# Thurston obstructions and tropical geometry

Rohini Ramadas<EMAIL_ADDRESS>Warwick Mathematics Institute,
University of Warwick, Coventry, UK

###### Abstract.

We describe an application of tropical moduli spaces to complex dynamics. A
post-critically finite branched covering $\varphi$ of $S^{2}$ induces a
pullback map on the Teichmüller space of complex structures of $S^{2}$; this
descends to an algebraic correspondence on the moduli space of point-
configurations of $\mathbb{P}^{1}$. We make a case for studying the action of
the tropical moduli space correspondence by making explicit the connections
between objects that have come up in one guise in tropical geometry and in
another guise in complex dynamics. For example, a Thurston obstruction for
$\varphi$ corresponds to a ray that is fixed by the tropical moduli space
correspndence, and scaled by a factor $\geq 1$. This article is intended to be
accessible to algebraic and tropical geometers as well as to complex
dynamicists.

##  1\. Introduction

Topologically speaking, every rational function
$f:\operatorname{\mathbb{C}\mathbb{P}^{1}}\to\operatorname{\mathbb{C}\mathbb{P}^{1}}$
is an orientation-preserving branched covering of the 2-sphere. In 1982,
Thurston addressed the question: which branched coverings are rational
functions? In order for this to be a well-posed question, we restrict to the
case of branched coverings that are post-critically finite, i.e. whose
postcritical set is finite. Given an orientation-preserving branched cover
$\varphi:S^{2}\to S^{2}$ with finite post-critical set $\mathbf{P}$, Thurston
showed that “pullback of complex structures along $\varphi$” is a well-defined
holomorphic self-map $\operatorname{TP}_{\varphi}$ of the Teichmüller space
$\mathcal{T}(S^{2},\mathbf{P})$ of complex structures on
$S^{2}\setminus\mathbf{P}$. Essentially tautologically, a fixed point of
$\operatorname{TP}_{\varphi}$ is an identification
$S^{2}\cong\mathbb{C}\mathbb{P}^{1}$ under which $\varphi$ is isotopic,
relative to $\mathbf{P}$, to a PCF rational function on
$\mathbb{C}\mathbb{P}^{1}$. Furthermore, $\operatorname{TP}_{\varphi}$ is
distance-nonincreasing with respect to the Teichmüller metric on
$\mathcal{T}(S^{2},\mathbf{P})$, and, under a mild additional hypothesis111The
hypothesis is that $\varphi$ has hyperbolic orbifold, see [DH93] for a
definition and complete classification of excluded $\varphi$. The only
examples that are excluded are certain examples with four or fewer post-
critical points. on $\varphi$, $\operatorname{TP}_{\varphi}^{2}$ is strictly
distance-decreasing. In the latter case, to which we restrict in this article,
either (A) $\operatorname{TP}_{\varphi}$ has fixed point, which is necessarily
unique, or (B) the branched covering $\varphi$ carries a topological object
called a Thurston obstruction. (This proof was written up by Douady and
Hubbard in [DH93].)

A Thurston obstruction is a multicurve — a collection of pairwise-disjoint
simple closed curves on $S^{2}\setminus\mathbf{P}$ — that has a type of
invariance called $\varphi$-stability, and that satisfies another condition
having to do with the spectral properties of its Thurston linear
transformation (defined in Section 3.1). In fact, Pilgrim [Pil01] showed that
if $\varphi$ is not isotopic to a PCF rational function, then it has a
canonical Thurston obstruction admitting an informal interpretation as
follows: for all $x\in\mathcal{T}(S^{2},\mathbf{P})$, as $r\to\infty$,
$\operatorname{TP}_{\varphi}^{r}(x)$ “goes to infinity”, i.e. leaves every
compact subset of $\mathcal{T}(S^{2},\mathbf{P})$. As
$\operatorname{TP}_{\varphi}^{r}(x)$ goes to infinity, the complex structure
on the sphere degenerates. As the complex structure degenerates, there is at
least one simple closed geodesic curve whose length goes to zero. The set of
all such curves is the canonical obstruction.

Thurston’s theorem (as well as the theory developed in its proof) is still
central in modern $1$-variable complex and arithmetic dynamics. For example,
one consequence of the theorem is that (with exceptions222The exceptions are
flexible Lattès maps: A flexible Lattès map is a rational map on
$\mathbb{C}\mathbb{P}^{1}$ that is descended from an affine map on an elliptic
curve $E$, via the hyperelliptic map $E\to\mathbb{C}\mathbb{P}^{1}$. Flexible
Lattès maps have square degree, and are PCF with exactly four post-critical
points.) PCF maps are “rigid”, i.e. do not deform in non-isotrivial families,
and are defined over $\overline{\mathbb{Q}}$. Teichmüller space
$\mathcal{T}(S^{2},\mathbf{P})$ is a nonalgebraic complex manifold; it is the
universal cover of the moduli space $\mathcal{M}_{0,P}$, an algebraic variety
parametrizing point-configurations on $\mathbb{C}\mathbb{P}^{1}$. In 2013,
Koch [Koc13] showed that $\operatorname{TP}_{\varphi}$ descends to an
algebraic multivalued self-map $\operatorname{MSC}_{\varphi}$ of
$\mathcal{M}_{0,\mathbf{P}}$ known as the moduli space correspondence. Fixed
points of $\operatorname{MSC}_{\varphi}$ are in bijection with PCF rational
functions that are Hurwitz equivalent to $\varphi$ (Defintion 5.1), but that
are not necessarily isotopic to $\varphi$.

There is a moduli space of tropical curves
$\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$, which is a polyhedral
object obtained by gluing convex cones together along faces ([Mik06, BHV01],
see Section 2.2). Very informally speaking,
$\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$ can be thought of as a
space parameterizing “paths to infinity” in $\mathcal{M}_{0,\mathbf{P}}$
[ACP15]. It follows directly from the existence and properties of the tropical
Hurwitz space constructed in [CMR16] that the moduli space correspondence
induces a tropical moduli space correspondence
$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$, a piece-wise linear,
multivalued action on $\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$.

In this article, we synthesize concepts well-understood in the field of
rational dynamics and Teichmüller theory with concepts well-understood in the
field of tropical curves and their moduli spaces. We believe that the
resulting synthesis results in conceptual clarity and valuable perspective. We
first explain how to think of the collection of Thurston linear
transformations as together comprising a tropical pullback map
$\operatorname{TP}^{\operatorname{trop}}$ (Section 4.6). We then show:

######  Theorem 1.1 (Informal summary of Section 7).

  1. (1)

(Proposition 7.6 and Diagram 5.)
$\operatorname{MSC}^{\operatorname{trop}}_{\varphi}$ descends from
$\operatorname{TP}^{\operatorname{trop}}_{\varphi}$.

  2. (2)

(Proposition 7.8.) Every locally-defined single-valued linear branch of
$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$ has the same matrix as
some Thurston linear transformation associated to $\varphi$.

  3. (3)

(Proposition 7.12 and Corollary 7.13.)
$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$-fixed rays correspond to
non-empty multicurves on $S^{2}$ that are $\varphi^{\prime}$-stable for some
$\varphi^{\prime}$ Hurwitz equivalent to $\varphi$.
$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$-fixed rays that are
scaled by a factor $\geq 1$ correspond to multicurves that are Thurston
obstructions for some $\varphi^{\prime}$ Hurwitz equivalent to $\varphi$.

Theorem 1.1 provides evidence that the tropical moduli space correspondence
captures important aspects of the dynamics near infinity of the algebraic
moduli space correspondence. It is therefore of interest to study the global
dynamics of the tropical correspondence. However, the technical tools needed
to undertake this study are currently under-developed. We explain some of
these difficulties in Remarks 7.1 and 7.15.

######  Remark 1.2.

The link of $\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$ is a compact
space obtained by gluing polyhedra together along faces, and (almost! — Remark
7.15) inherits a piece-wise affine multivalued action. Part (3) of Theorem 1.1
can be interpreted to mean that $\varphi$-stable multicurves on $S^{2}$
correspond to fixed points of the action on the link. Whether or not a
$\varphi$-stable multicurve is an obstruction is closely linked to the local
dynamical behaviour near the corresponding fixed point on the link. Making
this precise and studying the global dynamical properties of the action on the
link are currently the subject of investigation.

###  1.1. Organization

In Section 2 we give background on the moduli space
$\mathcal{M}_{0,\mathbf{P}}$ of point-configurations, its compactification
$\overline{\mathcal{M}}_{0,\mathbf{P}}$ and its tropicalization
$\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$. Section 3 mirrors Section
2, giving background on Teichmüller space, a partial compactification, and a
(non-standard) candidate for its tropicalization. In Section 4 we give
background on Thurston’s theorem, introducing the pullback map,
$\varphi$-stable multicurves, Thurston linear transformations and Thurston
obstructions. In Section 5 we give background on Hurwitz spaces and the moduli
space correspondence. In Section 6 we give background on compactifications and
tropicalizations of Hurwitz spaces. Section 7 contains the main content of
this article — we give a definition of the tropical moduli space
correspondence and prove Theorem 1.1. The reader will see that the proofs are
relatively straightforward once the set-up is appropriately made.

###  1.2. Acknowledgements

I am grateful for useful conversations with Xavier Buff, Charles Favre, Sarah
Koch, Curtis McMullen, Kevin Pilgrim, Dylan Thurston and Rob Silversmith. This
work was partially funded by UKRI EPSRC New Investigator Award EP/X026612/1.

##  2\. The moduli space of point-configurations

We introduce the moduli space of point-configurations, its compactification,
and its tropicalization. Throughout this article, we will assume that
$\mathbf{P}$ is a finite set of cardinality at least $3$. The moduli
$\mathcal{M}_{0,\mathbf{P}}$ is a smooth quasiprojective
$(\left\lvert\mathbf{P}\right\rvert-3)$-dimensional variety parametrizing
injective maps $\iota:\mathbf{P}\to\operatorname{\mathbb{C}\mathbb{P}^{1}}$,
up to post-composition by automorphisms of
$\operatorname{\mathbb{C}\mathbb{P}^{1}}$. If
$\mathbf{P}_{1}\subset\mathbf{P}_{2}\subset S^{2}$, there is a forgetful map
$\mu:\mathcal{M}_{0,\mathbf{P}_{2}}\to\mathcal{M}_{0,\mathbf{P}_{1}}$.

###  2.1. Stable curves and the Deligne-Mumford compactification.

A prestable $\mathbf{P}$-marked genus-$0$ curve is a pair $(C,\iota)$, where
$C$ is an at worst nodal curve of arithmetic genus $0$ (i.e. a tree of
$\operatorname{\mathbb{C}\mathbb{P}^{1}}$s attached at simple nodes), and
$\iota$ is an injective map from $\mathbf{P}$ into the smooth locus of $C$. In
this article, all prestable curves are assumed to be genus-$0$. A prestable
curve is called stable if each irreducible component has at least $3$ points
that are either nodes or marked. The dual graph of a nodal curve is the graph
which has a vertex for each irreducible component of the curve, and an edge
between two vertices if the corresponding irreducible components of the curve
meet at a node. Given a prestable $\mathbf{P}$-marked curve, its marked dual
tree $\operatorname{\tau}$ is the pair
$(\underline{\operatorname{\tau}},\widehat{\iota})$, where
$\underline{\operatorname{\tau}}$ is the dual graph of $C$ (which is
necessarily a tree), and $\widehat{\iota}$ is the map from $\mathbf{P}$ to the
set of vertices of $\underline{\operatorname{\tau}}$ sending $p\in\mathbf{P}$
to the vertex of $\underline{\operatorname{\tau}}$ corresponding to the
irreducible component of $C$ that is marked by $p$. We define the valence of a
vertex $v$ of $\operatorname{\tau}$ to be

$\operatorname{val}(v):=(\mbox{number of edges adjacent to
$v$})+\left\lvert\widehat{\iota}^{-1}(v)\right\rvert.$

A $\mathbf{P}$-marked tree $\operatorname{\tau}$ is called stable if every
vertex has valence at least $3$. A prestable $\mathbf{P}$-marked curve is
stable if and only if its marked dual tree is stable. The Deligne-Mumford
compactification $\overline{\mathcal{M}}_{0,\mathbf{P}}$ of
$\mathcal{M}_{0,\mathbf{P}}$ parametrizes, up to isomorphism, stable
$\mathbf{P}$-marked curves $(C,\iota)$.

Given a stable $\mathbf{P}$-marked tree $\operatorname{\tau}$, the locus in
$\overline{\mathcal{M}}_{0,\mathbf{P}}$ of curves with marked dual tree
isomorphic to $\operatorname{\tau}$ is a locally closed set
$\operatorname{X}_{\operatorname{\tau}}^{\circ}$ whose closure is an
irreducible smooth subvariety $\operatorname{X}_{\operatorname{\tau}}$ called
a boundary stratum of $\overline{\mathcal{M}}_{0,n}$. The codimension of
$\operatorname{X}_{\operatorname{\tau}}$ is equal to the number of edges of
$\operatorname{\tau}$. The containment
$\operatorname{X}_{\operatorname{\tau}_{1}}\subset\operatorname{X}_{\operatorname{\tau}_{2}}$
holds if and only if $\operatorname{\tau}_{2}$ can be obtained from
$\operatorname{\tau}_{1}$ by contracting a subset of edges of
$\operatorname{\tau}_{1}$.

