ and suppose that $g$ is a meromorphic function
quasiconformally conjugate to $g_{0}$. Then $g$ can be expressed as $g=f\circ
Q$ for some function $f\in{\cal F}_{p}$ and some polynomial $Q$ of degree $d$.

  2. 2.

Suppose $R_{0}\in Rat_{d}$ is rational of degree $d$ and $h_{0}=R_{0}\circ
f_{0}$. Then if $h(z)$ is quasiconformally conjugate to $h_{0}$, $h$ can be
expressed as $h=R\circ f$ for functions $R\in Rat_{d}$ and $f\in{\cal F}_{p}$.

######  Remark 8.4.

The proof of part 1 of the theorem works if $Q_{0}$ is replaced by a rational
function $R_{0}\in Rat_{d}$ but then the composed function $g_{0}$ has
essential singularities at the poles of $R_{0}$.

######  Remark 8.5.

Note that in part 1, $g=f\circ Q$ is a function with rational Schwarzian. For
each asymptotic value of $f$ of multiplicity $m$, $g$ has an asymptotic value
of multiplicity $dm$; moreover $g$ has the same critical points as $Q$, namely
$2d-2$ critical points counted with multiplicity, $d-1$ of which are at
infinity. In part 2, however, it is no longer true that the Schwarzian of the
composed function $h=R\circ f$ is a rational function. For example, it may
have infinitely many critical points.

######  Remark 8.6.

Observe that if $f\in{\cal F}_{p}$ and $Q$ is a degree $d$ polynomial then
$Q\circ f$ and $f\circ Q$ are semiconjugate by a degree $d$ polynomial – in
fact by $Q$.

###### Proof.

The proof of both parts of the theorem is essentially the same. We therefore
carry it out only for part 1.

In part 1, let $\phi^{\mu}$ be a quasiconformal homeomorphism with Beltrami
coefficient $\mu$ such that

$g_{\mu}=\phi^{\mu}\circ g_{0}\circ(\phi^{\mu})^{-1}$

is meromorphic. We can use $f_{0}$ to pull back the complex structure defined
by $\mu:=\bar{\partial}\phi^{\mu}/\partial\phi^{\mu}$ to obtain a complex
structure $\nu=f_{0}^{*}\mu$ such that the map

$f_{\mu}=\phi^{\mu}\circ f_{0}\circ(\phi^{\nu})^{-1}$

is meromorphic. Note that this is not a conjugacy since it involves two
different homeomorphisms.

We can now write

$g_{\mu}=\phi^{\mu}\circ f_{0}\circ(\phi^{\nu})^{-1}\circ\phi^{\nu}\circ
Q_{0}\circ(\phi^{\mu})^{-1}$

and set

$Q_{\mu}=\phi^{\nu}\circ Q_{0}\circ(\phi^{\mu})^{-1}.$

Again this is not a conjugacy but it is meromorphic since $g_{\mu}$ and
$f_{\mu}$ are homeomorphisms.

The main point here is that although $f_{\mu}$ is not a conjugate of $f_{0}$,
since the quasiconformal maps $\phi^{\mu}$ and $\phi^{\nu}$ are
homeomorphisms, the map $f_{\mu}$ is a meromorphic map with the same topology
as $f_{0}$; that is, it has $p$ asymptotic values and no critical values. By
Nevanlinna’s theorem, Theorem 8.1, $f_{\mu}$ belongs to ${\cal F}_{p}$.
Similarly, although $Q_{\mu}$ is not defined as a conjugate of $Q_{0}$, since
the quasiconformal maps $\phi^{\nu}$ and $\phi^{\eta}$ are homeomorphisms, the
map $Q_{\mu}$ is a meromorphic map with the same topology as $Q_{0}$; that is,
it is a degree $d$ branched covering of the Riemann sphere with the same
number of critical points and the same branching as $Q_{0}$ and thus it must
be a polynomial of degree $d$. ∎

Other families of holomorphic functions have been shown to be determined by
their topological covering properties. We state some of the theorems here and
refer the reader to the cited literature for proof.

######  Theorem 8.7.

([CJK17, Zak10].) Functions of type $Pe^{Q}$ for $P,Q$ polynomials are, up to
affine conjugacy, determined by their topological covering properties.

######  Theorem 8.8.

([BF10, Den13].) Any entire transcendental map of finite order with exactly
one critical point which is fixed and one asymptotic value with one finite
preimage, is affine conjugate to

$D_{a}(z)=a(e^{z}(z+1)+1),\quad a\in{\mathbb{C}}^{*}$

for some $a\in{\mathbb{C}}$. The affine conjugacy is used to make the
parameter unique by setting the critical point at $0$ and the asymptotic value
at $a$.

