# Matings of cubic polynomials with a fixed critical point. Part II:
Sufficiency of the limb condition

Thomas Sharland

###### Abstract.

In this article, a sequel to [13], we provide a combinatorial sufficient (and
conjecturally, necessary) condition (called the limb condition) for the mating
of two postcritically finite polynomials in $\mathcal{S}_{1}$ to be
obstructed. To do this, we study the rotation sets associated to the parameter
limbs in the connectedness locus of $\mathcal{S}_{1}$, which allows us to
determine when there exist ray classes in the formal mating which contain a
closed loop. We give a proof of the necessity of the limb condition for a
particular subset of postcritically finite maps in $\mathcal{S}_{1}$. Many
examples are given to illustrate the results of the paper.

###### Key words and phrases:

Complex Dynamics, Matings, Cubic Polynomials, Rotation Sets

###### 2010 Mathematics Subject Classification:

37F10,37F20,37F44

##  1\. Introduction

In this article we discuss the mating of two cubic polynomials with a fixed
critical point. Informally, the mating construction allows us to take two
complex polynomials $f$ and $g$ (along with their filled Julia sets $K(f)$ and
$K(g)$) and paste them together to construct branched covering of the sphere.
If this branched covering (or a suitable modification of it) is (Thurston)
equivalent to a rational map, we say that $f$ and $g$ are mateable.

In the quadratic case, it was shown through work of Rees, Shishikura and Tan
[16] that two postcritically finite polynomials with locally connected Julia
sets were mateable if and only if they do not belong to conjugate limbs of the
Mandelbrot set. In this case, the obstructions obtained are topological; the
space obtained from pasting together the filled Julia sets fails to be a
sphere. In other words, two such polynomials are mateable if and only if they
are topologically mateable.

However, Shishikura and Tan [15] showed that there exist cubic polynomials
which are topologically mateable, but the resulting branched covering fails to
be equivalent to a rational map. This indicates that the degree $3$ (and
higher) case is more complicated. Restricting attention to $\mathcal{S}_{1}$,
the space of cubic polynomials with a fixed critical point, the present author
showed in [13] that any obstruction to the mating of postcritically finite
polynomials in $\mathcal{S}_{1}$ is topological, paralleling the quadratic
case. However, in contrast to the quadratic case, we will prove the following.

######  Theorem 1.1.

There exist postcritically finite polynomials $F_{a}$, $F_{b}$ in
$\mathcal{S}_{1}$ which do not lie in conjugate limbs of $\mathcal{S}_{1}$ but
for which $F_{a}$ and $F_{b}$ are not mateable.

Thus the necessity of the conjugate limb condition does not carry over from
quadratic polynomials to the space $\mathcal{S}_{1}$. We provide a sufficient
(combinatorial) condition for two postcritically finite polynomials in
$\mathcal{S}_{1}$ to fail to be mateable (Proposition 3.7). After this, we
show that this condition can be satisfied even when the two polynomials do not
belong to conjugate limbs of $\mathcal{S}_{1}$ (Theorem 4.2). We conjecture
that the given sufficient condition is also necessary, and give a proof of
necessity in a special case (Theorem 4.5). The paper also contains a
discussion of the combinatorics of limbs in the space $\mathcal{S}_{1}$, which
is used to provide the combinatorial description mentioned above. Examples are
given to illuminate the main results.

##  2\. Preliminaries

We presume some familiarity with the dynamics of rational functions on the
Riemann sphere: for background, the reader is referred to [10].

###  2.1. Rotation sets

In this section we discuss the results we need on rotation sets. Much of the
background material is taken from [17] (see also [2]). Let
$\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the circle. Any continuous
$g\colon\mathbb{T}\to\mathbb{T}$ lifts, via the canonical projection, to a
continuous $G\colon\mathbb{R}\to\mathbb{R}$ which satisfies $G(x+1)=G(x)+d$
for some $d\in\mathbb{Z}$ called the degree of $g$. Furthermore, we say $g$ is
monotone if $G$ is monotone.

Given a continuous non-decreasing map $G\colon\mathbb{R}\to\mathbb{R}$ such
that $G(x+1)=G(x)+1$ for all $x\in\mathbb{R}$, the limit

$\tau(G)=\lim_{n\to\infty}\frac{G^{\circ n}(x)-x}{n}$

is independent of $x$, and is called the translation number of $G$. The
rotation number $\rho(g)$ of a degree $1$ monotone map
$g\colon\mathbb{T}\to\mathbb{T}$ is then defined as the residue class of
$\tau(G)$ modulo $\mathbb{Z}$, where $G$ is any lift of $g$.

For $d\geq 2$, we define $m_{d}\colon\mathbb{T}\to\mathbb{T}$ by

$m_{d}(t)=dt\pmod{\mathbb{Z}}.$

This paper will focus on $m_{2}$ and $m_{3}$, since both play an important
role in the dynamics of maps in $\mathcal{S}_{1}$. A non-empty compact set
$X\subseteq\mathbb{T}$ is called a rotation set if $m_{d}(X)=X$ and the
restriction $m_{d}|_{X}$ can be extended to a degree $1$ monotone map on
$\mathbb{T}$. Since any two such extensions $g$ and $h$ agree on $X$, it
follows from Theorem 1.8 of [17] that $\rho(g)=\rho(h)$. We denote this common
rotation number by $\rho(X)$ (or just $\rho$ if the set $X$ is clear in the
context) and call it the rotation number of $X$.

Given a rotation set $X$ for $m_{d}$, we call a component of
$\mathbb{T}\setminus X$ a gap of $X$. A gap is called a major gap if its
length $\ell$ satisfies $\ell\geq 1/d$. The multiplicity of a major gap is the
integer part of $d\ell$. If a gap is not major, then it is called a minor gap.
A minor gap maps homeomorphically onto its image (which must also be a gap of
$X$) under $m_{d}$. We collect some important consequences of the above.

######  Proposition 2.1 (Theorem 2.9, [17]).

A rotation set is uniquely determined by its major gaps.

######  Proposition 2.2 (Corollary 2.16, [17]).

Let $X\subset\mathbb{T}$ be a non-empty $m_{d}$-invariant set. Then $X$ is a
rotation set if and only if $\mathbb{T}\setminus X$ contains $d-1$ disjoint
intervals, each of length $1/d$.

We will refer to a periodic orbit under $m_{d}$ as a cycle. When $d=3$, a
rotation set can contain at most two cycles $C$ and $C^{\prime}$, each with
the same rotation number $\rho$ [17]. Furthermore, these cycles are
_superlinked_ in the sense that each gap of $C$ meets $C^{\prime}$, and each
gap of $C^{\prime}$ meets $C$. If $X$ is rotation set under $m_{3}$ with
$\rho(X)\in\mathbb{Q}$ made up of the union of two cycles, we will define the
_signature_ $s(X)$ to be vector $(s_{1},s_{2})$, where $s_{1}$ is the number
of elements of $X$ in $[0,1/2)$ and $s_{2}$ is the number of elements of $X$.
The following, a special case of a theorem of Goldberg, gives an important
specification of which signatures can occur when $X$ is the union of two
cycles under $m_{3}$.

######  Proposition 2.3 ([8]).

Suppose $0\leq p/q<1$ is a reduced fraction and $0\leq s_{1}\leq s_{2}=2q$.
Then there exists a rotation set under $m_{3}$ with rotation number $p/q$ and
signature $s(X)=(s_{1},s_{2})$ made up of the union of two cycles if and only
if $s_{1}$ is odd. Furthermore, if this rotation set exists, then it is
unique.

We remark that for $\rho=0$, the unique $m_{3}$-rotation set made up of the
union of two cycles is $X=\\{0,1/2\\}$, since $0$ and $1/2$ are the two fixed
points of $m_{3}$.

###  2.2. Cubic Polynomials with a fixed critical point

We now discuss some preliminaries on the dynamics of cubic polynomials. Much
of the material we need is from [11]. Denote by $\mathcal{S}_{1}$ the space of
cubic polynomials with a fixed critical point. Under a holomorphic change of
coordinates, any such polynomial is of the form

$F_{a}(z)=z^{3}-3a^{2}z+2a^{3}+a.$

Under this parameterisation $a$ is the fixed critical point and $-a$ is the
free critical point of $F_{a}$. We will call $v_{a}=F_{a}(-a)$ the free
critical value of $F_{a}$. The normal form above provides a conformal
isomorphism $\mathcal{S}_{1}\cong\mathbb{C}$ via $F_{a}\mapsto a$. The
connectedness locus $\mathcal{C}(\mathcal{S}_{1})$ is the set of polynomials
$F_{a}$ for which the Julia set is connected; equivalently, it is the set of
polynomials $F_{a}$ for which the free critical point $-a$ has bounded orbit.
For any map $F_{a}\in\mathcal{S}_{1}$, we denote by $U_{a}$ the Fatou
component containing the fixed critical point $a$ (thus $U_{a}$ is the
immediate basin of $a$). The main hyperbolic component $\mathcal{H}_{0}$ is
the hyperbolic component in $\mathcal{S}_{1}$ which contains $F_{0}$, the
function $z\mapsto z^{3}$ which has a double critical fixed point.

####  2.2.1. Structure of filled Julia sets

For all $F_{a}\in\mathcal{H}_{0}$, the Julia set of $F_{a}$ is the simple
closed curve $\partial U_{a}$ and $-a\in U_{a}$. Now suppose
$F=F_{a}\in\mathcal{C}(\mathcal{S}_{1})\setminus\mathcal{H}_{0}$. Then, by
Böttcher’s theorem, there exists a conformal isomorphism
$\text{B\"{o}t}=\text{B\"{o}t}_{a}\colon U_{a}\to\mathbb{D}$ given by

$\text{B\"{o}t}(F(z))=(\text{B\"{o}t}(z))^{2}.$

Since $\partial U_{a}$ is a simple closed curve, then by Carathéodory’s
Theorem, there exists a continuous extension of Böt to the boundary $\partial
U_{a}$. By abuse of notation, we will also use Böt for this homeomorphism from
$\overline{U}_{a}$ to $\overline{\mathbb{D}}$. Thus for each point
$z\in\partial U_{a}$, there exists a unique $t\in\mathbb{T}$ such that
$z=\text{B\"{o}t}(e^{2\pi it})$. For brevity we will write $\beta(t)$ instead
of $\text{B\"{o}t}(e^{2\pi it})$ and call $t$ the internal angle of $z$. The
following theorem then describes the structure of the (filled) Julia set of a
map in $\mathcal{C}(\mathcal{S}_{1})\setminus\mathcal{H}_{0}$.

######  Theorem 2.4 (Theorem 2.1, [11]).

If $F_{a}\in\mathcal{C}(\mathcal{S}_{1})$ with $-a\notin U_{a}$, then the
filled Julia set $K(F_{a})$ is equal to the union of the topological disk
$\overline{U}_{a}$ with a collection of compact connected sets $K_{t}$, where
$t$ ranges over a countable subset $\Lambda\subseteq\mathbb{T}$. Furthermore

  1. (i)

The $K_{t}$ are pairwise disjoint, and each $K_{t}$ intersects
$\overline{U}_{a}$ in the single boundary point $\beta(t)$.