A prestable $\mathbf{P}$-marked curve $(C,\iota)$ admits a unique
stabilization, which is a nodal curve $C^{\prime}$ together with a surjective
degree-$1$ map $\operatorname{St}:C\to C^{\prime}$ with the property that
$(C^{\prime},\operatorname{St}\circ\iota)$ is a stable $\mathbf{P}$-marked
curve. The map $\operatorname{St}$ may contract some irreducible components of
$C$ to points. Given $\mathbf{P}_{1}\subset\mathbf{P}_{2}$, the forgetful map
$\mu$ extends to a map
$\overline{\mathcal{M}}_{0,\mathbf{P}_{2}}\to\overline{\mathcal{M}}_{0,\mathbf{P}_{1}}$,
sending $(C,\iota:\mathbf{P}_{2}\hookrightarrow C)$ to the stabilization of
$(C,\iota|_{\mathbf{P}_{1}})$. The $\mathbf{P}_{1}$-marked dual tree of
$\mu((C,\iota))$ is determined by the $\mathbf{P}_{2}$-marked dual tree of
$(C,\iota)$ — we will denote by $\mu_{*}$ the induced map from the set of
(isomorphism classes of) $\mathbf{P}_{2}$-marked stable trees to the the set
of (isomorphism classes of) $\mathbf{P}_{1}$-marked stable trees.

###  2.2. Tropical curves and the tropical moduli space

The topological space of a $\mathbf{P}$-marked tree $\operatorname{\tau}$ is
the topological space obtained from the usual underlying topological space of
$\underline{\operatorname{\tau}}$ by attaching, for every $p\in\mathbf{P}$, a
ray $\cong[0,+\infty)$ to the vertex $\widehat{\iota}(p)$. The ray
corresponding to $p\in\mathbf{P}$ is often referred to as the leg of $p$. A
stable $\mathbf{P}$-marked genus-$0$ tropical curve is a metrization of the
topological space of a stable $\mathbf{P}$-marked tree $\operatorname{\tau}$
that obtained by assigning a positive edge length to every edge of
$\operatorname{\tau}$, and assigning the legs infinite length. All tropical
curves in this article are genus-$0$.

The space $\operatorname{Cone}^{\circ}(\operatorname{\tau})$ of
$\mathbf{P}$-marked tropical curves with underlying marked tree
$\operatorname{\tau}$ is a cone canonically isomorphic to
$\mathbb{R}_{>0}^{\operatorname{Edges}(\operatorname{\tau})}$. Here, the
coordinates in terms of the natural basis are exactly the edge-lengths. We
denote by $\operatorname{Cone}(\operatorname{\tau})$ the space
$\mathbb{R}_{\geq 0}^{\operatorname{Edges}(\operatorname{\tau})}$ of possibly
degenerate metrizations of $\operatorname{\tau}$. If $\operatorname{\tau}_{2}$
is obtained from $\operatorname{\tau}_{1}$ by contracting a subset $E$ of
edges of $\operatorname{\tau}_{1}$, then a degenerate metrization of
$\operatorname{\tau}_{1}$ that assigns length $0$ to every $e\in E$ also
yields a metrization of $\operatorname{\tau}_{2}$. This realizes the cone
$\operatorname{Cone}(\operatorname{\tau}_{2})$ as a face of
$\operatorname{Cone}(\operatorname{\tau}_{1})$. Note that the direction of
inclusion of cones is the reverse of the direction of inclusion of boundary
strata outlined in Section 2.1. The moduli space
$\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$ is the space of all stable
$\mathbf{P}$-marked tropical curves; it is a cone complex obtained by gluing
together the cones $\operatorname{Cone}(\operatorname{\tau})$, where
$\operatorname{\tau}$ ranges over all stable $\mathbf{P}$-marked trees, along
the inclusions
$\operatorname{Cone}(\operatorname{\tau}_{2})\hookrightarrow\operatorname{Cone}(\operatorname{\tau}_{1})$
mentioned above. Note that $\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$
can be identified with the cone over the boundary complex of
$\overline{\mathcal{M}}_{0,\mathbf{P}}$.

Suppose that $\mathbf{P}_{1}\subset\mathbf{P}_{2}$ and that $\Sigma$ is a
$\mathbf{P}_{2}$-marked tropical curve. Set
$\mu^{\operatorname{trop}}(\Sigma)$ to be the convex hull, within $\Sigma$, of
the legs corresponding to $p\in\mathbf{P}_{1}$; this is a
$\mathbf{P}_{1}$-marked tropical curve. The map
$\mu^{\operatorname{trop}}:\mathcal{M}_{0,\mathbf{P}_{2}}^{\operatorname{trop}}\to\mathcal{M}_{0,\mathbf{P}_{1}}^{\operatorname{trop}}$
is piecewise-linear, and is the tropicalization of the forgetful map $\mu$.

##  3\. Teichmüller space and its tropicalization

We introduce some objects from surface topology and propose a candidate for
the role of tropical Teichmüller space. Given $\mathbf{P}\subset S^{2}$, the
Teichmüller space $\mathcal{T}(S^{2},\mathbf{P})$ is a non-algebraic complex
manifold of dimension $\left\lvert\mathbf{P}\right\rvert-3$, parametrizing
complex structures on $(S^{2},\mathbf{P})$. More precisely, a point of
$\mathcal{T}(S^{2},\mathbf{P})$ is an equivalence class of homeomorphisms
$\psi:S^{2}\to\mathbb{C}\mathbb{P}^{1}$, where $\psi_{1}$ and $\psi_{2}$ are
equivalent if there is a Möbius transformation $\alpha$ such that $\psi_{1}$
and $\alpha\circ\psi_{2}$ are isotopic relative to $\mathbf{P}$. If
$\mathbf{P}_{1}\subset\mathbf{P}_{2}\subset S^{2}$, there is a holomorphic
forgetful map
$\widetilde{\mu}:\mathcal{T}(S^{2},\mathbf{P}_{2})\to\mathcal{T}(S^{2},\mathbf{P}_{1})$.

The pure mapping class group $\operatorname{PMod}(S^{2},\mathbf{P})$ is the
group of self-homeomorphisms of $S^{2}$ that fix every element of
$\mathbf{P}$, up to isotopy relative to $\mathbf{P}$.
$\operatorname{PMod}(S^{2},\mathbf{P})$ acts freely and properly on
$\mathcal{T}(S^{2},\mathbf{P})$ by precomposition [FM11]. There is a forgetful
map $\rho:\mathcal{T}(S^{2},\mathbf{P})\to\mathcal{M}_{0,\mathbf{P}}$ sending
$[\psi:S^{2}\to\operatorname{\mathbb{C}\mathbb{P}^{1}}]$ to
$[\psi|_{\mathbf{P}}]$; this map is the quotient by the action of
$\operatorname{PMod}(S^{2},\mathbf{P})$, and realizes
$\mathcal{T}(S^{2},\mathbf{P})$ as the universal cover of
$\mathcal{M}_{0,\mathbf{P}}$.

###  3.1. Multicurves and the curve complex

A simple closed curve on $S^{2}\setminus\mathbf{P}$ is called essential if it
does not bound a disc, and is called non-peripheral if it does not bound a
punctured disc. A multicurve on $S^{2}\setminus\mathbf{P}$ is a possibly empty
set of homotopy classes of pairwise disjoint, pairwise nonhomotopic, essential
and non-peripheral simple closed curves on $S^{2}\setminus\mathbf{P}$.

The curve complex $\operatorname{CC}(S^{2},\mathbf{P})$ of
$(S^{2},\mathbf{P})$ is the simplicial complex defined as follows: vertices of
$\operatorname{CC}(S^{2},\mathbf{P})$ are in bijection with homotopy classes
of simple closed curves on $S^{2}\setminus\mathbf{P}$ that are essential and
non peripheral. The vertices corresponding to
$[\gamma_{1}],\ldots,[\gamma_{k+1}]$ span a $k$-simplex if and only if there
exist pairwise disjoint representatives for
$[\gamma_{1}],\ldots,[\gamma_{k+1}]$, i.e. if and only if
$\\{[\gamma_{1}],\ldots,[\gamma_{k+1}]\\}$ is a multicurve on
$S^{2}\setminus\mathbf{P}$. The curve complex is locally infinite, and Gromov
hyperbolic [MM99]. The mapping class group
$\operatorname{PMod}(S^{2},\mathbf{P})$ acts on
$\operatorname{CC}(S^{2},\mathbf{P})$, but this action can have infinite
stabilizers.

###  3.2. Augmented Teichmüller space

In order to tell a more complete story we introduce augmented Teichmüller
space $\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})$, a partial
compactification of $\mathcal{T}(S^{2},\mathbf{P})$ parametrizing isotopy
classes of tuples $(C,\iota,\psi)$, where $(C,\iota)$ us a $\mathbf{P}$-marked
stable curve and $\psi:S^{2}\to C$ is a surjective continuous map satisfying:

  * •

$\psi$ is a homeomorphism away from nodes of $C$, and the preimage of every
node of $C$ is a simple closed curve of $S^{2}$, and

  * •

$\psi|_{\mathbf{P}}=\iota$.

The action of the mapping class group $\operatorname{PMod}(S^{2},\mathbf{P})$
on $\mathcal{T}(S^{2},\mathbf{P})$ extends continuously to an action on
$\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})$; the quotient is
$\overline{\mathcal{M}}_{0,\mathbf{P}}$. The quotient map
$\rho:\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})\to\overline{\mathcal{M}}_{0,\mathbf{P}}$
sends $[C,\iota,\psi]$ to the stable curve $[C,\iota]$. Note that given
$[C,\iota,\psi]\in\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})$,
the set $\Gamma$ of homotopy classes of simple closed curves of $S^{2}$
contracted to nodes by $\psi$ is a multicurve on $S^{2},\mathbf{P}$. The
boundary
$\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})\setminus\mathcal{T}(S^{2},\mathbf{P})$
is stratified, with strata indexed by multicurves. the boundary complex of
$\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})$ is exactly the curve
complex $\operatorname{CC}(S^{2},\mathbf{P})$.

###  3.3. “Tropical Teichmüller space”: the space of weighted multicurves

Given a multicurve $\Gamma$, we denote by $\operatorname{Cone}(\Gamma)$ the
cone $\mathbb{R}^{\Gamma}_{\geq 0}$. If there is a containment
$\Gamma_{2}\subset\Gamma_{1}$ of multicurves on $S^{2}\setminus\mathbf{P}$,
then there is a natural inclusion, as a face, of
$\operatorname{Cone}(\Gamma_{2})$ into $\operatorname{Cone}(\Gamma_{2})$. We
set $\operatorname{ConeCC}(S^{2},\mathbf{P})$ to be the union of the the cones
$\operatorname{Cone}(\Gamma)$, where $\Gamma$ ranges over all multicurves on
$S^{2}\setminus\mathbf{P}$, glued along the aforementioned inclusion maps.
This is the space of all “multicurves with positive weights”. Note that
$\operatorname{ConeCC}(S^{2},\mathbf{P})$ can be naturally identified with the
cone over $\operatorname{CC}(S^{2},\mathbf{P})$, the boundary complex of
$\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})$. We propose here
that $\operatorname{ConeCC}(S^{2},\mathbf{P})$ is the correct candidate for
the “tropical Teichmüller space”
$\mathcal{T}(S^{2},\mathbf{P})^{\operatorname{trop}}$.

Suppose that $\Gamma$ is a multicurve on $S^{2}\setminus\mathbf{P}$. We pick
pairwise disjoint representative curves $\gamma\in[\gamma]\in\Gamma$ and abuse
notation by writing “$S^{2}\setminus\Gamma$” to mean
$S^{2}\setminus\bigcup_{[\gamma]\in\Gamma}\gamma$. The dual tree of $\Gamma$
is the stable $\mathbf{P}_{1}$-marked tree
$\operatorname{\tau}(\Gamma)=(\underline{\operatorname{\tau}},\widehat{\iota})$
obtained as follows: vertices of $\underline{\operatorname{\tau}}$ correspond
to the connected components of $S^{2}\setminus\Gamma$. Edges of
$\underline{\operatorname{\tau}}$ are in canonical bijection with $\Gamma$:
two vertices of $\underline{\operatorname{\tau}}$ are joined by an edge
$e([\gamma])$ if the closures of the corresponding connected components of
$S^{2}\setminus\Gamma$ intersect exactly along $\gamma$. For $p\in\mathbf{P}$,
$\widehat{\iota}(p)$ is the vertex of $\underline{\operatorname{\tau}}$
corresponding to the connected component of $S^{2}\setminus\Gamma$ containing
$p$. Clearly, there is a canonical isomorphism of cones

$\displaystyle\rho^{\operatorname{trop}}:\operatorname{Cone}(\Gamma)=\mathbb{R}_{\geq
0}^{\Gamma}$
$\displaystyle\to\operatorname{Cone}(\operatorname{\tau}(\Gamma))=\mathbb{R}_{\geq
0}^{\operatorname{Edges}(\operatorname{\tau}(\Gamma))}$
$\displaystyle\sum_{[\gamma]\in\Gamma}a_{[\gamma]}[\gamma]$
$\displaystyle\mapsto\sum_{[\gamma]\in\Gamma}a_{[\gamma]}e([\gamma])$

These isomorphisms glue together to yield a piecewise-linear surjective map
$\rho^{\operatorname{trop}}:\operatorname{ConeCC}(S^{2},\mathbf{P})\to\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$.
This map realizes $\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$ as the
quotient of $\operatorname{ConeCC}(S^{2},\mathbf{P})$ by
$\operatorname{PMod}(S^{2},\mathbf{P})$.