######  Theorem 8.9.

([Bak84, Th. 6.2], [Tan01, Prop. 2]) Functions of type
$\int_{z_{0}}^{z}P(t)e^{Q}(t)dt$ for $P,Q$ polynomials are, up to affine
conjugacy, determined by their topological covering properties.

## References

  * [AB60] Lars Ahlfors and Lipman Bers, _Riemann’s mapping theorem for variable metrics_ , Ann. of Math. (2) 72 (1960), 385–404.
  * [Bak84] I. N. Baker, _Wandering domains in the iteration of entire functions_ , Proc. London Math. Soc. (3) 49 (1984), no. 3, 563–576. MR 759304
  * [BE95] W. Bergweiler and A. Eremenko, _On the singularities of the inverse of a meromorphic function of finite order_ , Rev. Mat. Iberoamericana 11 (1995), no. 2, 355–373.
  * [Ber93] Walter Bergweiler, _Iteration of meromorphic functions_ , Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151–188.
  * [Ber13] François Berteloot, _Bifurcation currents in holomorphic families of rational maps_ , Pluripotential theory, Lecture Notes in Math., vol. 2075, Springer, Heidelberg, 2013, pp. 1–93. MR 3089068
  * [BF10] R. Berenguel and N. Fagella, _An entire transcendental family with a persistent siegel disc_ , J. Diff. Eq. and App. 16 (2010), 523–553.
  * [BF14] Bodil Branner and Núria Fagella, _Quasiconformal surgery in holomorphic dynamics_ , Cambridge Studies in Advanced Mathematics, vol. 141, Cambridge University Press, Cambridge, 2014, With contributions by Xavier Buff, Shaun Bullett, Adam L. Epstein, Peter Haïssinsky, Christian Henriksen, Carsten L. Petersen, Kevin M. Pilgrim, Tan Lei and Michael Yampolsky. MR 3445628
  * [BK12] Walter Bergweiler and Janina Kotus, _On the Hausdorff dimension of the escaping set of certain meromorphic functions_ , Trans. Amer. Math. Soc. 364 (2012), no. 10, 5369–5394. MR 2931332
  * [BKY91a] Irvine N. Baker, Janina Kotus, and Lü Yinian, _Iterates of meromorphic functions. I_ , Ergodic Theory Dynam. Systems 11 (1991), no. 2, 241–248.
  * [BKY91b] , _Iterates of meromorphic functions. III. Preperiodic domains_ , Ergodic Theory Dynam. Systems 11 (1991), no. 4, 603–618.
  * [BKY92] I. N. Baker, J. Kotus, and Lü Yinian, _Iterates of meromorphic functions. IV. Critically finite functions_ , Results Math. 22 (1992), no. 3-4, 651–656. MR 1189754
  * [Bol99] A. Bolsch, _Periodic Fatou components of meromorphic functions_ , Bull. London Math. Soc. 31 (1999), no. 5, 543–555. MR 1703869 (2000e:30046)
  * [BR75] I. N. Baker and P. Rippon, _Iteration of exponential functions_ , Ann. Acad. Sci. Fenn. Math. Ser. A 1 (1975), no. 2, 277–283.
  * [CG93] Lennart Carleson and Theodore W. Gamelin, _Complex dynamics_ , Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383 (94h:30033)
  * [CJK17] T. Chen, Y. Jiang, and Linda Keen, _Bounded geometry and characterization of some transcendental maps._ , Indiana J. of Math. 66 (2017), no. 5, 1537–1571. MR MR3718435
  * [CJK18] , _Cycle doubling, merging and renormalization in the Tangent family._ , Conformal Geometry and Dynamical Systems. (2018), To appear, arxiv:1708.01808.
  * [CK] T. Chen and L. Keen, _Dynamics of generalized Nevanlinna functions_ , arxiv: 1805.10974.
  * [DB00] Adrien Douady and X. Buff, _Le théorème d’intégrabilité des structures presque complexes_ , The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 307–324.
  * [Den13] A. Deniz, _Entire transcendental maps with two singular values_ , Ph.D. thesis, Roskilde University / Universitat de Barcelona, 2013.
  * [DF08] R. Dujardin and C. Favre, _Distribution of rational maps with a preperiodic critical point_ , Am. J. of Math. 130 (2008), no. 4, 979–1032.
  * [DFJ02] R. L. Devaney, N. Fagella, and X. Jarque, _Hyperbolic components of the complex exponential family_ , Fundamenta Mathematicae 174 (2002), 193–215.