  2. (ii)

There is a distinguished element $t_{0}\in\Lambda$ such that the free critical
point $-a$ belongs to $K_{t_{0}}$.

  3. (iii)

If $t\in\mathbb{T}$, then $t\in\Lambda$ if and only if there exists some
$n\geq 0$ such that $2^{n}t\equiv t_{0}\pmod{\mathbb{Z}}$.

  4. (iv)

For $t\not\equiv t_{0}\pmod{\mathbb{Z}}$, the map $F_{a}$ carries $K_{t}$
homeomorphically onto $K_{2t}$. However, $F_{a}$ maps $K_{t_{0}}$ onto the
entire filled Julia set $K(F_{a})$.

We call $K_{t}$ a (dynamical) limb for the map $F$; the point $\beta(t)$ where
$K_{t}$ meets $U_{a}$ is called the root point of the limb $K_{t}$. The set
$K_{t_{0}}$ is called the critical limb of $F_{a}$.

Suppose a polynomial $F$ has a connected filled Julia set $K(F)$. By
Böttcher’s theorem, there exists a conformal isomorphism
$\text{B\"{o}t}_{F}\colon\widehat{\mathbb{C}}\setminus\overline{\mathbb{D}}\to\widehat{\mathbb{C}}\setminus
K(F)$ which conjugates the map $z\mapsto z^{d}$ on
$\widehat{\mathbb{C}}\setminus\overline{\mathbb{D}}$ to the map $F$ on
$\widehat{\mathbb{C}}\setminus K(F)$. Furthermore, there is a unique choice of
$\text{B\"{o}t}_{F}$ which is tangent to the identity at infinity.

######  Definition 2.5.

The external ray of angle $t$ for the polynomial $F$ is

$R_{F}(t)=\text{B\"{o}t}_{F}(r_{t})$

where $r_{t}=\\{r\exp(2\pi it):r>1\\}\subset\mathbb{C}\setminus\mathbb{D}$ is
the corresponding radial line.

We will often use the notation $R_{a}(t)$ instead of $R_{F_{a}}(t)$ for a
polynomial $F_{a}\in\mathcal{S}_{1}$. If the choice of polynomial is clear, we
sometimes drop the subscript altogether. By Carathéodory’s theorem, the filled
Julia set $K(F)$ (or equivalently, the Julia set $J(F)$) of $F$ is locally
connected if and only if $\text{B\"{o}t}_{F}$ extends continuously to the unit
circle. The induced semiconjugacy $\gamma_{F}\colon\mathbb{R}/\mathbb{Z}\to
J(F)$ is called the Carathéodory loop. The point $\gamma_{F}(t)\in J(F)$ is
called the landing point of the ray $R_{F}(t)$ and a point $z$ of the Julia
set is called biaccessible if it is the landing point of two external rays.
That is, there exists $s\neq t$ such that $z=\gamma_{F}(s)=\gamma_{F}(t)$.

####  2.2.2. Parameter Limbs

We will also need the notion of limbs in the parameter space. For a map
$F_{a}\in\mathcal{H}_{0}$, then similarly to the above, there exists a unique
conformal isomorphism Böt on a neighbourhood of $a$ conjugating $F_{a}$ to the
map $w\mapsto w^{2}$. By Lemma 3.6 in [11], Böt can be analytically continued
to a well-defined holomorphic function on a neighbourhood of the free critical
point $-a$. We may then define a holomorphic function
$\phi\colon\mathcal{H}_{0}\to\mathbb{D}$ by $\phi(a)=\text{B\"{o}t}_{a}(-a)$.
This map $\phi$ then extends to a continuous map
$\phi\colon\mathcal{C}(\mathcal{S}_{1})\to\overline{\mathbb{D}}$ such that if
$F\in\mathcal{C}(\mathcal{S}_{1})\setminus\mathcal{H}_{0}$ then
$\phi(F)=e^{2\pi it_{0}}$, where $t_{0}\in[0,1)$ is the internal argument of
the critical limb of $F$.

Let $t_{0}\in(0,1)$. We define the (parameter) limb $\mathcal{L}_{t_{0}}^{+}$
to be the set

$\mathcal{L}_{t_{0}}^{+}=\\{F_{a}\in\mathcal{C}(\mathcal{S}_{1})\mid\phi(F)=e^{2\pi
it_{0}}\text{ and }\mathrm{Re}(a)>0\\}$

and analogously

$\mathcal{L}_{t_{0}}^{-}=\\{F_{a}\in\mathcal{C}(\mathcal{S}_{1})\mid\phi(F)=e^{2\pi
it_{0}}\text{ and }\mathrm{Re}(a)<0\\}.$

For $t_{0}=0$ we define
$\mathcal{L}_{0}^{+}=\\{F_{a}\in\mathcal{C}(\mathcal{S}_{1})\mid\phi(F)=e^{2\pi
i0}\text{ and }\mathrm{Im}(a)<0\\}$ and
$\mathcal{L}_{0}^{-}=\\{F_{a}\in\mathcal{C}(\mathcal{S}_{1})\mid\phi(F)=e^{2\pi
i0}\text{ and }\mathrm{Im}(a)>0\\}$. Note that by the $180^{\circ}$ rotational
symmetry of $\mathcal{S}_{1}$, we have $F_{a}\in\mathcal{L}_{t_{0}}^{+}$ if
and only if $F_{-a}\in\mathcal{L}_{t_{0}}^{-}$. The root point of a limb
$\mathcal{L}_{t_{0}}^{\pm}$ is defined to be the unique point of
$\mathcal{L}_{t_{0}}^{\pm}\cap\partial\mathcal{H}_{0}$.

In much of this paper, we will have to deal with the special case $t_{0}=0$
separately from other periodic internal arguments. Fortunately, most of the
time the case for $t_{0}$ can be proved by inspection. When such a
consideration is made, it will be clearly indicated.

We will also need the notion of parameter rays in the complement of the
connectedness locus. They are closely related to the concept of external rays,
as the following definition shows.

######  Definition 2.6.

We say a map $F_{a}\in\mathcal{S}_{1}$ belongs to the parameter ray
$\mathcal{R}(\theta)$ if the external ray $R_{a}(\theta)$ passes through the
cocritical point $2a$ in the dynamical plane of $F_{a}$.

If $\theta$ is $m_{3}$-periodic, then the rays $\mathcal{R}(\theta-1/3)$ and
$\mathcal{R}(\theta+1/3)$ land on a parabolic map. In this case the angles
$\theta-1/3$ and $\theta+1/3$ are called _co-periodic_. The following
properties of parameter limbs come from [11] (Compare also [4] for a more in-
depth discussion). See Figure 1 for some examples of parameter limbs.

Figure 1. The connectedness locus of $\mathcal{S}_{1}$. Internal rays of the
main hyperbolic component $\mathcal{H}_{0}$ are shown, as well as some
associated limbs.

######  Proposition 2.7.

Suppose $t_{0}$ is periodic of period $q$ under $m_{2}$. Then there are two
external parameter rays $\mathcal{R}(\theta)$ and
$\mathcal{R}(\theta^{\prime})$, where $\theta=\frac{i}{3(3^{q}-1)}$ and
$\theta^{\prime}=\frac{i+1}{3(3^{q}-1)}$, which land at the root point of
$\mathcal{L}=\mathcal{L}_{t_{0}}^{\pm}$. Furthermore, for any map
$F_{a}\in\mathcal{L}$, the internal ray of angle $t_{0}+\frac{1}{2}$ in
$U_{a}$, as well as the external rays $R_{a}(\theta)$ and
$R_{a}(\theta^{\prime})$ land at a common pre-periodic point in $J(F_{a})$.

The above result implies the following.

######  Corollary 2.8.

Let $\theta=\frac{i}{3(3^{q}-1)}$ and
$\theta^{\prime}=\frac{i+1}{3(3^{q}-1)}$. If the two parameter angles
$\mathcal{R}(\theta)$ and $\mathcal{R}(\theta^{\prime})$ land at the root
point of $\mathcal{L}=\mathcal{L}_{t_{0}}^{\pm}$, then $R_{a}(\theta+1/3)$ and
$R_{a}(\theta^{\prime}-1/3)$ land at the root of the critical limb of any map
$F_{a}\in\mathcal{L}$.

###### Proof.

Let $F_{a}\in\mathcal{L}$. The landing point $z$ of the internal ray of angle
$t_{0}+\frac{1}{2}$ in $U_{a}$ has the same image as the landing point $r$ of
the internal ray of angle $t_{0}$ in $U_{a}$. By definition, $r$ is the root
point of the critical limb of $F_{a}$. Since by Proposition 2.7
$R_{a}(\theta)$ and $R_{a}(\theta^{\prime})$ land at $z$, then
$R_{a}(3\theta)$ and $R_{a}(3\theta^{\prime})$ land on $F_{a}(z)=F_{a}(r)$.
Thus one of the rays $R_{a}(\theta+1/3)$ and $R_{a}(\theta-1/3)$ must land on
$r$, and similarly one of the rays $R_{a}(\theta^{\prime}+1/3)$ and
$R_{a}(\theta^{\prime}-1/3)$ must land on $r$. But since the critical limb
maps onto the whole filled Julia set, the difference between the angles of the
rays landing on $r$ must be greater than $1/3$. But the only choice of angles
from the above options which achieve this are $\theta+1/3$ and
$\theta^{\prime}-1/3$. Thus $R_{a}(\theta+1/3)$ and
$R_{a}(\theta^{\prime}-1/3)$ land on $r$. ∎

In the case where $t_{0}$ is strictly preperiodic under $m_{2}$, the limb
$\mathcal{L}_{t_{0}}^{\pm}$ consists of a single function
$F_{a}\in\partial\mathcal{H}_{0}$, which is the landing point of exactly one
parameter ray. Furthermore, it follows from Theorem 2.4 that for this map
$F_{a}$, no periodic point on the boundary of $U_{a}$ is the landing point of
more than one external ray. Indeed, the Hubbard tree (the convex hull in the
Julia set of the critical orbits) for $F_{a}$ is contained in the closure of
$U_{a}$, and since all biaccessible periodic points belong to the Hubbard
tree, we see that $F_{a}$ has no biaccessible periodic points in its Julia
set.

###  2.3. Matings of postcritically finite polynomials

In this section we outline the mating construction.