If $\mathbf{P}_{1}\subset\mathbf{P}_{2}\subset S^{2}$, a simple closed curve
that is essential and non-peripheral on $S^{2}\setminus\mathbf{P}_{2}$ may be
non-essential or peripheral on $S^{2}\setminus\mathbf{P}_{1}$. For
$\Gamma^{\prime}$ a multicurve on $S^{2}\setminus\mathbf{P}_{2}$, we denote by
$\widetilde{\mu}{*}(\Gamma^{\prime})$ the (possibly empty) multicurve on
$S^{2}\setminus\mathbf{P}_{1}$ consisting of those
$[\gamma^{\prime}]\in\Gamma^{\prime}$ that are essential and non-peripheral on
$S^{2}\setminus\mathbf{P}_{1}$. The projection maps
$\operatorname{Cone}(\Gamma^{\prime})\to\operatorname{Cone}(\widetilde{\mu}{*}(\Gamma^{\prime}))$
glue together to determine a piecewise linear map
$\widetilde{\mu}^{\operatorname{trop}}:\operatorname{ConeCC}(S^{2},\mathbf{P}_{2})\to\operatorname{ConeCC}(S^{2},\mathbf{P}_{1})$.
We suggest that this map is a good candidate for the tropicalization of the
forgetful map $\widetilde{\mu}$. Note that there is no well-defined forgetful
map from $\operatorname{CC}(S^{2},\mathbf{P}_{2})$ to
$\operatorname{CC}(S^{2},\mathbf{P}_{1})$.

######  Remark 3.1.

Chan-Melo-Viviani [CMV13] have put forward a certain partial compactification
of Culler-Vogtmann [CV86] Outer Space as an alternative candidate for the role
of tropical Teichmüller space.

##  4\. Thurston’s topological characterization of PCF rational functions

In this section, we summarize the relevant parts of Thurston’s results
(written in [DH93]) giving a topological characterization of PCF rational
functions.

###  4.1. PCF branched coverings and PCF rational functions

A map $\varphi:S^{2}\to S^{2}$ is called a branched cover if there is a finite
set $\mathbf{B}\subset S^{2}$ such that
$\varphi:S^{2}\setminus\varphi^{-1}(\mathbf{B})\to S^{2}\setminus\mathbf{B}$
is a covering map. Here, we will only deal with orientation-preserving
branched covers. The restriction of $\varphi$ to a small punctured
neighbourhood of any point $x\in S^{2}$ is a covering map of some degree; this
is the local degree of $\varphi$ at that point. A critical point of $\varphi$
is a point at which the local degree is greater than $1$. Every rational
function $f:\mathbb{C}\mathbb{P}^{1}\to\mathbb{C}\mathbb{P}^{1}$ is an
orientation-preserving branched cover, and its critical points are exactly the
points at which the derivative vanishes. The local degree of $f$ at $x$ is the
unique $r>0$ such that the first $r-1$ derivatives of $f$ vanish at $f$ but
the $r$th does not.

The post-critical set of a branched cover is the set
$\\{\varphi^{n}(x)\thickspace|\thickspace n>0,x\mbox{ critical point of
}\varphi\\}$. A branched covering $\varphi:S^{2}\to S^{2}$ is called PCF
(post-critically finite) if its post-critical set is finite. Two PCF branched
covers $\varphi$ and $\varphi^{\prime}$ with post-critical sets $\mathbf{P}$
and $\mathbf{P}^{\prime}$ respectively are called combinatorially equivalent
(also referred to as Thurston equivalent) if there are orientation-preserving
homeomorphisms $\psi_{1},\psi_{2}:S^{2}\to S^{2}$ such that

  * •

$\psi_{1}$ and $\psi_{2}$ are isotopic relative to $\mathbf{P}$, and both map
$\mathbf{P}$ onto $\mathbf{P}^{\prime}$, and

  * •

$\varphi^{\prime}=\psi_{2}\circ\varphi\circ\psi_{1}^{-1}$.

Thurston’s influential result answers the question: given a PCF branched cover
$\varphi$, when is it combinatorially equivalent to a PCF rational function?
In fact, if $\varphi$ has hyperbolic orbifold (as in footnote 1), then either
(A) $\varphi$ is combinatorially equivalent to a rational map $f$, which is
unique up to Möbius conjugacy, or (B) $\varphi$ has a specific type of
topological obstruction in the form of an $\varphi$-stable multicurve, defined
in Section 4.3.

### Fixing a branched cover

We now fix $\varphi$, a PCF branched cover with hyperbolic orbifold, of degree
at least $2$, and whose post-critical set $\mathbf{P}$ has cardinality at
least $4$. Note that $\mathbf{P}\subset\varphi^{-1}(\mathbf{P})$. We denote by
$\widetilde{\mu}$ the forgetful map
$\mathcal{T}(S^{2},\varphi^{-1}(\mathbf{P}))\to\mathcal{T}(S^{2},\mathbf{P})$,
denote by $\mu$ the forgetful map from
$\mathcal{M}_{0,\varphi^{-1}(\mathbf{P}})$ to $\mathcal{M}_{0,\mathbf{P}}$,
denote by $\rho$ the map
$\mathcal{T}(S^{2},\mathbf{P})\to\mathcal{M}_{0,\mathbf{P}}$, and denote by
$\widetilde{\rho}$ the map
$\mathcal{T}(S^{2},\varphi^{-1}(\mathbf{P}))\to\mathcal{M}_{0,\varphi^{-1}(\mathbf{P})}$.

###  4.2. The pullback map on Teichmüller space

A homeomorphism $\psi:S^{2}\to\mathbb{C}\mathbb{P}^{1}$ determines a complex
structure on $S^{2}$. One can pull this complex structure back on charts along
the branched cover $\varphi$ to obtain a new complex structure on $S^{2}$, and
thus a global homeomorphism $\psi^{\prime}:S^{2}\to\mathbb{C}\mathbb{P}^{1}$;
by construction,
$\psi^{\prime}\circ\varphi\circ\psi:\mathbb{C}\mathbb{P}^{1}\to\mathbb{C}\mathbb{P}^{1}$
is holomorphic. Although there are choices involved, the class of
$\psi^{\prime}$ in $\mathcal{T}(S^{2},\varphi^{-1}(\mathbf{P}))$ is well-
defined and only depends on the class of $\psi$ in
$\mathcal{T}(S^{2},\mathbf{P})$. The map
$\widetilde{\operatorname{TP}}_{\varphi}:\mathcal{T}(S^{2},\mathbf{P})\to\mathcal{T}(S^{2},\varphi^{-1}(\mathbf{P}))$
that sends $[\psi]$ to $[\psi^{\prime}]$ is holomorphic. Composing with the
forgetful map $\widetilde{\mu}$, we obtain a holomorphic selfmap
$\operatorname{TP}_{\varphi}:=\widetilde{\mu}\circ\widetilde{\operatorname{TP}}_{\varphi}$
of $\mathcal{T}(S^{2},\mathbf{P})$, called Thurston’s pullback map.

###  4.3. $\varphi$-stable multicurves and Thurston obstructions

If $\Gamma$ is a multicurve on $S^{2}\setminus\mathbf{P}$, then its pullback
$\widetilde{\varphi}^{*}(\Gamma):=\\{[\varphi^{-1}(\gamma)]\thickspace|\thickspace[\gamma]\in\Gamma\\}$
is a multicurve on $S^{2}\setminus\varphi^{-1}(\mathbf{P})$. Some (or all) of
the components components of $\widetilde{\varphi}^{*}(\Gamma)$ may be
peripheral or non-essential on $S^{2}\setminus\mathbf{P}$. We set
$\varphi^{*}(\Gamma)$ to be the possibly empty multicurve
$\widetilde{\mu}_{*}([\Gamma])$.

There is a natural Thurston linear transformation
$\operatorname{\widetilde{TLT}}_{\varphi,\Gamma}$ from $\mathbb{R}^{\Gamma}$
to $\mathbb{R}^{\widetilde{\varphi}^{*}(\Gamma)}$, defined as follows in terms
on its action on the natural basis:

(1)
$\displaystyle\operatorname{\widetilde{TLT}}_{\varphi,\Gamma}([\gamma])=\sum_{\begin{subarray}{c}\gamma^{\prime}\text{simple
closed curve}\\\
\varphi(\gamma^{\prime})=\gamma\end{subarray}}\frac{1}{\deg(\varphi:\gamma^{\prime}\to\gamma)}[\gamma^{\prime}]$

The composition
$\operatorname{TLT}_{\varphi,\Gamma}:=\widetilde{\mu}^{\operatorname{trop}}\circ\operatorname{\widetilde{TLT}}_{\varphi,\Gamma}$
is also called the Thurston linear transformation; this is a map from
$\mathbb{R}^{\Gamma}$ to $\mathbb{R}^{\varphi^{*}(\Gamma)}$ acting on the
natural basis as follows:

(2)
$\displaystyle\operatorname{TLT}_{\varphi,\Gamma}([\gamma])=\sum_{\begin{subarray}{c}[\gamma^{\prime\prime}]\text{
essential and}\\\ \text{non-peripheral rel.
}\mathbf{P}\end{subarray}}\sum_{\begin{subarray}{c}\gamma^{\prime}\text{simple
closed}\\\ \varphi(\gamma^{\prime})=\gamma\\\ \gamma^{\prime}\text{ homotopic
to }\gamma^{\prime\prime}\text{ rel.
}\mathbf{P}\end{subarray}}\frac{1}{\deg(\varphi:\gamma^{\prime}\to\gamma)}[\gamma^{\prime\prime}]$

Both linear transformations (1) and (2) are commonly referred to as Thurston
linear transformations associated to the multicurve $\Gamma$.

A multicurve $\Gamma$ is called $\varphi$-stable if
$(\operatorname{TP}_{\varphi})^{*}(\Gamma)\subset\Gamma$. If $\Gamma$ is
$\varphi$-stable then the Thurston linear transformation associated to
$\Gamma$ can be considered to be a linear self-map of $\mathbb{R}^{\Gamma}$.
Given that the Thurston linear transformation has non-negative coefficients
with respect to the natural basis, its dominant eigenvalue
$\Lambda_{\varphi}(\Gamma)$ is non-negative and has an eigenvector in
$\mathbb{R}^{\Gamma}_{\geq 0}$. A $\varphi$-stable multicurve $\Gamma$ is
called a Thurston obstruction if $\Lambda_{\varphi}(\Gamma)\geq 1$.

###  4.4. Thurston’s theorem

The pullback map $\operatorname{TP}_{\varphi}$ admits a fixed point if and
only if $\varphi$ does not admit a Thurston obstruction. Also,
$\operatorname{TP}_{\varphi}$ has a fixed point if and only if $\varphi$ is
combinatorially equivalent to a rational function: if $[\psi]$ is a fixed
point of $\operatorname{TP}_{\varphi}$, then there is a homeomorphism
$\psi^{\prime}:S^{2}\to\operatorname{\mathbb{C}\mathbb{P}^{1}}$ that is
isotopic to $\psi$ rel. $\mathbf{P}$, and such that
$\psi^{\prime}\circ\varphi\circ\psi^{-1}$ is a PCF rational function. In fact,
$\operatorname{TP}_{\varphi}$ is distance-nonincreasing and
$\operatorname{TP}_{\varphi}^{2}$ is strictly distance-decreasing on
$\mathcal{T}(S^{2},\mathbf{P})$, and so a fixed point, if it exists, is
unique. We conclude that either (A) $\varphi$ is combinatorially equivalent to
a rational map, which is unique up to Möbius conjugacy, or (B) $\varphi$
admits a Thurston obstruction $\Gamma$.

###  4.5. The augmented pullback map

Selinger [Sel12] showed that Thurston’s pullback map extends continuously to
augmented Teichmüller space
$\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})$. We describe this
extension in more detail in the proof of Lemma 7.3.