  * [DK88] R. L. Devaney and L. Keen, _Dynamics of tangent_ , Dynamical Systems, Proceedings, University of Maryland, Springer-Verlag Lecture Notes in Mathematics. 1342 (1988), 105–111.
  * [Duj14] Romain Dujardin, _Bifurcation currents and equidistribution in parameter space_ , Frontiers in complex dynamics, Princeton Math. Ser., vol. 51, Princeton Univ. Press, Princeton, NJ, 2014, pp. 515–566. MR 3289920
  * [EG09] Alexandre Eremenko and Andrei Gabrielov, _Analytic continuation of eigenvalues of a quartic oscillator_ , Comm. Math. Phys. 287 (2009), no. 2, 431–457. MR 2481745 (2010b:34189)
  * [EL92] A. È. Erëmenko and M. Yu. Lyubich, _Dynamical properties of some classes of entire functions_ , Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020.
  * [Gau12] Thomas Gauthier, _Strong bifurcation loci of full Hausdorff dimension_ , Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), no. 6, 947–984 (2013). MR 3075109
  * [GK86] Lisa R. Goldberg and Linda Keen, _A finiteness theorem for a dynamical class of entire functions_ , Ergodic Theory Dynam. Systems 6 (1986), no. 2, 183–192.
  * [GK08] Piotr Galazka and Janina Kotus, _The straightening theorem for tangent-like maps_ , Pacific J. Math. 237 (2008), no. 1, 77–85. MR 2415208
  * [Hei57] Maurice Heins, _Asymptotic spots of entire and meromorphic functions_ , Ann. of Math. (2) 66 (1957), 430–439. MR 0094457 (20 #975)
  * [Her98] M. E. Herring, _Mapping properties of Fatou components_ , Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 2, 263–274. MR 1642181 (2000b:30032)
  * [Hil76] Einar Hille, _Ordinary differential equations in the complex domain_ , Wiley-Interscience, New York-London-Sydney, 1976, Pure and Applied Mathematics. MR 0499382
  * [Ive06] F. Iversen, _Recherches syr les fonctions inverses_ , Comptes Rendus 143 (1906), 877–879.
  * [Ive14] Felix Iversen, _Recherches sur les fonctions inverses_ , Ph.D. thesis, Helsingfors, 1914.
  * [KK97] Linda Keen and Janina Kotus, _Dynamics of the family $\lambda\tan z$_, Conformal geometry and dynamics 1 (1997), 28–57.
  * [KY06] Linda Keen and Shenglan Yuan, _Parabolic perturbation of the family $\lambda\tan z$_, Complex dynamics, Contemp. Math., vol. 396, Amer. Math. Soc., Providence, RI, 2006, pp. 115–128. MR 2209090
  * [McM94] Curtis T. McMullen, _Complex dynamics and renormalization_ , Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994\. MR 1312365 (96b:58097)
  * [Nev32] Rolf Nevanlinna, _Über Riemannsche Flächen mit endlich vielen Windungspunkten_ , Acta Math. 58 (1932), no. 1, 295–373. MR 1555350
  * [Nev70] , _Analytic functions_ , Translated from the second German edition by Phillip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. MR 0279280 (43 #5003)
  * [Pom92] Christian Pommerenke, _Boundary behaviour of conformal maps_ , Springer Verlag, 1992.
  * [RG03] Lasse Rempe-Guillen, _Dynamics of exponential maps_ , Ph.D. thesis, Christian-Albrechts-Universität Kiel, 2003.
  * [RS99] P. J. Rippon and G. M. Stallard, _Iteration of a class of hyperbolic meromorphic functions_ , Proc. Amer. Math. Soc. 127 (1999), no. 11, 3251–3258. MR 1610785
  * [Sch03] D. Schleicher, _Attracting dynamics of exponential maps_ , Ann. Acad. Sci. Fenn. Math. 28 (2003), 3–34.
  * [Sul85] Dennis Sullivan, _Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains_ , Ann. of Math. (2) 122 (1985), no. 3, 401–418.
  * [Tan01] Masahiko Taniguchi, _Explicit representation of structurally finite entire functions_ , Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 4, 68–70. MR 1829377
  * [Zak10] Saeed Zakeri, _On Siegel disks of a class of entire maps_ , Duke Math. J. 152 (2010), no. 3, 481–532. MR 2654221
  * [Zhe10] Jianhua Zheng, _Value distribution of meromorphic functions_ , Tsinghua University Press, Beijing; Springer, Heidelberg, 2010. MR 2757285 (2012d:30002)