####  2.3.1. Formal Mating

Let $f$ and $g$ be monic degree $d$ polynomials. We define

$\text{\textcopyright}=\mathbb{C}\cup\\{\infty\cdot e^{2\pi
it}:t\in\mathbb{R}/\mathbb{Z}\\},$

the complex plane compactified with the circle of directions at infinity. We
then continuously extend the two polynomials to the circle at infinity by
defining

$f(\infty\cdot e^{2\pi it})=\infty\cdot e^{2d\pi it}\quad\text{and}\quad
g(\infty\cdot e^{2\pi it})=\infty\cdot e^{2d\pi it}.$

Label this extended dynamical plane of $f$ (respectively $g$) by
$\text{\textcopyright}_{f}$ (respectively $\text{\textcopyright}_{g}$). We
create a topological $2$-sphere $\Sigma_{f,g}$ by gluing the two extended
planes together along the circle at infinity:

$\Sigma_{f,g}=\text{\textcopyright}_{f}\uplus\text{\textcopyright}_{g}/\sim$

where $\sim$ is the relation which identifies the point $\infty\cdot e^{2\pi
it}\in\text{\textcopyright}_{f}$ with the point $\infty\cdot e^{-2\pi
it}\in\text{\textcopyright}_{g}$. The _formal mating_ is then defined to be
the branched covering $f\uplus g\colon\Sigma_{f,g}\to\Sigma_{f,g}$ such that

$\displaystyle f\uplus g|_{\text{\textcopyright}_{f}}\,=$
$\displaystyle\,f\quad\textrm{and}$ $\displaystyle f\uplus
g|_{\text{\textcopyright}_{g}}\,=$ $\displaystyle\,g.$

####  2.3.2. Topological Mating

Let $f$ and $g$ be two monic degree $d$ polynomials with locally connected
Julia set. First, construct the space $\Sigma_{f,g}$ as above, and define the
_ray-equivalence relation_ $\approx$ as follows. We denote by $\sim_{f}$ the
smallest equivalence relation on $\text{\textcopyright}_{f}$ such that
$x\sim_{f}y$ if and only if $x,y\in\overline{R}_{f}(t)$ for some $t$; the
equivalence relation $\sim_{g}$ is defined analogously. Now $\approx$ is the
smallest equivalence relation on $\Sigma_{f,g}$ which is generated by
$\sim_{f}$ on $\text{\textcopyright}_{f}$ and $\sim_{g}$ on
$\text{\textcopyright}_{g}$. The class $[x]$ of a point $x\in\Sigma_{f,g}$
will be called a _ray-equivalence class_.

Evidently, there is a surjection $K(f)\uplus K(g)\to\Sigma_{f,g}/\approx$
given by the composition

$K(f)\uplus
K(g)\hookrightarrow\widetilde{\mathbb{C}}_{f}\uplus\widetilde{\mathbb{C}}_{g}\to\Sigma_{f,g}\to\Sigma_{f,g}/\approx.$

Furthermore, since $x\approx y$ implies $(f\uplus g)(x)\approx(f\uplus g)(y)$,
we can uniquely define the topological mating as the map $f\perp\\!\\!\\!\perp
g$ so that

|
---|---
$\textstyle{K(f)\uplus
K(g)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\uplus
g}$$\textstyle{K(f)\uplus
K(g)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma_{f,g}/\approx\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\perp\\!\\!\\!\perp
g}$$\textstyle{\Sigma_{f,g}/\approx}$

commutes. Under favourable conditions, the topological space
$\Sigma_{f,g}/\approx$ is a topological 2-sphere, in which case we say that
$f$ and $g$ are topologically mateable. In fact, this occurs if and only if
the equivalence relation $\approx$ is of Moore-type (see [9]). That is,
$\approx$ is a non-trivial closed relation such that each equivalence class is
closed and no equivalence class separates the sphere $\Sigma_{f,g}$.

####  2.3.3. Thurston’s Theorem

Let $F\colon\Sigma\to\Sigma$ be an orientation-preserving branched self-
covering of a topological $2$-sphere. We denote by $\Omega_{F}$ the critical
set of $F$ and define

$P_{F}=\bigcup_{n>0}F^{\circ n}(\Omega_{F})$

to be the postcritical set of $F$. We say that $F$ is postcritically finite if
$|P_{F}|<\infty$. We call $F\colon\Sigma\to\Sigma$ a _Thurston map_ if it is a
postcritically finite orientation-preserving branched self-covering of a
topological $2$-sphere.

######  Definition 2.9.

Let $F\colon\Sigma\to\Sigma$ and
$\widehat{F}\colon\widehat{\Sigma}\to\widehat{\Sigma}$ be Thurston maps. An
_equivalence_ is given by a pair of orientation-preserving homeomorphisms
$(\Phi,\Psi)$ from $\Sigma$ to $\widehat{\Sigma}$ such that

  * •

$\Phi|_{P_{F}}=\Psi|_{P_{F}}$

  * •

The following diagram commutes:

$\textstyle{(\Sigma,P_{F})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{F}$$\textstyle{(\widehat{\Sigma},P_{\widehat{F}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{F}}$$\textstyle{(\Sigma,P_{F})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{(\widehat{\Sigma},P_{\widehat{F}})}$

  * •

$\Phi$ and $\Psi$ are isotopic via a family of homeomorphisms
$t\mapsto\Phi_{t}$ which is constant on $P_{F}$.

If there exists an equivalence as above, we say that $F$ and $\widehat{F}$ are
equivalent. Note that in particular, a postcritically finite rational map
$R\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ on the Riemann sphere is
a Thurston map. Hence it is natural to ask when a general Thurston map is
equivalent to a rational map.

######  Definition 2.10.

Let $F$ be a Thurston map. A _multicurve_ is a collection
$\Gamma=\\{\gamma_{1},\ldots,\gamma_{n}\\}$ of simple, closed, non-peripheral
curves such that each $\gamma_{i}\in\Gamma$ is disjoint from each other
$\gamma_{j}$ and the $\gamma_{i}$ are pairwise non-homotopic relative to
$P_{F}$. A multicurve is called $F$-_stable_ if for all $\gamma_{i}\in\Gamma$,
all the non-peripheral components of $F^{-1}(\gamma_{i})$ are homotopic
relative to $P_{F}$ to elements of $\Gamma$. If $\Gamma$ is a multicurve (not
necessarily $F$-stable), we define the non-negative matrix
$F_{\Gamma}=(f_{ij})_{n\times n}$ as follows. Let $\gamma_{i,j,\alpha}$ be the
components of $F^{-1}(\gamma_{j})$ which are homotopic to
$\gamma_{i}\in\Gamma$ in $\Sigma\setminus P_{F}$. Now define

$F_{\Gamma}(\gamma_{j})=\sum_{i,\alpha}\frac{1}{\deg
F|_{\gamma_{i,j,\alpha}}\colon\gamma_{i,j,\alpha}\to\gamma_{j}}\gamma_{i}.$

where $\deg$ denotes the degree of the map. By standard results on non-
negative matrices (see [7]), this matrix $(f_{ij})$ will have a leading non-
negative eigenvalue $\lambda$. We write $\lambda(\Gamma)$ for the leading
eigenvalue associated to the multicurve $\Gamma$.

The importance of the above is due to the following rigidity theorem. A proof
can be found in [5].

######  Theorem 2.11 (Thurston).

  1. (i)

A Thurston map $F\colon\Sigma\to\Sigma$ of degree $d\geq 2$ with hyperbolic
orbifold is equivalent to a rational map
$R\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ if and only if there are
no $F$-stable multicurves with $\lambda(\Gamma)\geq 1$.

  2. (ii)

Any Thurston equivalence of rational maps $F$ and $\widehat{F}$ with
hyperbolic orbifolds is represented by a Möbius conjugacy.

An $F$-stable multicurve with $\lambda(\Gamma)\geq 1$ is called a _(Thurston)
obstruction_. The condition that $F$ has a hyperbolic orbifold is a purely
combinatorial one; this condition can be checked by inspecting the dynamics on
the union of the critical and postcritical sets of $F$. We remark that since
the maps in this article have two fixed critical points, the respective
orbifolds are guaranteed to be hyperbolic. For further details, see [5].

We observe that the formal mating of two postcritically finite polynomials is
a Thurston map. Thus one can ask if the mating is obstructed (that is, it has
a Thurston obstruction).

######  Definition 2.12.

A multicurve $\Gamma=\\{\gamma_{1},\ldots,\gamma_{n}\\}$ is a Levy cycle if
for each $i=1,\ldots,n$, the curve $\gamma_{i-1}$ (or $\gamma_{n}$ if $i=1$)
is homotopic to some component $\gamma_{i}^{\prime}$ of $F^{-1}(\gamma_{i})$
(rel $P_{F}$) and the map $F\colon\gamma_{i}^{\prime}\to\gamma_{i}$ is a
homeomorphism. Furthermore

  * •

We say $\Gamma$ is a degenerate Levy cycle if the connected components of
$\Sigma\setminus\bigcup_{i=1}^{n}\gamma_{i}$ are $D_{1},\ldots,D_{n},C$, where
each $D_{i}$ is a disk and moreover for each $i$ the preimage
$F^{-1}(D_{i+1})$ contains a component $D_{i+1}^{\prime}$ which is isotopic to
$D_{i}$ relative to $P_{F}$ and is such that $F\colon D_{i+1}^{\prime}\to
D_{i+1}$ is a homeomorphism.

  * •

A degenerate Levy cycle is called a removable Levy cycle if in addition, for
all $k\geq 1$ and all $i$, the components of $F^{-k}(D_{i})$ are disks.

There is a close relationship between Levy cycles and periodic cycles of ray
cycles, as given in the following theorem of Tan.

######  Proposition 2.13 ([15, 16]).

Let $F=f\uplus g$. Then every Levy cycle $\Gamma$ of $F$ corresponds to a
unique cycle of ray classes $[x_{0}],[x_{1}],\dotsc,[x_{n-1}]$ (called the
limit set of $\Gamma$) and finitely connected $X_{i}\subset[x_{i}]$ for each
$i$ such that $F\colon[x_{i}]\to[x_{i+1}]$ and $F\colon X_{i}\to X_{i+1}$ are
homeomorphisms. Moreover, $\Gamma$ is isotopic to a tubular neighbourhood of
$\bigcup_{i=0}^{n-1}X_{i}$. In particular

  1. (i)

If $\Gamma$ is degenerate, then each $X_{i}$ is a tree and contains at least
two elements of $P_{F}$.

  2. (ii)

If $\Gamma$ is not degenerate, each $X_{i}$ contains a closed loop.

Conversely, if $[x]$ is a periodic ray class which contains a closed loop or
at least two postcritical points, then each boundary curve of a tubular
neighbourhood of $[x]$ generates a Levy cycle.

######  Corollary 2.14.

Let $\Gamma$ be a non-removable degenerate Levy cycle of the mating $F=f\uplus
g$ of postcritically finite polynomials. Then there exists a ray class under
the mating which contains two critical values of $F$.

###### Proof.

First, note that by the Riemann-Hurwitz formula, if $D$ is a disk such that
$F^{-1}(D)$ is not a disk, then $D$ must contain (at least) two critical
values. Since $\Gamma$ is not removable, there exists a disk component
$\Delta$ of $\Sigma\setminus\Gamma$ and some $k$ such that $F^{-k}(\Delta)$
contains a non-disk component $C$.

By Proposition 2.13, there exists some $i$ such that $\partial\Delta$ is
isotopic to a tubular neighbourhood of $X_{i}$ which is a subset of a periodic
ray class $[x_{i}]$. Thus this tubular neighbourhood (isotopically) contains
$\Delta$. We claim that there exists a preimage of $[x_{i}]$ which contains
two critical values of $F$. If not, then no preimage of $X_{i}\subset[x_{i}]$
can contain two critical values of $F$. But by Riemann-Hurwitz, all preimages
of a tubular neighbourhood of $X_{i}$ would have to be disks. But then all
preimages of $\Delta$ would also be disks, which is a contradiction. Thus
$[x_{i}]$ must contain two critical values of $F$. ∎

We remark that the converse of the corollary is not true: it is possible that
two critical values belong to the same ray class but the mating is not
obstructed, see Example 4.7. The following theorem from [13] indicates that
all obstructions to matings of pairs of postcritically finite polynomials in
$\mathcal{S}_{1}$ are topological. This is not true for general cubic
polynomials: in [15], the authors constructed a mating of two postcritically
finite cubic polynomials which was obstructed, but this obstruction was not a
Levy cycle. Both the polynomials had critical points of period $3$.