###  4.6. “The tropical pullback map”

Observe that since the Thurston linear transformation (1) has non-negative
coefficients with respect to the natural bases, it restricts to a map
$\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}(\Gamma):\operatorname{Cone}(\Gamma)\to\operatorname{Cone}(\widetilde{\varphi}^{*}(\Gamma)).$
It is straightforward to check that the maps
$\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}(\Gamma)$ glue
together along faces to yield a piece-wise-linear map
$\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}:\operatorname{ConeCC}(S^{2},\mathbf{P})\to\operatorname{ConeCC}(S^{2},\varphi^{-1}(\mathbf{P})).$
Similarly, the linear transformation (2) restricts to a map
$\operatorname{TP}_{\operatorname{trop}}^{\operatorname{trop}}(\Gamma):\operatorname{Cone}(\Gamma)\to\operatorname{Cone}(\varphi^{*}(\Gamma)),$
and the maps $\operatorname{TP}_{\varphi}^{\operatorname{trop}}(\Gamma)$ glue
together along faces to yield a piece-wise-linear map
$\operatorname{TP}_{\varphi}^{\operatorname{trop}}:\operatorname{ConeCC}(S^{2},\mathbf{P})\to\operatorname{ConeCC}(S^{2},\varphi^{-1}(\mathbf{P})).$
We use the notation
$\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}$ and
$\operatorname{TP}_{\varphi}^{\operatorname{trop}}$ in order to propose that
these piece-wise linear maps ought to be considered to be tropicalizations of
$\widetilde{\operatorname{TP}}_{\varphi}$ and $\operatorname{TP}_{\varphi}$
respectively. Note that
$\operatorname{TP}_{\varphi}^{\operatorname{trop}}=\widetilde{\mu}^{\operatorname{trop}}\circ\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}.$

Under this interpretation, a $\varphi$-stable multicurve $\Gamma$ is exactly a
multicurve such that $\operatorname{Cone}(\Gamma)$ is invariant under the
action of $\operatorname{TP}_{\varphi}^{\operatorname{trop}}$, in which case
there is a ray in $\operatorname{Cone}(\Gamma)$ that is fixed and scaled by
the dominant eigenvalue $\Lambda_{\varphi}(\Gamma)$. A Thurston obstruction
therefore is the same as a ray in $\operatorname{ConeCC}(S^{2},\mathbf{P})$
that is fixed by $\operatorname{TP}_{\varphi}^{\operatorname{trop}}$, and
scaled by a factor $\geq 1$.

##  5\. Hurwtiz spaces and the moduli space correspondence

Here, we summarize the results of [Koc13] as they apply to this article. There
is Hurwitz space — a non-compact smooth variety
$\mathcal{H}_{\mathrm{prof}}(\varphi)$ parametrizing marked rational functions
on $\operatorname{\mathbb{C}\mathbb{P}^{1}}$ that have the same branching
profile as $\varphi$. More precisely, $\mathcal{H}_{\mathrm{prof}}(\varphi)$
parameterizes tuples $[f,\iota_{1},\iota_{2}]$, where

  * •

$\iota_{1}:\mathbf{P}\hookrightarrow\operatorname{\mathbb{C}\mathbb{P}^{1}}$
and
$\iota_{2}:\varphi^{-1}(\mathbf{P})\hookrightarrow\operatorname{\mathbb{C}\mathbb{P}^{1}}$
are injections,

  * •

$f:(\operatorname{\mathbb{C}\mathbb{P}^{1}},\iota_{2}(\varphi^{-1}(\mathbf{P})))\to(\operatorname{\mathbb{C}\mathbb{P}^{1}},\iota_{1}(\mathbf{P}))$
is a rational function that has the same degree as $\varphi$, and that maps
$\iota_{2}(\varphi^{-1}(\mathbf{P}))$ to $\iota_{1}(\mathbf{P})$ exactly as
$\varphi$ maps $\varphi^{-1}(\mathbf{P})$ to $\mathbf{P}$ (that is,
$f\circ\iota_{2}=\iota_{1}\circ\varphi$), including with the same local
degrees. (Note that this forces $f$ to have no critical values away from
$\iota_{1}(\mathbf{P})$.)

Two tuples $[f,\iota_{1},\iota_{2}]$ and
$[f^{\prime},\iota_{1}^{\prime},\iota_{2}^{\prime}]$ are equal in
$\mathcal{H}_{\mathrm{prof}}(\varphi)$ if they differ by changes of
coordinates on $\operatorname{\mathbb{C}\mathbb{P}^{1}}$, i.e. if there are
automorphisms $\alpha_{1}$ and $\alpha_{2}$ of
$\operatorname{\mathbb{C}\mathbb{P}^{1}}$ such that
$\iota_{1}^{\prime}=\alpha_{1}\circ\iota_{1}$,
$\iota_{2}^{\prime}=\alpha_{2}\circ\iota_{2}$, and $f^{\prime}=\alpha_{1}\circ
f\circ(\alpha_{2})^{-1}$. In general, $\mathcal{H}_{\mathrm{prof}}(\varphi)$
is disconnected, with connected components corresponding to a type of non-
dynamical topological equivalence known as Hurwitz equivalence, defined below.

######  Definition 5.1.

Given $\iota_{1}^{\prime},\iota_{1}^{\prime\prime}:\mathbf{P}\hookrightarrow
S^{2}$ and
$\iota_{2}^{\prime},\iota_{2}^{\prime\prime}:\varphi^{-1}(\mathbf{P})\hookrightarrow
S^{2}$ injections, and
$\varphi^{\prime}:(S^{2},\iota_{2}^{\prime}(\varphi^{-1}(\mathbf{P})))\to(S^{2},\iota_{1}^{\prime}(\mathbf{P}))$
and
$\varphi^{\prime\prime}:(S^{2},\iota_{2}^{\prime\prime}(\varphi^{-1}(\mathbf{P})))\to(S^{2},\iota_{1}^{\prime\prime}(\mathbf{P}))$
branched covers, we say that $\varphi^{\prime}$ and $\varphi^{\prime\prime}$
are Hurwitz equivalent if there exist homeomorphisms
$\psi_{1},\psi_{2}:S^{2}\to\operatorname{\mathbb{C}\mathbb{P}^{1}}$ such that
$(\psi_{1})\circ\iota_{1}^{\prime}=\iota_{1}^{\prime\prime}$ on $\mathbf{P}$,
$\psi_{2}\circ\iota_{2}^{\prime}=\iota_{2}^{\prime\prime}$ on
$\varphi^{-1}(\mathbf{P})$, and
$\varphi^{\prime\prime}=\psi_{1}\circ\varphi^{\prime}\circ(\psi_{2})^{-1}$.
Note that $\psi_{1}$ and $\psi_{2}$ are not required to be isotopic rel.
$\mathbf{P}$.

$\mathcal{H}_{\mathrm{prof}}(\varphi)$ has a unique connected component
$\mathcal{H}(\varphi)$ consisting of marked rational functions $f$ that are
Hurwitz equivalent to $\varphi$. If $\varphi^{\prime}$ is Hurwitz equivalent
to $\varphi$ then $\mathcal{H}(\varphi^{\prime})=\mathcal{H}(\varphi)$.

There are maps
$\pi_{1},\pi_{2}:\mathcal{H}_{\mathrm{prof}}(\varphi)\to\mathcal{M}_{0,\mathbf{P}}$
send $[f,\iota_{1},\iota_{2}]$ to $[\iota_{1}]$ and
$[\iota_{2}|_{\mathbf{P}}]$ respectively. The map $\pi_{1}$ is a covering map.
Note that we can factor $\pi_{2}$ as $\pi_{2}=\mu\circ\widetilde{\pi}_{2}$,
where
$\widetilde{\pi}_{2}:\mathcal{H}_{\mathrm{prof}}(\varphi)\to\mathcal{M}_{0,\varphi^{-1}(\mathbf{P})}$
sends $[f,\iota_{1},\iota_{2}]$ to $[\iota_{2}]$. The pair of maps
$\pi_{1},\pi_{2}:\mathcal{H}(\varphi)\to\mathcal{M}_{0,\mathbf{P}}$ as well as
the multivalued self-map

$\operatorname{MSC}_{\varphi}:=(\pi_{2})|_{\mathcal{H}(\varphi)}\circ(\pi_{1})|_{\mathcal{H}(\varphi)}^{-1}$

of $\mathcal{M}_{0,\mathbf{P}}$ are often referred to as the moduli space
correspondence333The Hurwitz space $\mathcal{H}(\varphi)$ defined here is a
finite cover of the Hurwitz space introduced in [Koc13], and so our moduli
space correspondence is off from the standard moduli space correspondence by a
global multiplicity factor. This minor technical modification doesn’t affect
any of the important properties but is better-behaved in the contexts of
compactifications and tropicalizations of Hurwitz spaces.. If
$\varphi^{\prime}$ is Hurwitz equivalent to $\varphi$ then
$\operatorname{MSC}_{\varphi^{\prime}}=\operatorname{MSC}_{\varphi}$ The
holomorphic Thurston pullback map $\operatorname{TP}_{\varphi}$ descends, via
the universal covering map $\rho$, to the algebraic moduli space
correspondence on $\mathcal{M}_{0,\mathbf{P}}$ in the following sense: there
is a holomorphic covering map
$\nu_{\varphi}:\mathcal{T}(S^{2},\mathbf{P})\to\mathcal{H}(\varphi)$ such that
$\rho=\pi_{1}\circ\nu_{\varphi}$, and
$\rho\circ\operatorname{TP}_{\varphi}=\pi_{2}\circ\nu_{\varphi}$. The
following diagram encapsulates the relationship between the moduli space
correspondence and Thurston’s pullback map.

(3)
${\mathcal{T}(S^{2},\mathbf{P})}$${\mathcal{T}(S^{2},\mathbf{P})}$${\mathcal{H}(\varphi)}$${\mathcal{M}_{0,\mathbf{P}}}$${\mathcal{M}_{0,\mathbf{P}}}$$\operatorname{TP}_{\varphi}$$\nu_{\varphi}$$\rho$$\rho$$\pi_{1}$$\pi_{2}$

A fixed point of the moduli space correspondence is a point
$x=[f,\iota_{1},\iota_{2}]\in\mathcal{H}(\varphi)$ such that
$[\iota_{1}]=\pi_{1}(x)=\pi_{2}(x)=[\iota_{2}|_{\mathbf{P}}]$. Note this means
that $[\iota_{1}]$ is only a “weakly” fixed point of the multivalued map
$\operatorname{MSC}_{\varphi}$ in the sense that it is contained in its own
image. In general, $\operatorname{MSC}_{\varphi}$ is unlikely to have a
“strongly” fixed point, i.e. a point whose image contains itself and no other
point. Fixed points of the moduli space correspondence correspond to PCF
rational functions that are Hurwitz equivalent to $\varphi$.

##  6\. Compactifications and tropicalizations of Hurwitz spaces

###  6.1. Admissible covers

An admissible cover is a finite map $f:C_{2}\to C_{1}$ of pre-stable curves
satisfying

  1. (1)

smooth points of $C_{2}$ map to smooth points of $C_{1}$, and nodes of $C_{2}$
maps to nodes of $C_{1}$,

  2. (2)

(balancing at nodes) the two branches at a node $x$ of $C_{2}$ map to distinct
branches at $f(x)$, with equal local degrees at $x$.

The above conditions ensure that $f$ is can be obtained as a limit of a family
of maps between smooth algebraic curves. An admissible cover with the same
profile as $\varphi$ is a tuple $(f,C_{1},\iota_{1},C_{2},\iota_{2})$, where

  * •

$(C_{1},\iota_{1})$ is a $\mathbf{P}$-marked stable curve,

  * •

$(C_{2},\iota_{2})$ is a $\varphi^{-1}(\mathbf{P})$-marked stable curve,

  * •

$f:C_{2}\to C_{1}$ is an admissible cover of the same degree as $\varphi$,
such that

    * –

$f$ maps $\iota_{2}(\varphi^{-1}(\mathbf{P}))$ to $\iota_{1}(\mathbf{P})$
exactly as $\varphi$ maps $\varphi^{-1}(\mathbf{P})$ to $\mathbf{P}$ (that is,
$f\circ\iota_{2}=\iota_{1}\circ\varphi$), including with the same local
degrees,

    * –

$f$ has no critical points away from $\iota_{2}(\varphi^{-1}(\mathbf{P}))$ and
the set of nodes of $C_{2}$.

Due to works of Harris-Mumford [HM82], there is a possibly reducible
projective variety $\overline{\mathcal{H}}_{\mathrm{prof}}(\varphi)$, with
mild singularities, that parametrizes admissible covers with the same profile
as $\varphi$. $\overline{\mathcal{H}}_{\mathrm{prof}}(\varphi)$ contains
$\mathcal{H}(\varphi)$ as an open subset. Set
$\overline{\mathcal{H}}(\varphi)$ to be the closure of $\mathcal{H}(\varphi)$
in $\overline{\mathcal{H}}_{\mathrm{prof}}(\varphi)$ – this is an irreducible
component of $\overline{\mathcal{H}}_{\mathrm{prof}}(\varphi)$. If
$[(f,C_{1},\iota_{1},C_{2},\iota_{2})]\in\overline{\mathcal{H}}(\varphi)$ we
say that $f$ is an admissible cover of topological type $\varphi$, because it
can be obtained as a limit of a family of rational maps that are Hurwitz
equivalent to $\varphi$444There is in principle a way to characterize $f$ of
topological type $\varphi$ in terms of the Hurwitz equivalence classes of the
restrictions of $f$ to the irreducible
$\operatorname{\mathbb{C}\mathbb{P}^{1}}$ components of $C_{2}$. However, it
is subtle and, as far as we are aware, has not been carefully worked out in
the literature.. We set $\widetilde{\mathcal{H}}_{\mathrm{prof}}(\varphi)$ to
the the normalization of $\overline{\mathcal{H}}_{\mathrm{prof}}(\varphi)$ –
by [HM82] $\widetilde{\mathcal{H}}_{\mathrm{prof}}(\varphi)$ is smooth, and by
Abramovich-Corti-Vistoli [ACV03] it admits an interpretation as a moduli space
of covers of orbifold curves.
$\widetilde{\mathcal{H}}_{\mathrm{prof}}(\varphi)$ contains as a connected
component the normalization $\widetilde{\mathcal{H}}(\varphi)$ of
$\overline{\mathcal{H}}(\varphi)$ — this is a smooth compactification of
$\mathcal{H}(\varphi)$.