######  Theorem 2.15.

Let $f$ and $g$ be monic postcritically finite polynomials in
$\mathcal{S}_{1}$. Then any obstruction to the formal mating $f\uplus g$
contains a Levy cycle.

######  Corollary 2.16.

Let $t_{0}$ be strictly preperiodic under $m_{2}$ and suppose $F_{a}$ is the
unique element of the limb $\mathcal{L}_{t_{0}}^{\pm}$. If $F_{b}$ is a
postcritically finite polynomial in $\mathcal{S}_{1}$ and the mating
$F_{a}\uplus F_{b}$ is obstructed, then this obstruction contains a degenerate
Levy cycle.

###### Proof.

By Theorem 2.15, any obstruction of the mating contains a Levy cycle. By
Proposition 2.13, if the obstruction was non-degenerate Levy cycle, then there
would have to be a periodic ray cycle containing a closed loop. But $F_{a}$
has no biaccessible periodic points, so this is impossible. ∎

####  2.3.4. Degenerate Matings

We now discuss a technique due to Shishikura [14] which allows us to modify a
branched covering with removable Levy cycles so that it is no longer
obstructed. This allows us to deal with cases where two polynomials are
topologically mateable, but the formal mating is obstructed. Let $F$ be a
Thurston map. We need to set up a partial ordering on the (possibly empty) set
$\mathfrak{R}(F)$ of isotopy classes of removable Levy cycles for $F$. First,
note that if $\Lambda$ is a removable Levy cycle, then the set
$\Sigma\setminus\Lambda$ consists of $n$ disks $D_{1},\dotsc,D_{n}$ and an
additional component $C$, such that for each $j$, the preimage $F^{-1}(D_{j})$
has a component $D_{j}^{\prime}$ isotopic to $D_{j-1}$ and the map $F\colon
D_{j}^{\prime}\to D_{j}$ is a homeomorphism. Also, by definition, for each
$k$, all components of $F^{-k}(D_{1})$ are disks. Define
$\mathcal{D}(\Lambda)=\bigcup_{j=1}^{n}D_{j}$.

We need to construct our partial order on $\mathfrak{R}(F)$. For two elements
$\lambda_{1},\lambda_{2}\in\mathfrak{R}$, we say $\lambda_{1}<\lambda_{2}$ if
there exists representatives $\Lambda_{1}$ and $\Lambda_{2}$ of $\lambda_{1}$
and $\lambda_{2}$ respectively such that
$\mathcal{D}(\Lambda_{1})\subset\mathcal{D}(\Lambda_{2})$.

Define $\mathfrak{R}^{\prime}$ to be the set of $\lambda\in\mathfrak{R}$ which
are maximal under the partial order and such that for any Levy cycle $\Gamma$,
the geometric intersection number $\lambda\cdot\Gamma=0$ (see e.g [6, p. 28]
for the definition of geometric intersection number). In the case of matings,
we have the following.

######  Proposition 2.17 ([15]).

Let $F=f\uplus g$ be the formal mating of two postcritically finite
polynomials. The set $\mathfrak{R}^{\prime}(F)$ consists of boundary curves of
tubular neighbourhoods of periodic cycles
$\\{[x_{1}],[x_{2}],\dotsc,[x_{m}]\\}$ of ray classes such that each $[x_{i}]$
contains at least two points of $P_{F}$ and no ray class in $\bigcup_{n\geq
0}F^{-n}([x_{i}])$ contains a closed loop. Furthermore if $\Sigma/\approx$ is
homeomorphic to $S^{2}$, then all Levy cycles for $F$ are removable.

The above result allows us to define (essential) mateability of two
polynomials $f$ and $g$ in $\mathcal{S}_{1}$. This adapts the definition of
mateability given in [15] (Compare [14], where this was called the degenerate
mating), which uses the notion of being _weakly equivalent_ to a rational map.
One can think of this weak equivalence as being obtained by shrinking
(simultaneously) all removable Levy cycles of the branched covering $F=f\uplus
g$ to points. If the original two polynomials were topologically mateable,
then this new branched covering will no longer have any Levy cycles.

######  Definition 2.18.

We say that two postcritically finite polynomials $f,g\in\mathcal{S}_{1}$ are
(essentially) mateable if all irreducible obstructions for $F=f\uplus g$ are
removable Levy cycles.

Since the matings of postcritically finite polynomials in $\mathcal{S}_{1}$
have hyperbolic orbifolds, the above definition is equivalent to the
topological mating of $f$ and $g$ being Thurston equivalent to a rational map.

##  3\. Combinatorics of Limbs

In this section, we collect together some combinatorial properties of the
parameter limbs defined in Section 2.2.2.

######  Lemma 3.1.

Let $t_{0}\neq 0$ be periodic under $m_{2}$ and suppose $F_{a}$ belongs to a
limb with internal argument $t_{0}$.

  1. (i)

$F_{a}\in\mathcal{L}^{+}_{t_{0}}$, if and only if $R_{a}(0)$ lands on
$\partial U_{a}$ and $R_{a}(1/2)$ lands on the critical limb of $F_{a}$.

  2. (ii)

$F_{a}\in\mathcal{L}^{-}_{t_{0}}$ if and only if the ray of angle $R_{a}(1/2)$
lands on $\partial U_{a}$ and $R_{a}(0)$ lands on the critical limb of
$F_{a}$.

###### Proof.

Notice that for any $F_{a}\in\mathcal{C}(\mathcal{S}_{1})$ there exists a
fixed point for $F_{a}$ on the boundary of $U_{a}$, and this point must be the
landing point of either $R_{a}(0)$ or $R_{a}(1/2)$. Thus to prove the first
claim, it suffices to show that $R_{a}(1/2)$ lands in the critical limb of
$F_{a}$ whenever $F_{a}\in\mathcal{L}_{t_{0}}^{+}$. In this case, since
$t_{0}\neq 0$, if $\theta<\theta^{\prime}$ and $\mathcal{R}(\theta)$ lands
with $\mathcal{R}(\theta^{\prime})$ at the root of the parameter limb
$\mathcal{L}^{+}_{t_{0}}$ then
$\\{\theta,\theta^{\prime}\\}\subset(5/6,1/6)\subset\mathbb{T}$. By Corollary
2.8, the $R_{a}(\theta+1/3)$ and $R_{a}(\theta^{\prime}-1/3)$ land on the root
of the critical limb of $F_{a}$. But $\theta+1/3\in(1/6,1/2)\subset\mathbb{T}$
and $\theta^{\prime}-1/3\in(1/2,5/6)\subset\mathbb{T}$, and so
$\theta+1/3<1/2<\theta^{\prime}-1/3$. Hence $R_{a}(1/2)$ must land in the
critical limb of $F_{a}$. The proof for the second case is similar. ∎

In the special case that $t_{0}=0$, the external rays of angle $0$ and $1/2$
both land on $\partial U_{a}$ at the root point of the critical limb of
$F_{a}$. Together with their landing point, these rays separate the critical
limb from the rest of Julia set.

Recall that for $F_{a}\in\mathcal{S}_{1}$, we denote by
$\beta(t)=\text{B\"{o}t}(e^{2\pi it})$ the point on $U_{a}$ which is the
landing point of the internal ray of angle $t$.

######  Definition 3.2.

Let $t_{0}\in\mathbb{Q}/\mathbb{Z}$ be periodic of period $p$ under $m_{2}$
and suppose $F_{a}$ belongs to the limb $\mathcal{L}_{t_{0}}^{\pm}$. Then the
points $\beta(t_{0}),\,\beta(2t_{0}),\,\dotsc,\,\beta(2^{p-1}t_{0})$ form a
period $p$ cycle under $F_{a}$ on the boundary of $U_{a}$, and these points
are each the landing point of two external rays. This periodic cycle will be
called the $\alpha$-_periodic cycle_ of $F$. The rotation number of the
$\alpha$-periodic cycle is defined to be the rotation number of $t_{0}$ under
angle doubling.

The terminology is chosen to draw a parallel with the notion of the
$\alpha$-periodic point of quadratic polynomials. By Proposition 2.7, this
periodic orbit persists within a limb. That is, if $\mathcal{L}_{t_{0}}^{\pm}$
is a limb with periodic internal argument, then for all maps in this limb, the
internal and external rays landing on the $\alpha$-periodic cycle have the
same angles. See Figure 2 for examples of two maps in the same limb. Thus when
we refer to the rotation number of a limb $\mathcal{L}^{\pm}_{t_{0}}$, we mean
the rotation number of the associated $\alpha$-periodic orbit. By the above
discussion, this is well-defined. An equivalent definition is to call the
$\alpha$-periodic cycle the unique periodic cycle of biaccessible points on
the boundary of $U_{a}$.

Figure 2. Two maps belonging to the limb $\mathcal{L}_{2/3}^{+}$ in
$\mathcal{S}_{1}$. The rays (drawn by hand) landing on the $\alpha$-periodic
cycle for both maps are shown. Note the angles of these rays are the same for
both maps. The critical points are indicated by $\ast$. The figures show that
$\Theta(\mathcal{L}_{2/3}^{+})=\\{\frac{1}{8},\frac{2}{8},\frac{3}{8},\frac{6}{8}\\}$.

We remark that we do not define the notion of an $\alpha$-periodic cycle when
the internal argument of the limb is strictly preperiodic. Let $t_{0}$ be
$m_{2}$-periodic and consider the limb
$\mathcal{L}=\mathcal{L}_{t_{0}}^{\pm}$. Denote by $\Theta(\mathcal{L})$ the
set of all arguments of external rays landing on the $\alpha$-periodic cycle
of the maps in $\mathcal{L}$.

######  Proposition 3.3.

Let $t_{0}$ be periodic under $m_{2}$ and $\mathcal{L}_{t_{0}}^{\pm}$ a limb
of $\mathcal{S}_{1}$.

  1. (i)

If the orbit $t_{0}$ under $m_{2}$ has rotation number $\rho$, then
$\Theta(\mathcal{L}_{t_{0}}^{\pm})$ is a rotation set under $m_{3}$, with
rotation number $\rho$. Furthermore, $\Theta(\mathcal{L}_{t_{0}}^{\pm})$ has
exactly two major gaps.

  2. (ii)

If the orbit of $t_{0}$ under $m_{2}$ is not a rotation set, then
$\Theta(\mathcal{L}_{t_{0}}^{\pm})$ is not a rotation set and has exactly one
major gap.

Furthermore, if $k$ is the period of $t_{0}$ under $m_{2}$, then for each
$0\leq i\leq k-1$, $\Theta(\mathcal{L}_{t_{0}}^{\pm})$ has a gap of length
$\frac{3^{i}}{3^{k}-1}$. In particular, the two angles $\theta$ and
$\theta^{\prime}$ for which the corresponding rays land at the root of the
critical limb of $F_{a}$ bound a major gap of length
$\frac{3^{k-1}}{3^{k}-1}$.

###### Proof.

The proof follows easily from the fact that (external) rays cannot cross and
Proposition 2.2. For part (ii), the angles of the rays landing at the root of
the critical limb bound the unique major gap. The existence of the gaps of the
prescribed lengths follows from considering the forward images of the sectors
bounded by the rays of angle $\theta$ and $\theta^{\prime}$ in Proposition
2.7. Details are left to the reader. ∎

We see that if $\Theta(\mathcal{L})$ is a rotation set, then is made up of the
union of two distinct cycles under $m_{3}$, each with the same rotation number
as $\Theta(\mathcal{L})$. By Proposition 2.3, such a set is completely
described by its rotation number $p/q$ and $s_{1}(X)$, the number of elements
of $X$ in $[0,1/2)$. Furthermore, since $s_{1}$ must be odd, we get the
following.

######  Lemma 3.4 (Compare Corollary 3.15, [17]).

Given a rotation number $\rho=p/q$, there exists exactly $q$ rotation sets
with rotation number $p/q$ under $m_{3}$ which are the union of two cycles.

###### Proof.

The admissible values of $s_{1}$ are $1,3,\dotsc 2q-1$, and each is associated
to a unique rotation set. ∎

We now show that each of these $q$ rotation sets is realised as
$\Theta(\mathcal{L})$ for some limb $\mathcal{L}\subseteq\mathcal{S}_{1}$.

######  Corollary 3.5.

Suppose $A$ is a rotation set which is the union of two cycles and
$\rho(A)=p/q$. Then there exists a unique limb $\mathcal{L}^{+}_{t_{0}}$ of
$\mathcal{S}_{1}$ such that $A=\Theta(\mathcal{L}^{+}_{t_{0}})$ and a unique
limb $\mathcal{L}_{t_{0}^{\prime}}^{-}$ such that
$A=\Theta(\mathcal{L}^{-}_{t_{0}^{\prime}})$.

###### Proof.

Clearly there are exactly $q$ limbs of the form $\mathcal{L}^{+}_{t_{0}}$ for
which the internal argument $t_{0}$ belongs to the unique rotation set of
$m_{2}$ that has rotation number $p/q$. By Lemma 3.4, there are $q$ candidates
for $\Theta(\mathcal{L})$ for such a limb. So it is sufficient to prove that
if $t_{0}\neq s_{0}$, then
$\Theta(\mathcal{L}_{t_{0}}^{+})\neq\Theta(\mathcal{L}_{s_{0}}^{+})$.

Assume $t_{0}$ belongs to a $m_{2}$-rotation set $X$. If $s_{0}$ does not
belong to an $m_{2}$-rotation set, or belongs to a rotation set with a
different rotation number than $X$, then by Proposition 3.3 we must have
$\Theta(\mathcal{L}_{t_{0}}^{+})\neq\Theta(\mathcal{L}_{s_{0}}^{+})$. So
suppose $t_{0},s_{0}\in X$ but $t_{0}\neq s_{0}$. Then the parameter rays
landing at the root of $\mathcal{L}_{t_{0}}^{+}$ must be different than those
landing at the root of $\mathcal{L}_{s_{0}}^{+}$. Hence, by Corollary 2.8, the
angles of the rays landing on the root of the critical limb of maps in
$\mathcal{L}_{t_{0}}^{+}$ must be different from the angles of the rays
landing on the root of the critical limb of maps in $\mathcal{L}_{s_{0}}^{+}$.
But these angles bound a major gap for the rotation sets
$\Theta(\mathcal{L}_{t_{0}}^{+})$ and $\Theta(\mathcal{L}_{s_{0}}^{+})$
respectively. Since the two rotation sets have different major gaps, it
follows from Proposition 2.1 that
$\Theta(\mathcal{L}_{t_{0}}^{+})\neq\Theta(\mathcal{L}_{s_{0}}^{+})$.

The proof for limbs of the form $\mathcal{L}_{t_{0}}^{-}$ is similar. ∎

######  Corollary 3.6.

If $A$ is a rotation cycle under $m_{3}$ and is the union of two distinct
cycles, then there exactly two limbs of $\mathcal{S}_{1}$ for which
$A=\Theta(\mathcal{L})$.

The above result indicates that the dynamics of the limbs in $\mathcal{S}_{1}$
which have a rotation number are “combinatorially full”, in the following
sense. Given a rotation set $A$ under $m_{3}$, which consists of two periodic
cycles, then there exists a limb $\mathcal{L}\subset\mathcal{S}_{1}$ for which
$A=\Theta(\mathcal{L})$. In fact, if one considers the moduli space
$\mathcal{S}_{1}/\mathcal{I}$, where $\mathcal{I}$ is the canonical involution
$z\mapsto-z$, then there is precisely one limb corresponding to each rotation
set $A$ of the form given above. This is analogous to the case for the
Mandelbrot set; given a rotation set $A$ under $m_{2}$, there exists a limb in
the Mandelbrot set for which the external rays landing on the associated
$\alpha$-fixed point is the set $A$. However, it should be noted that, in
contrast to the Mandelbrot set, there are limbs of $\mathcal{S}_{1}$ which
don’t have an associated rotation number. It would be interesting to study the
combinatorics of the sets $\Theta(\mathcal{L})$ for these limbs. Petersen and
Zakeri have studied the combinatorics of periodic orbits under $m_{d}$ in
[12].

In what follows we will use the notation $-\Theta(\mathcal{L})$ for the set
$\\{-\theta\mid\theta\in\Theta(\mathcal{L})\\}$.

######  Proposition 3.7.

Let $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ be limbs of $\mathcal{S}_{1}$ such
that $\Theta(\mathcal{L}_{1})=-\Theta(\mathcal{L}_{2})$. Then if $F_{a}$,
$F_{b}$ are postcritically finite polynomials such that
$F_{a}\in\mathcal{L}_{1}$ and $F_{b}\in\mathcal{L}_{2}$, then $F_{a}$ and
$F_{b}$ are not mateable.

###### Proof.

Consider the ray class $R$ containing the $\alpha$-periodic cycle of $F_{a}$.
Since $\Theta(\mathcal{L}_{1})=-\Theta(\mathcal{L}_{2})$, $R$ is also contains
the $\alpha$-periodic cycle of $F_{b}$, and these points are the only elements
of $J(F_{a})\cup J(F_{b})$ belonging to $R$. The points of an
$\alpha$-periodic cycle are biaccessible, so this ray class may be thought of
a (not necessarily connected) finite graph where every vertex has (at least)
two edges landing on it. Such a graph must contain a closed loop. Thus the ray
equivalence relation separates the sphere, and so by Proposition 2.17, there
exists a non-removable Levy cycle for the mating. ∎

We will say that if two limbs $\mathcal{L}$ and $\mathcal{L}^{\prime}$ are
such that $\Theta(\mathcal{L})=-\Theta(\mathcal{L}^{\prime})$ then the pair
$\\{\mathcal{L},\mathcal{L}^{\prime}\\}$ satisfies the limb condition. Given a
limb $\mathcal{L}\subseteq\mathcal{S}_{1}$, we will say that the conjugate
limb $\overline{\mathcal{L}}$ is the set of maps $F_{a}$ such that
$F_{\overline{a}}\in\mathcal{L}$. In terms of internal arguments, if
$t_{0}\neq 0$ then the conjugate limb to $\mathcal{L}_{t_{0}}^{\pm}$ is the
limb $\mathcal{L}_{-t_{0}}^{\pm}$. For $t_{0}=0$, the limbs
$\mathcal{L}_{0}^{+}$ and $\mathcal{L}_{0}^{-}$ are mutually conjugate. In all
cases, if the parameter ray $\mathcal{R}(\theta)$ lands on the root of
$\mathcal{L}$, then $\mathcal{R}(-\theta)$ lands on the root of
$\overline{\mathcal{L}}$.

######  Lemma 3.8.

Let $\mathcal{L}$ be a limb of $\mathcal{S}_{1}$ with periodic internal angle.
Then $\Theta(\overline{\mathcal{L}})=-\Theta(\mathcal{L})$.

###### Proof.

Suppose $\mathcal{R}(\theta)$ and $\mathcal{R}(\theta^{\prime})$ (where
$\theta<\theta^{\prime}$) land on the root of $\mathcal{L}$. Then
$\mathcal{R}(-\theta)$ and $\mathcal{R}(-\theta^{\prime})$ land on the root of
$\overline{\mathcal{L}}$. Hence by Corollary 2.8, if
$F_{b}\in\overline{\mathcal{L}}$ then the rays $R_{b}(-(\theta^{\prime}-1/3))$
and $R_{b}(-(\theta+1/3))$ land on the root of the critical limb of $F_{b}$.
But this means $-(\theta^{\prime}-1/3)$ and $-(\theta+1/3)$ bound a major gap
of $\Theta(\overline{\mathcal{L}})$ and in particular
$\\{-(\theta^{\prime}-1/3),-(\theta+1/3)\\}\subseteq\Theta(\overline{\mathcal{L}})$.
Taking forward iterates under $m_{3}$, it follows that
$\Theta(\overline{\mathcal{L}})=-\Theta(\mathcal{L})$. ∎

In the special case where the internal argument is $0$, we have
$\Theta(\mathcal{L}_{0}^{+})=\Theta(\mathcal{L}_{0}^{-})=\\{0,1/2\\}$.

######  Proposition 3.9.

Let $F_{a}$ and $F_{b}$ be postcritically finite polynomials that lie in
conjugate limbs of $\mathcal{S}_{1}$. Then $F_{a}$ and $F_{b}$ are not
mateable.

###### Proof.

If $\mathcal{L}$ and $\mathcal{L}^{\prime}$ are conjugate limbs with periodic
internal argument then $\Theta(\mathcal{L})=-\Theta(\mathcal{L}^{\prime})$ and
so by Proposition 3.7, the mating of $F_{a}$ and $F_{b}$ must be obstructed.

Now suppose $t_{0}$ is strictly preperiodic and $F_{a}$ is the unique element
of $\mathcal{L}=\mathcal{L}^{\pm}_{t_{0}}$. Then $F_{a}$ is the landing point
of a parameter ray $\mathcal{R}(\theta)$, and the dynamical ray
$R_{a}(\theta)$ lands on the cocritical point $2a\in J(F_{a})$. This implies
that the rays $R_{a}(\theta+1/3)$ and $R_{a}(\theta-1/3)$ land on the free
critical point $-a\in J(F_{a})$. Similarly if $F_{b}$ is the unique map in
$\overline{\mathcal{L}}$, then the rays $R_{b}(-(\theta+1/3))$ and
$R_{b}(-(\theta-1/3))$ land on the free critical point $-b\in J(F_{b})$. But
then the ray class under the mating containing the two critical points $-a$
and $-b$ contains a loop, and so the mating is obstructed. ∎

We provide some examples showing the obstructions that occur in the previous
proposition. We use laminations (using only the leaves corresponding to the
obstruction we are considering) to illustrate the ray classes that form an
obstruction in the matings.