###  6.2. The moduli space correspondence on
$\overline{\mathcal{M}}_{0,\mathbf{P}}$

The maps $\pi_{1}$ and $\pi_{2}$ from $\mathcal{H}_{\mathrm{prof}}(\varphi)$
to $\mathcal{M}_{0,\mathbf{P}}$ extend to regular maps from both
$\overline{\mathcal{H}}_{\mathrm{prof}}(\varphi)$ and its normalization
$\widetilde{\mathcal{H}}_{\mathrm{prof}}(\varphi)$ to
$\overline{\mathcal{M}}_{0,\mathbf{P}}$. The extension of $\pi_{1}$ to
$\widetilde{\mathcal{H}}_{\mathrm{prof}}(\varphi)$ is a ramified cover. As a
result, the multivalued moduli space correspondence
$\operatorname{MSC}_{\varphi}$ extends to
$\overline{\mathcal{M}}_{0,\mathbf{P}}$ “without indeterminacy” [Ram18]. Also,
just as $\pi_{2}$ factors as $\mu\circ\widetilde{\pi_{2}}$, so does its
extension to $\widetilde{\mathcal{H}}_{\mathrm{prof}}(\varphi)$. By [HV10],
the holomorphic map
$\nu_{\varphi}:\mathcal{T}(S^{2}\mathbf{P})\to\mathcal{H}(\varphi)$ extends
continuously to a surjective map
$\nu_{\varphi}:\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})\to\overline{\mathcal{H}}(\varphi)$,
which in turn lifts to a continuous surjective map
$\widetilde{\nu_{\varphi}}:\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})\to\widetilde{\mathcal{H}}(\varphi)$.
We describe the extension $\nu_{\varphi}$ in more detail in the proof of Lemma
7.3. The upshot is that Diagram (3) Section 5 extends to partial/full
compactifications of the spaces involved as depicted below, and the
compactified moduli space correspondence descends from the augmented pullback
map.

(4)
${\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})}$${\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})}$${\overline{\mathcal{H}}(\varphi)}$${\overline{\mathcal{M}}_{0,\mathbf{P}}}$${\overline{\mathcal{M}}_{0,\mathbf{P}}}$$\operatorname{TP}_{\varphi}$$\nu_{\varphi}$$\rho$$\rho$$\pi_{1}$$\pi_{2}$

###  6.3. Combinatorial types of admissible covers

Suppose $(f,C_{1},\iota_{1},C_{2},\iota_{2})$ is an admissible cover with the
same profile as $\varphi$. The combinatorial type
$\Lambda=(\operatorname{\tau}_{1},\operatorname{\tau}_{2},F,\operatorname{EdgeDeg},\varphi|_{\varphi^{-1}(\mathbf{P})},\operatorname{LegDeg})$
of $f$ records

  * •

the marked dual trees
$\operatorname{\tau}_{1}=(\underline{\operatorname{\tau}_{1}},\widehat{\iota_{1}})$
and
$\operatorname{\tau}_{2}=(\underline{\operatorname{\tau}_{2}},\widehat{\iota_{2}})$
of $C_{1}$ and $C_{2}$ respectively,

  * •

the graph homomorphism
$F:\underline{\operatorname{\tau}_{2}}\to\underline{\operatorname{\tau}_{1}}$
of dual trees that encodes on vertices how irreducible components of $C_{2}$
map to irreducible components of $C_{1}$, and on edges how nodes of $C_{2}$
map to nodes of $C_{1}$,

  * •

the edge-degree map
$\operatorname{EdgeDeg}:\operatorname{Edges}(\operatorname{\tau}_{2})\to\mathbb{Z}_{>0}$:
for $e$ an edge of $\operatorname{\tau}_{2}$ corresponding to a node $\eta$ of
$C_{2}$, $\operatorname{EdgeDeg}(e)$ is the local degree of $f$ at $\eta$
(this is well-defined by the balancing condition),

  * •

the restriction $\varphi|_{\varphi^{-1}(\mathbf{P})}$ recording how legs of
$\operatorname{\tau}_{2}$ map to legs of $\operatorname{\tau}_{1}$, and

  * •

the leg-degree map
$\operatorname{LegDeg}:\varphi^{-1}(\mathbf{P})\to\mathbb{Z}_{>0}$ sending $p$
to the local degree of $\varphi$ at $p$ (equivalently, that of $f$ at
$\iota_{2}(p)$).

Note that the final two entries of the combinatorial type of $f$ depend only
on $\varphi$. If $\Lambda$ is the combinatorial type of some admissible cover
with the same profile as $\varphi$, we will say that $\Lambda$ is a
realizeable combinatorial type of profile $\varphi$. If $\Lambda$ is the
combinatorial type of some admissible cover with the same topological type as
$\varphi$, we will say that $\Lambda$ is a realizeable combinatorial type of
topological type $\varphi$.

The locus
$\mathcal{H}(\Lambda)\subset\overline{\mathcal{H}}_{\mathrm{prof}}(\varphi)$
consisting of admissible covers with fixed combinatorial type $\Lambda$ is a
locally-closed subset that is isomorphic to a product of Hurwitz spaces. Its
closure $\overline{\mathcal{H}}(\Lambda)$ is a not-necessarily-irreducible
subvariety called a boundary stratum. The boundary stratum
$\overline{\mathcal{H}}(\Lambda)$ has a component contained in
$\overline{\mathcal{H}}(\varphi)$ if and only if $\Lambda$ has topological
type $\varphi$.

Suppose that
$\Lambda=(\operatorname{\tau}_{1},\operatorname{\tau}_{2},F,\operatorname{EdgeDeg},\varphi|_{\varphi^{-1}(\mathbf{P})},\operatorname{LegDeg})$
is a realizeable combinatorial type of profile $\varphi$, and that $E$ is a
(possibly empty) subset of edges of $\operatorname{\tau}_{1}$. Set
$\operatorname{\tau}_{1}^{\prime}$ to be the tree obtained from
$\operatorname{\tau}_{1}$ by contracting all the edges not in $E$, and set
$\operatorname{\tau}_{2}^{\prime}$ to be the tree obtained from
$\operatorname{\tau}_{1}$ by contracting all the edges $e$ such that
$F(e)\not\in E$. There is an induced graph homomorphism
$F^{\prime}:\underline{\operatorname{\tau}_{2}^{\prime}}\to\underline{\operatorname{\tau}_{1}^{\prime}}$.
Set
$\Lambda^{\prime}=(\operatorname{\tau}_{1}^{\prime},\operatorname{\tau}_{2}^{\prime},F^{\prime},\operatorname{EdgeDeg}|_{E},\varphi|_{\varphi^{-1}(\mathbf{P})},\operatorname{LegDeg})$.
Then $\Lambda^{\prime}$ is also a realizeable combinatorial type of profile
$\varphi$; in fact, any admissible cover $f$ with combinatorial type $\Lambda$
can be obtained as a limit of a family of admissible covers with combinatorial
type $\Lambda^{\prime}$ [HM82]. In this setting we say that $\Lambda^{\prime}$
is obtained from $\Lambda$ by edge-contraction. The containment
$\overline{\mathcal{H}}(\Lambda)\subset\overline{\mathcal{H}}(\Lambda^{\prime})$
holds if and only if $\Lambda^{\prime}$ is obtained from $\Lambda$ by edge-
contraction. If $\Lambda$ has topological type $\varphi$ and
$\Lambda^{\prime}$ is obtained from $\Lambda$ by edge-contraction then
$\Lambda^{\prime}$ also has topological type $\varphi$.

######  Remark 6.1.

It is possible but computationally difficult to enumerate the realizeable
combinatorial types of profile $\varphi$. Doing so involves solving cases of
the Hurwitz realizeability problem of deciding whether there exist branched
covers with a specified ramification profile, or, equivalently, whether one
can factor the identity element in a symmetric group as a product of elements
with specified cycle types. However, it is so far not well-understood how to
determine whether or not a realizeable combinatorial type has topological type
$\varphi$.

###  6.4. The tropical Hurwitz space

In this section, we summarize some of [CMR16] as it applies to the moduli
space correspondence. Suppose that
$\Lambda=(\operatorname{\tau}_{1},\operatorname{\tau}_{2},F,\operatorname{EdgeDeg},\varphi|_{\varphi^{-1}(\mathbf{P})},\operatorname{LegDeg})$
is a realizeable combinatorial type of profile $\varphi$. A tropical
admissible cover of combinatorial type $\Lambda$ is a tuple
$(\Sigma_{1},\Sigma_{2},\mathbf{F})$, where:

  * •

$\Sigma_{1}$ and $\Sigma_{2}$ are tropical curves ($\mathbf{P}$\- and
$\varphi^{-1}(\mathbf{P})$\- marked respectively) with underlying trees
$\operatorname{\tau}_{1}$ and $\operatorname{\tau}_{2}$ respectively,

  * •

$\mathbf{F}:\Sigma_{2}\to\Sigma_{1}$ is a piece-wise affine map realizing the
combinatorial graph homomorphism $F$, such that the leg of $\Sigma_{2}$
corresponding to $p\in\varphi^{-1}$ maps to the leg of $\Sigma_{1}$
corresponding to $\varphi(p)$, and such that slopes encode local degrees as
follows:

    * –

for $e$ an edge of $\Sigma_{2}$,
$\operatorname{Slope}(\mathbf{F}|_{e})=\operatorname{EdgeDeg}(e)$, and

    * –

for $l$ a leg of $\Sigma_{2}$ corresponding to $p\in\varphi^{-1}(\mathbf{P})$,
$\operatorname{Slope}(\mathbf{F}|_{l})=\text{(local deg. of $\varphi$ at
$p$)}$.

Given $\Lambda$ together with $\Sigma_{1}$ a tropical curve with underlying
tree $\operatorname{\tau}_{1}$, there is a unique tropical admissible cover
$(\Sigma_{1},\Sigma_{2},\mathbf{F})$ with combinatorial type $\Lambda$: the
edge-lengths of $\Sigma_{2}$ and the piece-wise affine structure of
$\mathbf{F}$ are determined by $F$, $\operatorname{EdgeDeg}$ and
$\operatorname{LegDeg}$. This tells us that the space
$\operatorname{Cone}^{\circ}(\Lambda)$ of tropical admissible covers of
combinatorial type $\Lambda$ is a cone isomorphic to
$\mathbb{R}_{>0}^{\left\lvert\operatorname{Edges}(\operatorname{\tau}_{1})\right\rvert}$.
However, unlike the case of a cone
$\operatorname{Cone}^{\circ}(\operatorname{\tau})=\mathbb{R}_{>0}^{\operatorname{Edges}(\operatorname{\tau})}$
of $\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$, where the cone
coordinates are exactly the edge lengths of the tropical curve, in the case of
$\operatorname{Cone}^{\circ}(\Lambda)$ the relationship between the cone
coordinates and the edge-lengths of $\Sigma_{1}$ is more complicated, as we
describe. In order to avoid confusion, we will use the notation $\check{e}$
for the cone coordinate on $\operatorname{Cone}^{\circ}(\Lambda)$
corresponding to $e\in\operatorname{Edges}(\operatorname{\tau}_{1})$. Now, for
$e\in\operatorname{Edges}(\operatorname{\tau}_{1})$, set
$\operatorname{LcmDeg}(e):=\operatorname{lcm}\\{\operatorname{EdgeDeg}(e^{\prime})\\}_{F(e^{\prime})=e}$.
Then, from
$\sum_{e\in\operatorname{Edges}(\operatorname{\tau}_{1})}a_{e}\check{e}\in\operatorname{Cone}^{\circ}(\Lambda)$
we obtain a tropical curve $\Sigma_{1}$ by assigning, to edge $e$ of
$\operatorname{\tau}_{1}$, the length $\operatorname{LcmDeg}(e)\cdot a_{e}$.
This edge-length assignment determines a unique tropical admissible cover of
combinatorial type $\Lambda$. It follows that an edge $e^{\prime}$ of
$\operatorname{\tau}_{2}$ mapping via $F$ to $e$ is assigned length
$\frac{\operatorname{LcmDeg}(e)}{\operatorname{EdgeDeg}(e^{\prime})}\cdot
a_{e}$ ([CMR16] Section 5.4). The closed cone
$\operatorname{Cone}(\Lambda)=\mathbb{R}_{\geq 0}^{\\{\check{e}\\}}$ can be
canonically identified with the space of of tropical admissible covers whose
combinatorial types can be obtained from $\Lambda$ by edge-contraction. Set
$\mathcal{H}_{\mathrm{prof}}^{\operatorname{trop}}(\varphi)$ to be the
tropical Hurwitz space parameterizing tropical admissible covers of profile
$\varphi$; this is a complex constructed by gluing the cones
$\\{\operatorname{Cone}(\Lambda)\\}_{\Lambda\mbox{ realizeable}}$ together
along the inclusions of faces
$\operatorname{Cone}(\Lambda^{\prime})\hookrightarrow\operatorname{Cone}(\Lambda)$,
where $\Lambda^{\prime}$ is obtained from $\Lambda$ by edge-contraction.