######  Example 3.10.

Consider a map $F_{a}$ in the parameter limb $\mathcal{L}_{4/7}^{+}$. The
parameter rays landing at the root of $\mathcal{L}_{4/7}^{+}$ have angles
$\frac{1}{78}$ and $\frac{2}{78}$, from which it follows that the rays
$R_{a}(1/26)$ and $R_{a}(2/26)$ land together in the dynamical plane of
$F_{a}$. For a map $F_{b}$ in the conjugate limb $\mathcal{L}_{3/7}^{-}$, the
rays $R_{b}(24/26)$ and $R_{b}(25/26)$ land together. Hence we have the
lamination for the mating as given in Figure 3.

Figure 3. The lamination for the mating in Example 3.10.

######  Example 3.11.

We consider a case where the internal argument of the limb is strictly
preperiodic, so that the limb consists of a single map on
$\partial\mathcal{H}_{0}$. So let $F_{a}$ be the map in
$\mathcal{L}_{5/6}^{+}$. This is the landing point of the ray
$\mathcal{R}(1/8)$, which means $R_{a}(11/24)$ and $R_{a}(19/24)$ land
together at the free critical point $-a$. For the conjugate map $F_{b}$, which
is the landing point of $\mathcal{R}(7/8)$, the rays $R_{b}(7/24)$ and
$R_{b}(13/24)$ both land on the free critical point $-b$. The ray class
containing $-a$ and $-b$ is a closed loop and so the mating is obstructed. See
Figure 4 for the lamination of the mating and the Julia set of $F_{a}$.

Figure 4. An obstructed mating for a map in a limb with strictly preperiodic
internal argument. The two free critical points belong to the same ray class,
and this ray class contains a closed loop. On the right is the Julia set for
the “inner map” in the lamination.

In the first example where the internal argument is periodic each of the ray
classes in the obstruction contain precisely one point from each of the
$\alpha$-periodic cycles of $F_{a}$ and $F_{b}$. The obstruction is thus made
up of $3$ closed loops, since the $\alpha$-periodic cycles of $F_{a}$ and
$F_{b}$ both have period $3$. Furthermore, the “smallest” loop bounds a disk
which contains both free critical values. When the internal argument is
strictly preperiodic, the ray class containing a closed loop contains the free
critical points of the polynomials.

######  Definition 3.12.

Let $t_{0}$ be $m_{2}$-periodic and $\mathcal{L}^{\pm}_{t_{0}}$ a limb in
$\mathcal{S}_{1}$. We say $\mathcal{L}^{\prime}$ is a complementary limb to
$\mathcal{L}$ if the following two conditions hold.

  1. (i)

$\mathcal{L}^{\prime}$ is not the conjugate limb to $\mathcal{L}$.

  2. (ii)

$\Theta(\mathcal{L}^{\prime})=-\Theta(\mathcal{L})$.

The limbs $\mathcal{L}_{0}^{\pm}$ are self-complementary; they are their own
complementary limbs. Indeed is clear from the definition that in general limbs
are mutually complementary; that is, $\mathcal{L}^{\prime}$ is a complementary
limb for $\mathcal{L}$ if and only if $\mathcal{L}$ is a complementary limb
for $\mathcal{L}^{\prime}$.

######  Lemma 3.13.

If $F_{a}$ and $F_{b}$ are postcritically finite polynomials which lie in
complementary limbs of $\mathcal{S}_{1}$, then $F_{a}\uplus F_{b}$ is
obstructed.

###### Proof.

If $\mathcal{L}$ and $\mathcal{L}^{\prime}$ are complementary limbs then
$\Theta(\mathcal{L}^{\prime})=-\Theta(\mathcal{L})$. Thus by Proposition 3.7,
the mating of maps in complementary limbs of $\mathcal{S}_{1}$ is obstructed.
∎

In the next section, we prove that this statement is not vacuous - some limbs
do have complementary limbs. By Corollary 3.6 and Lemma 3.8, a complementary
limb, if it exists, is unique.

###  3.1. Combinatorial data of a limb

We wish to prove the existence of complementary limbs for limbs whose internal
argument has a rotation number. We will also give a combinatorial description
of how a limb and a complementary limb are related.

######  Proposition 3.14.

Let $\mathcal{L}$ be a limb of $\mathcal{S}_{1}$. Then $\mathcal{L}$ has a
complementary limb if and only if $\mathcal{L}$ has a rotation number.

###### Proof.

First suppose $\mathcal{L}$ does not have a rotation number. By Proposition
3.3, the set $\Theta(\mathcal{L})$ has only one major gap, bounded by the
angles $\theta$ and $\theta^{\prime}$. In the dynamical plane for a map
$F_{a}\in\mathcal{L}$, this major gap is bounded by the external rays
$R_{a}(\theta)$ and $R_{a}(\theta^{\prime})$ landing on the root point of the
critical limb of $F_{a}$. Now suppose $\mathcal{L}^{\prime}$ is a limb with
$\Theta(\mathcal{L}^{\prime})=-\Theta(\mathcal{L})$. Then, since for any map
$F_{b}\in\mathcal{L}^{\prime}$ the rays landing at the root of the critical
limb have to bound a major gap, we see that these rays must be angles
$R_{b}(-\theta)$ and $R_{b}(-\theta^{\prime})$. But then we see that
$\mathcal{L}^{\prime}$ must be the conjugate limb to $\mathcal{L}$, and so
$\mathcal{L}$ has no complementary limb.

Now suppose $\mathcal{L}$ has a rotation number. Then the set
$\Theta(\mathcal{L})$ has two major gaps and thus the set
$-\Theta(\mathcal{L})$ also has two major gaps and so is a rotation set. But
by Corollary 3.6, there are two limbs satisfying
$\Theta(\mathcal{L}^{\prime})=-\Theta(\mathcal{L})$. Since only one can be a
conjugate limb, we see that $\mathcal{L}$ must have a complementary limb. ∎

We provide a combinatorial description of parameter limbs which will allow us
to find the complementary limb for any limb which has a rotation number. We
first need to discuss the combinatorics of rotation sets under $m_{2}$. Let
$t_{0}$ belong to the unique rotation set $X_{p/q}$ under $m_{2}$ with
rotation number $\rho=p/q\in\mathbb{Q}-\\{0\\}$. Write $X_{p/q}$ in cyclic
order as $\\{t_{1},t_{2},\dotsc,t_{q}\\}$, so that $0\in(t_{q},t_{1})$. Then
there exists a unique $1\leq k\leq q$ such that $t_{0}=t_{k}$. We call $k$ the
(combinatorial) position of $t_{0}$ in $X_{p/q}$. We define the _combinatorial
data_ for the limb $\mathcal{L}_{t_{0}}^{\pm}$ to be the triple
$(\rho,k,\pm)$.

######  Example 3.15.

Consider the $m_{2}$-rotation set $X_{2/5}$ with rotation number
$\rho=\frac{2}{5}$. Then $X_{2/5}=\\{5/31,9/31,10/31,18/31,20/31\\}$ and the
combinatorial position of $18/31$ is $4$. Thus the Limb
$\mathcal{L}_{18/31}^{+}$ has combinatorial data
$\left(\frac{2}{5},4,+\right)$.

It is clear that, given a triple $\mathcal{D}=(p/q,k,\pm)$ where $1\leq k\leq
q$ , there exists a unique limb $\mathcal{L}$ which realises the combinatorial
data $\mathcal{D}$. The combinatorial data above allows us to easily recognise
complementary limbs. As an aside, we remark that if a limb has combinatorial
data $(p/q,k,\pm)$ then the conjugate limb has combinatorial data
$(-p/q,q+1-k,\pm)$.

######  Lemma 3.16.

Suppose the combinatorial position of $t_{0}$ is $k$.

  1. (i)

The number of elements in $\Theta(\mathcal{L}_{t_{0}}^{+})\cap[0,1/2)$ is
equal to $2k-1$.

  2. (ii)

The number of elements in $\Theta(\mathcal{L}_{t_{0}}^{-})\cap[1/2,1)$ is
equal to $2k-1$.

###### Proof.

As usual, we only prove the first claim, since the proof of the second is
similar. For $t_{0}=0$, the only possible position is $k=1$ and the result
follows since the only rotation set consisting of two cycles with rotation
number $0/1$ is $\\{0,1/2\\}$. Suppose $t_{0}\neq 0$ has combinatorial
position $k$ and let $F_{a}\in\mathcal{L}_{t_{0}}^{+}$. Then by Lemma 3.1, the
angles of the external rays $R_{a}(\theta)$, $R_{a}(\theta^{\prime})$ landing
on the root of the critical limb of $F_{a}$ satisfy
$\theta<1/2<\theta^{\prime}$. Since $\beta(0)$ is the landing point of the ray
$R_{a}(0)$, then for $1\leq i\leq k-1$, the two external rays landing on
$\beta(t_{i})$ lie in $[0,1/2)$, and for $i>k$, the angles of the external
rays landing on $\beta(t_{i})$ lie in $[1/2,1)$. The result follows. ∎

######  Lemma 3.17.

Suppose $\mathcal{L}$ is the limb with combinatorial data $(p/q,k,\pm)$. Then
the limb with data $\mathcal{D}^{\prime}=(-p/q,k,\mp)$ is the complementary
limb to $\mathcal{L}$.

###### Proof.

The result clearly holds for $t_{0}$ = 0. So suppose
$\mathcal{L}=\mathcal{L}_{t_{0}}^{+}$ has data $(p/q,k,+)$ with $p/q\neq 0$.
The complementary limb $\mathcal{L}^{\prime}$, which we know exists by
Proposition 3.14, must then satisfy
$\Theta(\mathcal{L}^{\prime})=-\Theta(\mathcal{L})$ and so will have rotation
number $-p/q$. By Corollary 3.6, there are two limbs for which this equality
holds. One must be the conjugate limb to $\mathcal{L}$, which is the limb
$\mathcal{L}_{-t_{0}}^{+}$, and the other must be a limb
$\mathcal{L}_{s_{0}}^{-}$ for some $s_{0}$.

By Lemma 3.16, there are $2k-1$ elements in $\Theta(\mathcal{L})\cap(0,1/2)$.
Thus the complementary limb $\mathcal{L}_{s_{0}}^{-}$ must have $2k-1$
elements in $\Theta(\mathcal{L}_{s_{0}}^{-})\cap(1/2,1)$. But again by Lemma
3.16, this means the combinatorial position of $s_{0}$ is $k$. Hence the
complementary limb has data $(-p/q,k,-)$. The proof for a limb with data
$(p/q,k,-)$ is analogous. ∎

We give some examples of how we find complementary limbs using the above
lemma.

######  Example 3.18.

Consider a map $F_{a}$ in the parameter limb $\mathcal{L}_{4/7}^{+}$. We saw
before in Example 3.10 that that the rays $R_{a}(\frac{1}{26})$ and
$R_{a}(\frac{2}{26})$ land together in the dynamical plane of $F_{a}$. The
combinatorial data of this limb is $\left(\frac{1}{3},3,+\right)$. The
complementary limb with combinatorial data $\left(\frac{2}{3},3,-\right)$ is
the limb $\mathcal{L}_{6/7}^{-}$. The associated external rays for
$\mathcal{L}_{6/7}^{-}$ have angles $\frac{49}{78}$ and $\frac{50}{78}$,
meaning if $F_{b}\in\mathcal{L}_{6/7}^{-}$ then the dynamical rays
$R_{b}(\frac{23}{26})$ and $R_{b}(\frac{24}{26})$ land together in the
dynamical plane. It is then not hard to see we have the lamination of the
mating as given in Figure 5.

Figure 5. The lamination for the mating of a map in $\mathcal{L}_{4/7}^{+}$
and a map in $\mathcal{L}_{6/7}^{-}$. The ray classes of the $\alpha$-periodic
cycles form a single closed loop.

The ray class corresponding to the two $\alpha$-periodic cycles come together
to form one large loop, thus the mating is obstructed by this Levy cycle.

######  Example 3.19.

Again we consider the limb $\mathcal{L}_{18/31}^{+}$. Since this limb has
combinatorial data $\left(\frac{2}{5},4,+\right)$, Lemma 3.17 implies that the
complementary limb has combinatorial data $\left(\frac{3}{5},4,-\right)$. Let
$X_{3/5}$ be the unique rotation set under $m_{2}$ with rotation number $3/5$.
Then

$X_{3/5}=-X_{2/5}=\\{11/31,13/31,21/31,22/31,26/31\\}$

The element in this set with position $4$ is $22/31$. Thus the complementary
limb to $\mathcal{L}_{18/31}^{+}$ is $\mathcal{L}_{22/31}^{-}$. Using results
of Zakeri [17], one can compute that the rays $\mathcal{R}(19/726)$ and
$\mathcal{R}(20/726)$ land at the root of $\mathcal{L}_{18/31}^{+}$, while the
rays $\mathcal{R}(427/726)$ and $\mathcal{R}(428/726)$ land on the root of
$\mathcal{L}_{22/31}^{-}$. The major gaps for
$\Theta(\mathcal{L}_{18/31}^{+})$ are the arcs $(87/242,168,242)$ and
$(180/242,19/242)$. The major gaps for $\Theta(\mathcal{L}_{22/31}^{-}$ are
the arcs $(74/242,155/242)$ and $(223/242,62/242)$. One can easily check that
the ray class formed create a single closed loop.

Before giving the next example, we present some results from [3]. Let
$F_{a}\in\mathcal{S}_{1}$ and denote by $F_{a}^{\ast}$ the map
$F_{-\overline{a}}$. It was shown that, if the mating
$F_{a}\perp\\!\\!\\!\perp F_{a}^{\ast}$ was not obstructed, then the resulting
rational map belongs to the space $\mathcal{A}$ of antipode preserving cubic
rational maps (That is, maps which commute with the map
$z\mapsto-1/\overline{z}$) which have a pair of fixed critical points. In
particular, if $F_{a}\in\mathcal{H}_{0}\subseteq\mathcal{S}_{1}$, then it was
shown that the mating $F_{a}\perp\\!\\!\\!\perp F_{a}^{\ast}$ exists and
belongs to the main hyperbolic component $\widetilde{\mathcal{H}}_{0}$ in the
space of $\mathcal{A}$ (this component has the map $z\mapsto-z^{3}$ as its
centre). As with $\mathcal{H}_{0}\subseteq\mathcal{S}_{1}$, one can assign an
internal argument to points on $\partial\widetilde{\mathcal{H}}_{0}$.

Now suppose that $\Theta_{p/q}$ is the (unique) $m_{2}$-rotation set with
rotation number $p/q$. When $q$ is odd, there exists a unique
$\theta\in\Theta_{p/q}$ which has combinatorial position $(q+1)/2$ in
$\Theta_{p/q}$. Following [3], we say the angle $\theta$ is _balanced_. It was
shown that if an angle $\theta$ is balanced, then the corresponding point on
$\partial\widetilde{\mathcal{H}}_{0}$ with internal argument $\theta$ is an
ideal point on the circle at infinity in $\mathcal{A}$. This suggests that
matings of maps in the $\mathcal{L}_{\theta}^{\pm}$ with those in the limb
$(\mathcal{L}_{\theta}^{\pm})^{\ast}$ is obstructed. But observe that if
$\theta\in\Theta_{p/q}$ is balanced then the combinatorial position of
$\theta$ is $(q+1)/2$ and so $\mathcal{L}_{\theta}^{+}$ has data
$(p/q,(q+1)/2,+)$. Lemma 3.17 then asserts that the complementary limb will
have data $(-p/q,(q+1)/2,-)$. But the limb with this data is
$\mathcal{L}_{-\theta}^{-}$, which is precisely the limb
$(\mathcal{L}_{\theta}^{\pm})^{\ast}$. Thus the balanced case corresponds
exactly to the case where $\mathcal{L}^{\ast}$ is the complementary limb to
$\mathcal{L}$. We give an example below.

######  Example 3.20.

Consider the limb $\mathcal{L}_{5/7}^{+}$. The associated external (parameter)
angles are $\frac{7}{78}$ and $\frac{8}{78}$, and the combinatorial data of
$\mathcal{L}$ is $\left(\frac{2}{3},2,+\right)$, and so the angle $5/7$ is
balanced. For maps in this limb, the rays of angles $\frac{7}{26}$ and
$\frac{8}{26}$ land at a common period $3$ point. The limb with combinatorial
data $\left(\frac{1}{3},2,-\right)$ has internal angle $\frac{2}{7}$ and the
associated external angles are $\frac{31}{78}$ and $\frac{32}{38}$, from which
it follows that the rays of angles $\frac{5}{26}$ and $\frac{6}{26}$ land
together in the dynamical plane. This time we have the following picture.

Figure 6. An example when the limb is balanced in the sense of [3]. Notice
that picture has a symmetry (inversion in the unit circle followed by
$180^{\circ}$ rotation), which occurs because the complementary limb is equal
to the “negative conjugate” of the original limb in the balanced case.

Again, the ray class corresponding to the $\alpha$-periodic cycles form one
loop.

We remark that in each case, the obstruction consists of a single closed loop.

##  4\. Main results

We now are ready to state the main theorem. Recall that in the quadratic case,
the culmination of work by Rees, Shishikura and Tan gave the following.

######  Theorem 4.1 ([16]).

Let $f$ and $g$ be postcritically finite quadratic polynomials in the
Mandelbrot set. Then $f\perp\\!\\!\\!\perp g$ is obstructed if and only if $f$
and $g$ belong to conjugate limbs of the Mandelbrot set.

The work carried out so far means we can easily prove the following, which is
a slightly stronger version of Theorem 1.1.

######  Theorem 4.2.

Let $t_{0}$ be $m_{2}$-periodic such that the associated orbit has a rotation
number. Let $F_{a}$ be a postcritically finite polynomial in the limb
$\mathcal{L}_{t_{0}}^{\pm}$. Then there exists a postcritically finite
polynomial $F_{b}$ which does not belong to the conjugate limb
$\mathcal{L}_{-t_{0}}^{\pm}$ such that $F_{a}\uplus F_{b}$ is obstructed by
Lemma 3.13.

###### Proof.

Since $t_{0}$ has a rotation number under $m_{2}$, then by Proposition 3.14,
the limb $\mathcal{L}_{t_{0}}^{\pm}$ has a complementary limb
$\mathcal{L}^{\prime}$. Given $F_{a}\in\mathcal{L}_{t_{0}}^{\pm}$, take
$F_{b}$ to be any postcritically finite map in $\mathcal{L}^{\prime}$. Then
$F_{a}\uplus F_{b}$ is obstructed. ∎

The proof of Theorem 1.1 follows immediately. We believe that the limb
condition is also necessary, and so conjecture the following.

######  Conjecture 4.3.

Let $f,g\in\mathcal{S}_{1}$ be postcritically finite polynomials in
$\mathcal{S}_{1}$. Then $f$ and $g$ are not mateable if and only if one of the
following conditions hold.

  1. (i)

The polynomials $f$ and $g$ belong to conjugate limbs in $\mathcal{S}_{1}$.

  2. (ii)

The polynomials $f$ and $g$ belong to complementary limbs in
$\mathcal{S}_{1}$.

Thus, we conjecture that the mateability question in $\mathcal{S}_{1}$ can be
completely understood from the combinatorics of the $\alpha$-periodic cycles
of the two polynomials, just as the mateability question for quadratic
polynomials can be completely understood from the combinatorics of the rays
landing on the $\alpha$-fixed points of the polynomials being mated. Numerical
experiments, involving looking at various parts of the space of cubic rational
maps which have two fixed critical points (for published examples, see [1, 3])
certainly suggest this is the case.

In the quadratic case, the mating of $f$ and $g$ is obstructed if and only if
the ray classes under the mating of the $\alpha$-fixed points of $f$ and $g$
coincide. The generalisation for the $\alpha$-periodic cycles does not hold:
it is possible for the two $\alpha$-periodic cycles to belong to the same ray
class (or a union of ray classes, each of which contains an element of both
$\alpha$-periodic cycles), but for this ray class (or union of ray classes)
not to contain a loop. In terms of the sets $\Theta(\mathcal{L})$ and
$\Theta(\mathcal{L}^{\prime})$, this means that
$\Theta(\mathcal{L})\cap-\Theta(\mathcal{L}^{\prime})\neq\varnothing$ but
$\Theta(\mathcal{L})\neq-\Theta(\mathcal{L}^{\prime})$. The reason is that
$m_{2}$-rotation sets are uniquely defined by their rotation sets; this is not
the case for $m_{3}$-rotation sets.

######  Example 4.4.

We give an example where
$\Theta(\mathcal{L})\cap-\Theta(\mathcal{L}^{\prime})\neq\varnothing$ but
$\\{\mathcal{L},\mathcal{L}^{\prime}\\}$ does not satisfy the limb condition.
Let $F_{a}$ be the unique map in $\mathcal{L}=\mathcal{L}_{2/3}^{+}$ for which
the free critical point $-a$ has period $2$ (the Julia set is shown on the
left of Figure 2). Since $\Theta(\mathcal{L})=\\{1/8,2/8,3/8,6/8\\}$, we see
that $\Theta(\mathcal{L})\cap-\Theta(\mathcal{L})=\\{2/8,6/8\\}$. But the
mating is not obstructed; the only biaccessible periodic points for $F_{a}$
belong to the $\alpha$-periodic cycle, and so there is no Levy cycle for this
mating. $F_{a}\uplus F_{a}$ is equivalent to the rational map given in Figure
7.

Figure 7. An example where
$\Theta(\mathcal{L})\cap-\Theta(\mathcal{L}^{\prime})\neq\varnothing$ but the
mating is not obstructed. On the left is the lamination showing the rays
landing on the $\alpha$-periodic cycles, and on the right is the Julia set of
the rational map constructed by the mating.

###  4.1. Necessity of the limb condition for preperiodic internal arguments

In the case where the internal argument of the limb
$\mathcal{L}_{t_{0}}^{\pm}$ $t_{0}$ is strictly preperiodic, we can prove
necessity of the limb condition. The goal of this section is to prove the
following theorem.

######  Theorem 4.5.

Let $t_{0}$ be strictly preperiodic under $m_{2}$ and suppose $F_{a}$ is the
unique element of the limb $\mathcal{L}_{t_{0}}^{\pm}$. If $F_{b}$ is a
postcritically finite polynomial in $\mathcal{S}_{1}$ then the mating
$F_{a}\perp\\!\\!\\!\perp F_{b}$ is obstructed if and only if $F_{b}$ is the
unique element of the conjugate limb $\mathcal{L}_{-t_{0}}^{\pm}$.

The following lemma contains important properties of the ray classes in the
matings considered in the proof of Theorem 4.5.