There are natural piece-wise linear maps $\pi_{1}^{\operatorname{trop}}$ and
$\pi_{2}^{\operatorname{trop}}$ from
$\mathcal{H}_{\mathrm{prof}}^{\operatorname{trop}}(\varphi)$ to
$\mathcal{M}_{0,\mathbf{P}^{\operatorname{trop}}}$, sending
$(\Sigma_{1},\Sigma_{2},\mathbf{F})$ to $[\Sigma_{1}]$ and
$[\mu^{\operatorname{trop}}(\Sigma_{2})]$ respectively; clearly
$\pi_{2}^{\operatorname{trop}}$ factors as
$\mu^{\operatorname{trop}}\circ\widetilde{\pi_{2}}^{trop}$, where
$\widetilde{\pi_{2}}^{\operatorname{trop}}:\mathcal{H}_{\mathrm{prof}}^{\operatorname{trop}}(\varphi)\to\mathcal{M}_{0,\varphi^{-1}(\mathbf{P})}$
sends $(\Sigma_{1},\Sigma_{2},\mathbf{F})$ to $[\Sigma_{2}]$. The map
$\pi_{1}^{\operatorname{trop}}$ is finite-to-one. More specifically, given any
cone $\operatorname{Cone}(\operatorname{\tau}_{1})$ of
$\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$,
$(\pi_{1}^{trop})^{-1}(\operatorname{Cone}(\operatorname{\tau}_{1}))$ consists
of finitely many cones of
$\mathcal{H}_{\mathrm{prof}}^{\operatorname{trop}}(\varphi)$. Each such cone
is of the form $\operatorname{Cone}(\Lambda)$, where
$\Lambda=(\operatorname{\tau}_{1},\ldots)$. In terms of the generators
$(\check{e})_{e\in\operatorname{Edges}(\operatorname{\tau}_{1})}$ for
$\operatorname{Cone}(\Lambda)$ and
$\operatorname{Edges}(\operatorname{\tau}_{1})$ for
$\operatorname{Cone}(\operatorname{\tau}_{1})$, the map
$\pi_{1}^{\operatorname{trop}}$ is diagonal with scaling factors
$(\operatorname{LcmDeg}(e))$. This reflects the local geometry of the map
$\pi_{1}:\widetilde{\mathcal{H}}_{\mathrm{prof}}(\varphi)\to\overline{\mathcal{M}}_{0,\mathbf{P}}$
as follows: there are local coordinates $(x_{e})$ on
$\widetilde{\mathcal{H}}_{\mathrm{prof}}(\varphi)$ along the boundary stratum
$\overline{\mathcal{H}}(\Lambda)$ and local coordinates $(y_{e})$ on
$\overline{\mathcal{M}}_{0,\mathbf{P}}$ along
$\operatorname{X}_{\operatorname{\tau}_{1}}$ such that
$\pi_{1}^{*}(y_{e})=x_{e}^{\operatorname{LcmDeg}(e)}$.

######  Remark 6.2 (Relationship with the boundary complex of admissible
covers).

Unlike $\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$, which is the cone
over the boundary complex of $\overline{\mathcal{M}}_{0,\mathbf{P}}$, the
tropical Hurwitz space
$\mathcal{H}_{\mathrm{prof}}^{\operatorname{trop}}(\varphi)$ is not the cone
over the boundary complex of
$\widetilde{\mathcal{H}}_{\mathrm{prof}}(\varphi)$. There is a possibly
disconnected cone complex
$\mathcal{H}^{\operatorname{trop}}_{\mathrm{prof,true}}$, the “true tropical
Hurwitz space”, that has a $k$-dimensional cone for every irreducible
component of some codimension-$k$ boundary stratum of
$\widetilde{\mathcal{H}}_{\mathrm{prof}}(\varphi)$. By [CMR16], there is a
finite-to-one map
$\chi:\mathcal{H}^{\operatorname{trop}}_{\mathrm{prof,true}}\to\mathcal{H}_{\mathrm{prof}}^{\operatorname{trop}}(\varphi)$
whose restriction to every open cone is an isomorphism onto its image, but
which might send distinct cones of
$\mathcal{H}^{\operatorname{trop}}_{\mathrm{prof,true}}$ to the same cone of
$\mathcal{H}_{\mathrm{prof}}^{\operatorname{trop}}(\varphi)$.

##  7\. The tropical moduli space correspondence and Thurston’s pullback map

There is a closed sub-cone-complex (of full dimension)
$\mathcal{H}^{\operatorname{trop}}(\varphi)\subset\mathcal{H}_{\mathrm{prof}}^{\operatorname{trop}}(\varphi)$
consisting of tropical admissible covers whose combinatorial types also have
topological type $\varphi$. However, as explained in Remark 6.1 it is not
well-understood how to in general enumerate the cones of, and to explicitly
describe $\mathcal{H}^{\operatorname{trop}}(\varphi)$. We will refer to pair
of maps
$\pi_{1}^{\operatorname{trop}},\pi_{2}^{\operatorname{trop}}:\mathcal{H}^{\operatorname{trop}}(\varphi)\to\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$
as well as to the multivalued map

$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}:=(\pi_{2}^{\operatorname{trop}})|_{\mathcal{H}^{\operatorname{trop}}(\varphi)}\circ(\pi_{1}^{\operatorname{trop}})|_{\mathcal{H}^{\operatorname{trop}}(\varphi)}^{-1}:\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}\to\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$

as the naive tropical moduli space correspondence.

######  Remark 7.1.

The “true” tropical Hurwitz space
$\mathcal{H}^{\operatorname{trop}}_{\mathrm{prof,true}}$ (Remark 6.2) has a
connected component
$\mathcal{H}^{\operatorname{trop}}_{\mathrm{true}}:=\chi^{-1}(\widetilde{\mathcal{H}}_{\mathrm{prof}}(\varphi))$
that is the cone over the boundary complex of
$\widetilde{\mathcal{H}}(\varphi)$. The pair of maps
$\pi_{1}^{\operatorname{trop}}\circ\chi,\pi_{2}^{\operatorname{trop}}\circ\chi:\mathcal{H}^{\operatorname{trop}}_{\mathrm{true}}(\varphi)\to\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$
ans well as the multivalued map
$\pi_{2}^{\operatorname{trop}}\circ\chi\circ((\pi_{1}^{\operatorname{trop}}\circ\chi)|_{\mathcal{H}^{\operatorname{trop}}_{\mathrm{true}}(\varphi)})^{-1}$
ought to be thought of as the “true” tropical moduli space correspondence.
Given that the difference between
$\mathcal{H}^{\operatorname{trop}}_{\mathrm{true}}$ and
$\widetilde{\mathcal{H}}(\varphi)$ lies purely in the multiplicity of
occurrence of cones, the difference between the true and the naive tropical
moduli space correspondence also lies purely in multiplicities. If the global
dynamics of the tropical moduli space correspondence are to be studied in any
substantive way, it is absolutely necessary to get the multiplicities correct.
This study therefore ought to be carried out in the context of the true
correspondence – we believe that this is the object that best captures the
algebraic dynamics “near infinity” of the moduli space correspondence on
$\mathcal{M}_{0,\mathbf{P}}$.

In the case that $\varphi$ has topological degree $2$, the situation is
particularly good. The Hurwitz space $\mathcal{H}_{\mathrm{prof}}(\varphi)$ is
connected, so in fact
$\mathcal{H}(\varphi)=\mathcal{H}_{\mathrm{prof}}(\varphi)$. The
compactifications $\overline{\mathcal{H}}_{\mathrm{prof}}(\varphi)$,
$\overline{\mathcal{H}}(\varphi)$,
$\widetilde{\mathcal{H}}_{\mathrm{prof}}(\varphi)$, and
$\widetilde{\mathcal{H}}(\varphi)$ all coincide, and have irreducible boundary
strata. This implies that the tropical spaces
$\mathcal{H}^{\operatorname{trop}}(\varphi)$,
$\mathcal{H}_{\mathrm{prof}}^{\operatorname{trop}}(\varphi)$,
$\mathcal{H}^{\operatorname{trop}}_{\mathrm{true}}(\varphi)$, and
$\mathcal{H}^{\operatorname{trop}}_{\mathrm{prof,true}}$ coincide as well, and
can be concretely described. In the context that $\varphi$ is degree-$2$, one
is comparatively well-placed to study the dynamics if the true moduli space
correspondence.

###  7.1. Tropical admissible covers from weighted multicurves

In this section, we establish that the relationship between the tropical
moduli space correspondence and the the tropical Thurston pullback map exactly
mirrors the relationship between the algebraic moduli space correspondence and
the holomorphic Thurston pulllback map. First, we show how to obtain a
combinatorial type of admissible cover given a multicurve on
$S^{2}\setminus\mathbf{P}$.

######  Definition 7.2.

Suppose that $\Gamma=\\{\gamma_{1},\ldots,\gamma_{k}\\}$ is a multicurve on
$S^{2}\setminus\mathbf{P}$. We obtain a combinatorial type

$\Lambda_{\varphi}(\Gamma)=(\operatorname{\tau}_{1},\operatorname{\tau}_{2},F,\operatorname{EdgeDeg},\varphi|_{\varphi^{-1}(\mathbf{P})},\operatorname{LegDeg})$

of admissible cover from $\Gamma$ and $\varphi$ as follows. Set
$\operatorname{\tau}_{1}=\operatorname{\tau}(\Gamma)$ to be its marked dual
tree as defined in Section 3.3. Set
$\Gamma_{2}:=\widetilde{\varphi}^{*}(\Gamma)$ to be its pullback (a multicurve
on $S^{2}\setminus\varphi^{-1}(\mathbf{P})$), and set
$\operatorname{\tau}_{2}$ to be the $\varphi^{-1}(\mathbf{P})$-marked dual
tree of $\Gamma_{2}$. Set
$F:\underline{\operatorname{\tau}_{2}}\to\underline{\operatorname{\tau}_{1}}$
to be the graph homomorphism determined by $\varphi$ in the natural way,
specifically:

  * •

For $v$ a vertex of $\operatorname{\tau}_{2}$ corresponding to connected
component $Y$ of $S^{2}\setminus(\gamma)_{\gamma\in\Gamma_{2}}$, set $F(v)$ to
be the vertex of $\operatorname{\tau}_{1}$ corresponding to the connected
component $\varphi(Y)$ of $S^{2}\setminus(\gamma)_{\gamma\in\Gamma}$.

  * •

For $e$ an edge of $\operatorname{\tau}_{2}$ corresponding to simple closed
curve $\gamma\in\Gamma_{2}$, set $F(e)$ to be the edge of
$\operatorname{\tau}_{1}$ corresponding to simple closed curve
$\varphi(\gamma)\in\Gamma$.

Set
$\operatorname{EdgeDeg}:\operatorname{Edges}(\operatorname{\tau}_{2})\to\mathbb{Z}_{\>0}$
be the map that sends the edge $e$ of $\operatorname{\tau}_{2}$ corresponding
to $\gamma\in\Gamma_{2}$ to $\deg(\varphi|_{\gamma})$. As explained in Section
6.3, $\varphi|_{\varphi^{-1}(\mathbf{P})}$ and $\operatorname{LegDeg}$ are
determined by just $\varphi$.

######  Lemma 7.3 ([HV10, Sel12]).

For any multicurve $\Gamma$, the combinatorial type
$\Lambda_{\varphi}(\Gamma)$ in Definition 7.2 has topological type $\varphi$.

###### Proof.

By [Koc13] the map
$\nu_{\varphi}:\mathcal{T}(S^{2},\mathbf{P})\to\mathcal{H}(\varphi)$ can be
described in terms of the pullback map
$\widetilde{\operatorname{TP}}_{\varphi}$ as follows. Given
$[\psi]\in\mathcal{T}(S^{2},\mathbf{P})$, under the pair of complex structures
$[\psi]$ and $\widetilde{\operatorname{TP}}_{\varphi}([\psi])$ on
$(S^{2},\mathbf{P})$ and $(S^{2},\varphi^{-1}(\mathbf{P}))$ respectively, the
branched cover $\varphi$ is identified with a holomorphic map $f$;
$\nu_{\varphi}$ sends $[\psi]$ to $[f]$.

The extension of $\widetilde{\operatorname{TP}}_{\varphi}$ to
$\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})$ constructed in
[Sel12] as well as the extension of
$\nu_{\varphi}:\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})\to\overline{\mathcal{H}}(\varphi)$
constructed in [HV10] can be described as follows. Given
$[C,\iota,\psi]\in\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})$,
set $\Gamma$ to be the multicurve on $S^{2}\setminus\mathbf{P}$ consisting of
curves contracted by $\psi:S^{2}\to C$, set
$\Gamma_{2}:=\widetilde{\varphi}^{*}(\Gamma)$ to be its pullback (a multicurve
on $S^{2}\setminus\varphi^{-1}(\mathbf{P})$), set $X_{2}$ to be the
$\varphi^{-1}(\mathbf{P})$-marked topological surface obtained from $S^{2}$ by
contracting to a point each curve in $\Gamma_{2}$ and set $\psi_{2}:S^{2}\to
X_{2}$ to be the quotient map. The branched cover $\varphi$ descends to a
branched cover $\underline{\varphi}:X_{2}\to C$. One can lift the complex
structure on $C$ along $\underline{\varphi}$ to obtain an identification
$X_{2}\cong C_{2}$, where $C_{2}$ is a $\varphi^{-1}(\mathbf{P})$-marked
stable curve. We set
$\widetilde{\operatorname{TP}}_{\varphi}(C,\iota,\psi)=(C_{2},(\psi_{2})|_{\varphi^{-1}(\mathbf{P})},\psi_{2})\in\operatorname{\overline{\mathcal{T}}}(S^{2},\varphi^{-1}(\mathbf{P}))$.
By construction, under the identification $X_{2}\cong C_{2}$, the branched
cover $\underline{\varphi}:C_{2}\to C$ is an admissible cover $f$ of profile
$\varphi$. We set
$\nu_{\varphi}([C,\iota,\psi])=[f]\in\overline{\mathcal{H}}(\varphi)$. The
combinatorial type of $f$ is clearly $\Lambda_{\varphi}(\Gamma)$, and since
$[f]\in\overline{\mathcal{H}}(\varphi)$, $\Lambda_{\varphi}(\Gamma)$ has
topological type $\varphi$. Finally we note that any multicurve $\Gamma$ is
exactly the contracted multicurve for some
$[C,\iota,\psi]\in\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})$ –
the lemma follows. ∎

######  Definition 7.4 (Definition-Lemma).

We define a map
$\nu_{\varphi}^{\operatorname{trop}}:\operatorname{ConeCC}(S^{2},\mathbf{P})\to\mathcal{H}^{\operatorname{trop}}(\varphi)$
defined on cones as follows. Given a multicurve $\Gamma$ on
$S^{2}\setminus\mathbf{P}$, let
$\Lambda_{\varphi}(\Gamma)=(\operatorname{\tau}_{1},\operatorname{\tau}_{2},F,\operatorname{EdgeDeg},\varphi|_{\varphi^{-1}(\mathbf{P})},\operatorname{LegDeg})$
be the combinatorial type of admissible cover as in Definition 7.2. Recall
that $\operatorname{\tau}_{1}=\operatorname{\tau}(\Gamma)$ has an edge
$e([\gamma])$ for every $[\gamma]\in\Gamma$. Given
$\sum_{[\gamma]\in\Gamma}a_{[\gamma]}[\gamma]\in\operatorname{Cone}(\Gamma)\subset\operatorname{ConeCC}(S^{2},\mathbf{P})$,
we obtain a possibly degenerate metrization of $\operatorname{\tau}_{1}$ by
assigning length $a_{[\gamma]}$ to $e([\gamma])$, yielding a tropical curve
$\Sigma_{1}$. This tropical curve is the target curve of a unique tropical
admissible cover
$(\Sigma_{1},\Sigma_{2},\mathbf{F})\in\operatorname{Cone}(\Lambda_{\varphi}(\Gamma))$.
We set
$\nu_{\varphi}^{\operatorname{trop}}(\sum_{[\gamma]\in\Gamma}a_{[\gamma]}[\gamma])=(\Sigma_{1},\Sigma_{2},\mathbf{F})$.
Note that $\nu_{\varphi}^{\operatorname{trop}}$ is well-defined and piecewise
linear, and its restriction to every cone .

######  Lemma 7.5.

The image of
$\nu_{\varphi}^{\operatorname{trop}}:\operatorname{ConeCC}(S^{2},\mathbf{P})\to\mathcal{H}_{\mathrm{prof}}^{\operatorname{trop}}(\varphi)$
is exactly $\mathcal{H}^{\operatorname{trop}}(\varphi)$.

###### Proof.

By [HV10],
$\nu_{\varphi}:\operatorname{\overline{\mathcal{T}}}(S^{2},\mathbf{P})\to\overline{\mathcal{H}}(\varphi)$
is surjective, so every combinatorial type of admissible cover $\Lambda$ that
has the topological type of $\varphi$ arises as $\Lambda_{\varphi}(\Gamma)$
for some multicurve $\Gamma$. Finally, we note that
$\nu_{\varphi}^{\operatorname{trop}}:\operatorname{Cone}(\Gamma)\to\operatorname{Cone}(\Lambda_{\varphi}(\Gamma))$
is a bijection. ∎

######  Proposition 7.6.

We have

  1. (1)

$\rho^{\operatorname{trop}}=\pi_{1}^{\operatorname{trop}}\circ\nu_{\varphi}^{\operatorname{trop}}$,

  2. (2)

$\widetilde{\rho}^{\operatorname{trop}}\circ\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}=\widetilde{\pi}_{2}^{\operatorname{trop}}\circ\nu_{\varphi}^{\operatorname{trop}}$,

  3. (3)

$\rho^{\operatorname{trop}}\circ\operatorname{TP}_{\varphi}^{\operatorname{trop}}=\pi_{2}^{\operatorname{trop}}\circ\nu_{\varphi}^{\operatorname{trop}}.$

######  Remark 7.7.

Proposition 7.6 says that the tropical moduli space correspondence descends
from the tropical Thurston pullback map on weighted multicurves, in the
following sense: Parts (1) and (3) of the proposition together imply that
Diagram (5) below commutes. This is the third — tropical — companion of
Diagram (3) in Section 5 and its compactified/augmented Diagram (4) in Section
6.2.

(5)
${\operatorname{ConeCC}(S^{2},\mathbf{P})}$${\operatorname{ConeCC}(S^{2},\mathbf{P})}$${\mathcal{H}^{\operatorname{trop}}(\varphi)}$${\mathcal{M}^{\operatorname{trop}}_{0,\mathbf{P}}}$${\mathcal{M}^{\operatorname{trop}}_{0,\mathbf{P}}}$$\operatorname{TP}_{\varphi}^{\operatorname{trop}}$$\nu_{\varphi}^{\operatorname{trop}}$$\rho^{\operatorname{trop}}$$\rho^{\operatorname{trop}}$$\pi_{1}^{\operatorname{trop}}$$\pi_{2}^{\operatorname{trop}}$

###### Proof of Proposition 7.6.

The first equality (1) follows directly from the definition (Definition 7.4)
of $\nu_{\varphi}^{\operatorname{trop}}$ together with the definition of
$\rho^{\operatorname{trop}}$ given in Section 3.3. For (2), suppose that
$\Gamma$ is any multicurve on $S^{2}\setminus\mathbf{P}$, and
$\sum_{[\gamma]\in\Gamma}a_{[\gamma]}[\gamma]\in\operatorname{Cone}^{\circ}(\Gamma)$.
Set

$\Lambda_{\varphi}(\Gamma)=(\operatorname{\tau}_{1},\operatorname{\tau}_{2},F,\operatorname{EdgeDeg},\varphi|_{\varphi^{-1}(\mathbf{P})},\operatorname{LegDeg})$

to be the combinatorial type of admissible cover as in Definition 7.2. Recall
that $\operatorname{\tau}_{1}=\operatorname{\tau}(\Gamma)$, so the edges of
$\operatorname{\tau}_{1}$ are in bijection with simple closed curves
$[\gamma]\in\Gamma$. Also
$\operatorname{\tau}_{2}=\operatorname{\tau}(\widetilde{\varphi}^{*}(\Gamma))$,
so the edges of $\operatorname{\tau}_{2}$ are in bijection with simple closed
curves $[\gamma^{\prime}]$ that map to $[\gamma]\in\Gamma$. For any
$[\gamma^{\prime}_{0}]\in\widetilde{\varphi}^{*}(\Gamma)$ mapping via
$\varphi$ to $[\gamma_{0}]$ with degree $d_{\gamma_{0}^{\prime}}$. By (1), the
coefficient of $[\gamma^{\prime}]$ in
$\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}(\sum_{[\gamma]\in\Gamma}a_{[\gamma]}[\gamma]\in\operatorname{Cone}(\Gamma))$
is $\frac{a_{[\gamma_{0}]}}{d_{\gamma^{\prime}_{0}}}$.

To understand
$\widetilde{\pi}_{2}^{\operatorname{trop}}\circ\nu_{\varphi}^{\operatorname{trop}}(\sum_{[\gamma]\in\Gamma}a_{[\gamma]}[\gamma])$,
we observe that first
$\nu_{\varphi}^{\operatorname{trop}}(\sum_{[\gamma]\in\Gamma}a_{[\gamma]}[\gamma])$
produces a tropical admissible cover $(\Sigma_{1},\Sigma_{2},\mathbf{F})$ with
combinatorial type $\Lambda_{\varphi}(\Gamma)$, then, applying
$\widetilde{\pi_{2}}^{\operatorname{trop}}$, we record the tropical curve
$\Sigma_{2}$, which has underlying tree $\operatorname{\tau}_{2}$. For any
$[\gamma_{0}]\in\Gamma$, the length, on the tropical curve $\Sigma_{1}$, of
the edge $e([\gamma_{0}])$ of its underlying tree $\operatorname{\tau}_{1}$,
is $a_{[\gamma_{0}]}$. For any
$[\gamma^{\prime}_{0}]\in\widetilde{\varphi}^{*}(\Gamma)$ mapping via
$\varphi$ to $[\gamma_{0}]$ with degree $d_{\gamma_{0}^{\prime}}$, the slope
of $\mathbf{F}$ along $e([\gamma^{\prime}])$ equals $d_{[\gamma^{\prime}]}$,
so the length of $e([\gamma^{\prime}])$ on the tropical curve $\Sigma_{2}$
equals $\frac{a_{[\gamma_{0}]}}{d_{\gamma^{\prime}_{0}}}$.

To understand
$\widetilde{\rho}^{\operatorname{trop}}\circ\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}(\sum_{[\gamma]\in\Gamma}a_{[\gamma]}[\gamma])$,
we note that
$\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}(\sum_{[\gamma]\in\Gamma}a_{[\gamma]}[\gamma])$
produces a non-negative weighting on the multicurve
$\widetilde{\varphi}^{*}(\Gamma)$, then, applying
$\widetilde{\rho}^{\operatorname{trop}}$, we obtain a tropical curve
$\Sigma_{2}^{\prime}$ with underlying tree
$\operatorname{\tau}_{2}=\operatorname{\tau}(\widetilde{\varphi}^{*}(\Gamma))$,
and edge-lengths corresponding to the weights in
$\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}(\sum_{[\gamma]\in\Gamma}a_{[\gamma]}[\gamma])$.
By (1), for any $[\gamma^{\prime}_{0}]\in\widetilde{\varphi}^{*}(\Gamma)$
mapping via $\varphi$ to $[\gamma_{0}]$ with degree $d_{\gamma_{0}^{\prime}}$,
the coefficient of $[\gamma^{\prime}]$ in
$\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}(\sum_{[\gamma]\in\Gamma}a_{[\gamma]}[\gamma]\in\operatorname{Cone}(\Gamma))$
is $\frac{a_{[\gamma_{0}]}}{d_{\gamma^{\prime}_{0}}}$. So the length of
$e([\gamma^{\prime}])$ on the tropical curve $\Sigma_{2}^{\prime}$ equals
$\frac{a_{[\gamma_{0}]}}{d_{\gamma^{\prime}_{0}}}$.

We conclude that $\Sigma_{1}=\Sigma_{2}$, and so
$\widetilde{\pi}_{2}^{\operatorname{trop}}\circ\nu_{\varphi}^{\operatorname{trop}}(\sum_{[\gamma]\in\Gamma}a_{[\gamma]}[\gamma])=\widetilde{\rho}^{\operatorname{trop}}\circ\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}(\sum_{[\gamma]\in\Gamma}a_{[\gamma]}[\gamma])$.