######  Lemma 4.6.

Let $t_{0}$ be strictly preperiodic under $m_{2}$ and suppose $F_{a}$ is the
unique element of the limb $\mathcal{L}_{t_{0}}^{\pm}$. Let $F_{b}$ be a
postcritically finite polynomial in $\mathcal{S}_{1}$ such that, in the mating
$F_{a}\uplus F_{b}$, the free critical values $v_{a}$ and $v_{b}$ of $F_{a}$
and $F_{b}$ respectively belong to the same ray class $[v_{a}]$. Then:

  1. (i)

if $R_{a}(\theta)$ is the ray landing on $v_{a}$, then $R_{b}(-\theta)$ lands
on $v_{b}$.

  2. (ii)

there exists an angle $\eta$ such that $R_{a}(\eta)$ lands on the free
critical point $-a$ of $F_{a}$ and $R_{b}(-\eta)$ lands on the free critical
point $-b$ of $F_{b}$.

  3. (iii)

$v_{b}$ is the only point in the ray class $[v_{a}]$ which belongs to
$J(F_{b})$.

  4. (iv)

the free critical points $-a$ and $-b$ of $F_{a}$ and $F_{b}$ respectively
belong to the same ray class $[-a]$.

  5. (v)

the ray class $[-a]$ containing the two free critical points contains a closed
loop if and only if there exist distinct $\eta$, $\eta^{\prime}$ such that
$R_{a}(\eta)$ lands together with $R_{a}(\eta^{\prime})$ on $-a$ and
$R_{b}(-\eta)$ lands together with $R_{b}(\eta^{\prime})$ on $-b$.

###### Proof.

  1. (i)

Suppose $R_{F_{b}}(-\theta)$ does not land on $v_{b}$. Since $R_{a}(\theta)$
is the unique external ray landing on $v_{a}$, it follows that there must be a
biaccessible point $z$ in $[v_{a}]\cap J(F_{a})$. By assumption, the free
critical point $-a$ of $F_{a}$ is not periodic, and so the ray class $[v_{a}]$
maps forward homeomorphically under iterates of the mating. In particular,
there is a forward iterate of $z$ which is a biaccessible periodic point in
$J(F_{a})$. But since $F_{a}$ has no biaccessible periodic point, this is a
contradiction.

  2. (ii)

Let $R_{a}(\theta)$ be the ray landing on $v_{a}$. By part (i),
$R_{b}(-\theta)$ lands on $v_{b}$. Consider the three preimages of
$R_{a}(\theta)$ under $F_{a}$. Two of these preimage rays must land on $-a$.
Similarly, two of the three preimages of $R_{b}(-\theta)$ must land on $-b$.
By the pigeonhole principle, there must then exist some $\eta$ such that
$R_{a}(\eta)$ lands on $-a$ and $R_{b}(-\eta)$ lands on $-b$.

  3. (iii)

To see that $v_{b}$ is the only element of the ray class that belongs to
$J(F_{b})$, first observe that if there is another element of $J(F_{b})$ in
the ray class, then there must also be a biaccessible point of $J(F_{a})$ in
the ray class. Similarly to the above, we can then map the ray class forward
and obtain a biaccessible periodic point in $J(F_{a})$. Again this is a
contradiction.

  4. (iv)

Suppose that $[-a]$ and $[-b]$ are distinct ray classes. Both of these ray
classes map onto $[v_{a}]$ by a degree $2$ mapping, and so elements of
$[v_{a}]$ have at least four preimages under $F_{a}\uplus F_{b}$. But the
mating has degree $3$, so this is a contradiction and so the ray classes
$[-a]$ and $[-b]$ are equal.

  5. (v)

If there exist $\theta$ and $\theta^{\prime}$ such that $R_{a}(\theta)$ and
$R_{a}(\theta^{\prime})$ land on $-a$ and $R_{b}(-\theta)$ and
$R_{b}(\theta^{\prime})$ land on $-b$, then (the closure of) these rays form a
closed loop within the ray class $[-a]$.

Now suppose that there is only one angle $\eta$ such that $R_{a}(\eta)$ lands
on $-a$ and $R_{b}(\eta)$ lands on $-b$ (there must be at least one such
$\eta$ by part (ii)). Then we see that both $-b$ and $2b$ both belong to
$[-a]$, and these are the only two elements in $[-a]\cap J(F_{b})$, since
$v_{b}$ is the only element of $J(F_{b})$ in $[v_{a}]$ (part (iii)).
Furthermore, suppose $z\in J(F_{a})\cap[-a]$ is an element distinct from $-a$.
We claim $z$ is the landing point of only one external ray. For if not,
similar reasoning as that used in part (i) would imply that a forward iterate
of $z$ is a biaccessible periodic point for $F_{a}$, which is impossible.

We now think of the ray class $[-a]$ as a connected bipartite graph with parts
$A=[-a]\cap J(F_{a})$ and $B=[-a]\cap J(F_{b})=\\{-b,2b\\}$, and such that
there is an edge between $x\in A$ and $y\in B$ if and only if there exists $t$
such that $R_{a}(t)$ lands on $x$ and $R_{b}(-t)$ lands on $y$. Suppose the
number of vertices in this graph is $n$. Then since $B=\\{-b,2b\\}$,we see
there are $n-2$ elements of $A$. Since the graph is bipartite, the number of
edges in the graph is equal to the sum of the degrees (in the graph theory
sense) of the vertices in $A$. But $-a$ has degree $2$, and it was shown above
that each of the other $n-3$ vertices in $A$ has degree $1$. Thus there are
$n-1$ edges in the graph. Since a connected graph with $n$ vertices is a tree
if and only it has $n-1$ edges, we see that the graph does not contain a
closed loop.

∎

###### Proof of Theorem 4.5.

If $F_{b}$ is the conjugate map to $F_{a}$ then the mating is obstructed by
Proposition 3.9.

To prove the converse, we will show that if $F_{b}$ is not the conjugate map
to $F_{a}$, then any obstruction to the mating $F_{a}\uplus F_{b}$ is a
removable Levy cycle. First note that by Corollary 2.16, any obstruction to
this mating must be a degenerate Levy cycle, and if this Levy cycle is not
removable, it follows from Corollary 2.14 that there must be a ray class which
contains both $v_{a}$ and $v_{b}$, the free critical values of $F_{a}$ and
$F_{b}$ respectively. Let $\mathcal{R}(\theta)$ be the unique parameter ray
landing on $F_{a}$. Then $R_{a}(\theta)$ is the only ray landing on the
cocritical point $2a\in J(F_{a})$. Taking the forward image, we see that
$R_{a}(3\theta)$ is the only ray which lands on the free critical value
$v_{a}$ of $F_{a}$ and so $R_{a}(\theta-1/3)$ and $R_{a}(\theta+1/3)$ are the
only rays landing on the free critical point $-a$. According to Lemma 4.6
(part (i)), since $v_{a}$ and $v_{b}$ to belong to the same ray class
$[v_{a}]$, then $v_{b}$ is the landing point of $R_{b}(-3\theta)$. This means
that the cocritical point $2b$ of $F_{b}$ must be the landing point of one of
$R_{b}(-\theta)$, $R_{b}(-(\theta-1/3))$ or $R_{b}(-(\theta+1/3))$. Hence, if
$v_{a}$ and $v_{b}$ belong to the same ray class, then $F_{b}$ is equal to the
landing point of one of $\mathcal{R}(-\theta)$, $\mathcal{R}(-(\theta-1/3))$
or $\mathcal{R}(-(\theta+1/3))$. Furthermore, again by Lemma 4.6 (part (iii)),
the only element of $J(F_{b})$ in $[v_{a}]$ is $v_{b}$, and the preimage of
the ray class $[v_{a}]$ contains a ray class $[-a]$ which contains both free
critical points (part (iv)).

If $F_{b}$ is not the conjugate map to $F_{a}$, then $F_{b}$ is not the
landing point of $\mathcal{R}(-\theta)$. Thus it remains to show that if
$F_{b}$ is the landing point of $\mathcal{R}(-(\theta-1/3))$ or
$\mathcal{R}(-(\theta+1/3))$, then no ray class in the mating $F_{a}\uplus
F_{b}$ which contains a closed loop. In fact, it suffices to show that this is
the case for the ray class $[-a]$ which contains both $-a$ and $-b$, since all
other ray classes map homeomorphically onto their image, and the ray class
$[v_{a}]$ does not contain a closed loop by Proposition 2.13. We will assume
$F_{b}$ is the landing point of $\mathcal{R}(-(\theta-1/3))$; the proof of the
other case is similar. We may also assume $F_{b}$ is not parabolic, since if
$F_{b}$ is a parabolic map then $v_{b}$ does not belong to the Julia set of
$F_{b}$.

Thus $R_{b}(-(\theta-1/3))$ lands on the cocritical point $2b$ and therefore
the rays $R_{b}(-\theta)$ and $R_{b}(-(\theta+1/3))$ land on the free critical
point $-b$. Since $R_{a}(\theta)$ lands on $2a$ and $R_{a}(\theta-1/3)$ and
$R_{a}(\theta+1/3)$ are the only rays which land on $-a$, we see that there is
only the angle $\eta=\theta+1/3$ for which $R_{a}(\eta)$ lands on $-a$ and
$R_{b}(-\eta)$ lands on $-b$. By Lemma 4.6 (part (v)), the ray class does not
contain a closed loop, and so $F_{a}$ and $F_{b}$ are mateable. ∎

Figure 8. An example where the two critical points are identified under the
mating. The upper images show the two polynomials $F_{a}$ and $F_{b}$ in the
mating of Example 4.7, and the second row of images shows the lamination and
the corresponding rational map, which has a double critical point (which can
be seen just to the left of the centre of the image).

As the above proof suggests, it is possible that $F_{a}$ and $F_{b}$ to be
(essentially) mateable even if there is a ray equivalence class containing
both the free critical points $-a\in J(F_{a})$ and $-b\in J(F_{b})$. In this
case, these simple critical points combine to become a double critical point
for the resulting rational map. The author is grateful to Daniel Meyer for
pointing out that this phenomenon is quite common when one considers matings
involving polynomials having critical points in their Julia set.

######  Example 4.7.

Consider the map $F_{a}$ which is the landing point of $\mathcal{R}(1/36)$.
This is the unique map in the limb $\mathcal{L}_{7/12}^{+}$. Now consider the
mating of this map with $F_{b}$, the map which is the landing point of
$\mathcal{R}(11/36)$, which belongs to the limb $\mathcal{L}_{0}^{-}$ (see
Figure 8). A simple calculation shows that $R_{a}(13/36)$ and $R_{a}(25/36)$
land on $-a$, the free critical point of $F_{a}$, while $R_{b}(23/36)$ and
$R_{b}(35/36)$ land on $-b$, the free critical point of $F_{b}$. Furthermore,
$R_{a}(1/12)$ lands on $v_{a}$ and $R_{b}(11/12)$ lands on $v_{b}$, so the two
critical values belong to the same ray class (and the boundary of a tubular
neighbourhood of this ray class generates a removable Levy cycle). However,
the topological mating of these two polynomials exists, and is Thurston
equivalent to the rational map whose Julia set is shown in the bottom right
image of Figure 8.

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