For the final equality (3), observe that since
$\pi_{2}^{\operatorname{trop}}=\mu^{\operatorname{trop}}\circ\widetilde{pi}_{2}^{\operatorname{trop}}$,
we can write

$\displaystyle\pi_{2}^{\operatorname{trop}}\circ\nu_{\varphi}^{\operatorname{trop}}$
$\displaystyle=\mu^{\operatorname{trop}}\circ\widetilde{pi}_{2}^{\operatorname{trop}}\circ\nu_{\varphi}^{\operatorname{trop}}$
$\displaystyle=\mu^{\operatorname{trop}}\circ\widetilde{\rho}^{\operatorname{trop}}\circ\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}$
$\displaystyle=\rho^{\operatorname{trop}}\circ\widetilde{\mu}^{\operatorname{trop}}\circ\widetilde{\operatorname{TP}}_{\varphi}^{\operatorname{trop}}$
$\displaystyle=\rho^{\operatorname{trop}}\circ\operatorname{TP}^{\operatorname{trop}}_{\varphi}$

The second equality follows from Part (2) of this Lemma. ∎

###  7.2. Local properties of the tropical moduli space correspondence

Picking a locally-defined inverse branch of
$\pi_{1}:\mathcal{H}(\varphi)\to\mathcal{M}_{0,\mathbf{P}}$ gives us a
locally-defined single-valued branch of the multivalued map
$\operatorname{MSC}_{\varphi}=(\pi_{2})|_{\mathcal{H}(\varphi)}\circ(\pi_{1})|_{cH(\varphi)}$
on $\mathcal{M}_{0,\mathbf{P}}$. This local single-valued branch has exactly
the same local properties as the pullback map $\operatorname{TP}_{\varphi}$ on
$\mathcal{T}(S^{2},\mathbf{P})$, for example the derivative at a fixed point.
There is a tropical analog as follows:
$(\pi_{1}^{\operatorname{trop}})_{\mathcal{H}^{\operatorname{trop}}(\varphi)}$
is finite-to-one and can be thought of as a branched cover. Choosing a cone
$\operatorname{Cone}(\Lambda)$ of $\mathcal{H}^{\operatorname{trop}}(\varphi)$
yields an inverse branch
$(\pi_{1}^{\operatorname{trop}}|_{\operatorname{Cone}(\Lambda)})^{-1}$ of
$\pi_{1}$, defined on $\pi_{1}(\operatorname{Cone}(\Lambda))$, which is a cone
of $\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$. In fact, if
$\Lambda=(\operatorname{\tau}_{1},\operatorname{\tau}_{2},\ldots)$, then the
image, under
$\pi_{1}^{\operatorname{trop}}(\operatorname{Cone}(\Lambda))=\operatorname{Cone}(\operatorname{\tau}_{1})$
and
$\pi_{2}^{\operatorname{trop}}(\operatorname{Cone}(\Lambda))\subset\operatorname{Cone}(\mu_{*}(\operatorname{\tau}_{2}))$.
Then

$\operatorname{MSC}_{\varphi,\Lambda}^{\operatorname{trop}}:=(\pi_{2})|_{\operatorname{Cone}(\Lambda)}\circ(\pi_{1}^{\operatorname{trop}}|_{\operatorname{Cone}(\Lambda)})^{-1}:\operatorname{Cone}(\operatorname{\tau}_{1})\to\operatorname{Cone}(\mu_{*}(\operatorname{\tau}_{2}))$

is the single-valued linear branch associated to $\Lambda$ of the tropical
moduli space correspondence
$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$. If
$\operatorname{Cone}(\Lambda)$ is a maximal cone of
$\mathcal{H}^{\operatorname{trop}}(\varphi)$, then we will refer to
$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$ as a local branch of
$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$. Proposition 7.8 says
that every branch of $\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$ is
given by some Thurston linear transformation (Section 4.6 Equation (1)).

######  Proposition 7.8.

Suppose $\operatorname{Cone}(\Lambda)$ is any cone of
$\mathcal{H}^{\operatorname{trop}}(\varphi)$, that
$\Lambda=\Lambda_{\varphi}(\Gamma)=(\operatorname{\tau}_{1},\operatorname{\tau}_{2},\ldots)$,
and that $\Gamma$ is any multicurve on $S^{2}\setminus\mathbf{P}$ such that
$\Lambda=\Lambda_{\varphi}(\Gamma)$. Let $M_{1}$ be the matrix of the branch
$\operatorname{MSC}_{\varphi,\Lambda_{\varphi}(\Gamma)}^{\operatorname{trop}}$,
written with respect to the natural generators
$\operatorname{Edges}(\operatorname{\tau}_{1})$ of
$\operatorname{Cone}(\operatorname{\tau}_{1})$ and
$\operatorname{Edges}(\mu_{*}(\operatorname{\tau}_{2}))$ of
$\operatorname{Cone}(\mu_{*}(\operatorname{\tau}_{2}))$. Let $M_{2}$ be the
matrix of Thurston linear transformation
$\operatorname{TLT}_{\varphi,\Gamma}$((1)), written with respect to the
natural bases of $\mathbb{R}^{\Gamma}$ and $\mathbb{R}^{\varphi^{*}(\Gamma)}$.
Then $M_{1}=M_{2}$.

###### Proof.

Write $\Gamma=\\{[\gamma_{1}],\ldots,[\gamma_{r}]\\}$, which means that
$\operatorname{\tau}_{1}=\operatorname{\tau}(\Gamma)$ has edges
$e([\gamma_{1}],\ldots,e([\gamma_{r}])$. Write
$\varphi^{*}(\Gamma)=\\{[\gamma^{\prime}_{1}],\ldots,[\gamma^{\prime}_{s}]\\}$,
which means that
$\mu_{*}(\operatorname{\tau}_{2})=\operatorname{\tau}(\varphi^{*}(\Gamma))$
has edges $e([\gamma^{\prime}_{1}]),\ldots,e([\gamma^{\prime}_{s}])$. Recall
that

  * •

$\operatorname{Cone}(\Gamma)\subset\operatorname{ConeCC}(S^{2},\mathbf{P})$ is
generated by $[\gamma_{1}],\ldots,[\gamma_{r}]$,

  * •

$\operatorname{Cone}(\operatorname{\tau}_{1})$ is generated by
$e([\gamma_{1}],\ldots,e([\gamma_{r}])$,

  * •

$(\rho^{\operatorname{trop}})|_{\operatorname{Cone}(\Gamma)}:\operatorname{Cone}(\Gamma)\to\operatorname{Cone}(\operatorname{\tau}_{1})$
is an isomorphism taking $[\gamma_{i}]$ to $e([\gamma_{i}])$,

  * •

$\operatorname{Cone}(\varphi^{*}(\Gamma))\subset\operatorname{ConeCC}(S^{2},\mathbf{P})$
is generated by $[\gamma^{\prime}_{1}],\ldots,[\gamma^{\prime}_{s}]$,

  * •

$\operatorname{Cone}(\mu_{*}(\operatorname{\tau}_{2}))$ is generated by
$e([\gamma^{\prime}_{1}],\ldots,e([\gamma^{\prime}_{s}])$,

  * •

$(\rho^{\operatorname{trop}})|_{\operatorname{Cone}(\varphi^{*}(\Gamma))}:\operatorname{Cone}(\varphi^{*}(\Gamma))\to\operatorname{Cone}(\mu_{*}(\operatorname{\tau}_{2}))$
is an isomorphism taking $[\gamma^{\prime}_{j}]$ to
$e([\gamma^{\prime}_{j}])$,

  * •

The matrix of
$(\operatorname{TP}_{\varphi}^{\operatorname{trop}})|_{\operatorname{Cone}(\Gamma)}:\operatorname{Cone}(\Gamma)\to\operatorname{Cone}(\varphi^{*}(\Gamma))$,
with respect to the generators $[\gamma_{1}],\ldots,[\gamma_{r}]$ and
$[\gamma^{\prime}_{1}],\ldots,[\gamma^{\prime}_{s}]$ equals $M_{2}$.

By Proposition 7.6 (3),

$\displaystyle(\rho^{\operatorname{trop}})_{\operatorname{Cone}(\Gamma)}=(\pi_{1}^{\operatorname{trop}})|_{\operatorname{Cone}(\Lambda_{\varphi}(\Gamma))}\circ(\nu_{\varphi}^{\operatorname{trop}})|_{\operatorname{Cone}(\Gamma)}$

By Proposition 7.6 (1),

$\displaystyle(\rho^{\operatorname{trop}})|_{\operatorname{Cone}(\varphi^{*}(\Gamma))}\circ(\operatorname{TP}_{\varphi}^{\operatorname{trop}})|_{\operatorname{Cone}(\Gamma)}=(\pi_{2}^{\operatorname{trop}})|_{\operatorname{Cone}(\Lambda_{\varphi}(\Gamma))}\circ\nu_{\varphi}^{\operatorname{trop}})|_{\operatorname{Cone}(\Gamma)}$

So, as maps from $\operatorname{Cone}(\operatorname{\tau}_{1})$ to
$\operatorname{Cone}(\mu_{*}(\operatorname{\tau}_{2}))$:

(6)
$\displaystyle(\rho^{\operatorname{trop}})|_{\operatorname{Cone}(\varphi^{*}(\Gamma))}\circ(\operatorname{TP}_{\varphi}^{\operatorname{trop}})|_{\operatorname{Cone}(\Gamma)}\circ((\rho^{\operatorname{trop}})|_{\operatorname{Cone}(\Gamma)})^{-1}$
$\displaystyle=$
$\displaystyle(\pi_{2}^{\operatorname{trop}})|_{\operatorname{Cone}(\Lambda_{\varphi}(\Gamma))}\circ\nu_{\varphi}^{\operatorname{trop}})|_{\operatorname{Cone}(\Gamma)}\circ((\rho^{\operatorname{trop}})|_{\operatorname{Cone}(\Gamma)})^{-1}$
$\displaystyle=$
$\displaystyle(\pi_{2}^{\operatorname{trop}})|_{\operatorname{Cone}(\Lambda_{\varphi}(\Gamma))}\circ\nu_{\varphi}^{\operatorname{trop}})|_{\operatorname{Cone}(\Gamma)}\circ((\nu_{\varphi}^{\operatorname{trop}})|_{\operatorname{Cone}(\Gamma)})^{-1}\circ((\pi_{1}^{\operatorname{trop}})|_{\operatorname{Cone}(\Lambda_{\varphi}(\Gamma))})^{-1}$
(7) $\displaystyle=$
$\displaystyle(\pi_{2}^{\operatorname{trop}})|_{\operatorname{Cone}(\Lambda_{\varphi}(\Gamma))}\circ((\pi_{1}^{\operatorname{trop}})|_{\operatorname{Cone}(\Lambda_{\varphi}(\Gamma))})^{-1}$

In terms of the generators $\\{e([\gamma_{i}])\\}$ for
$\operatorname{Cone}(\operatorname{\tau}_{1})$ and $\\{e([\gamma_{j}])\\}$ for
$\operatorname{Cone}(\mu_{*}(\operatorname{\tau}_{2}))$, it is clear that (6)
has matrix $M_{2}$ and (by definition) (7) has matrix $M_{1}$. The proposition
follows.

∎

###  7.3. Weakly fixed cones

A weakly fixed cone of $\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$ is
a cone $\operatorname{Cone}(\Lambda)$ of
$\mathcal{H}^{\operatorname{trop}}(\varphi)$ such that
$\pi_{2}^{\operatorname{trop}}(\operatorname{Cone}(\Lambda))\subset\pi_{1}^{\operatorname{trop}}(\operatorname{Cone}(\Lambda))$.
If $\Lambda=(\operatorname{\tau}_{1},\operatorname{\tau}_{2},\ldots)$ then
$\operatorname{Cone}(\Lambda)$ is weakly fixed if and only if
$\mu_{*}(\operatorname{\tau}_{2})$ is obtained from $\operatorname{\tau}_{1}$
by edge-contraction (possibly of the empty set of edges). In this case, via
the inclusion of cones
$\operatorname{Cone}(\mu_{*}(\operatorname{\tau}_{2}))\hookrightarrow\operatorname{Cone}(\operatorname{\tau}_{1})$,
the branch $\operatorname{MSC}_{\varphi,\Lambda}^{\operatorname{trop}}$ of
$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$ can be considered to be a
self-map of $\operatorname{Cone}(\operatorname{\tau}_{1})$. By the Perron-
Frobenius theorem on non-negative matrices:

  * •

The dominant eigenvalue $\operatorname{\lambda_{\mathrm{dom}}}(\Lambda)$ of
$\operatorname{MSC}_{\varphi,\Lambda}^{\operatorname{trop}}$ is real and non-
negative, and has an eigenvector in
$\operatorname{Cone}(\operatorname{\tau}_{1})$, and

  * •

if $\operatorname{MSC}_{\varphi,\Lambda}^{\operatorname{trop}}$ has an
eigenvector in $\operatorname{Cone}^{\circ}(\operatorname{\tau}_{1})$, then
its eigenvalue is $\operatorname{\lambda_{\mathrm{dom}}}(\Lambda)$.

######  Proposition 7.9.

  1. (1)

If $\Gamma$ is a $\varphi$-stable multicurve then
$\operatorname{Cone}(\Lambda_{\varphi}(\Gamma))$ is a weakly fixed cone of
$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$, and the dominant
eigenvalue $\operatorname{\lambda_{\mathrm{dom}}}(\Lambda_{\varphi}(\Gamma))$
equals the Thurston eigenvalue
$\operatorname{\lambda_{\mathrm{Thurst}}}(\Gamma)$ associated to $\Gamma$.

  2. (2)

If $\operatorname{Cone}(\Lambda)$ is weakly fixed cone of
$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$ then there exists
$\varphi^{\prime}$ Hurwitz equivalent to $\varphi$ together with a
$\varphi^{\prime}$-stable multicurve $\Gamma$ such that
$\Lambda=\Lambda_{\varphi^{\prime}}(\Gamma)$.

###### Proof.

For (1), write
$\Lambda_{\varphi}(\Gamma)=(\operatorname{\tau}_{1},\operatorname{\tau}_{2},\ldots)$.
If $\Gamma$ is $\varphi$-stable then $\varphi^{*}(\Gamma)\subset(\Gamma)$,
which implies that
$\mu_{*}(\operatorname{\tau}_{2})=\operatorname{\tau}(\varphi^{*}(\Gamma))$ is
obtained from $\operatorname{\tau}_{1}=\operatorname{\tau}(\Gamma)$ by edge-
contraction, which in turn implies that $\Lambda_{\varphi}(\Gamma)$ is weakly
fixed by $\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$. By Proposition
7.8, the branch $\operatorname{MSC}_{\varphi,\Lambda}^{\operatorname{trop}}$
and the Thurston linear transformation $\operatorname{TLT}_{\varphi,\Gamma}$
have the same matrix representative therefore have the same dominant
eigenvalue. So
$\operatorname{\lambda_{\mathrm{dom}}}(\Lambda_{\varphi}(\Gamma))=\operatorname{\lambda_{\mathrm{Thurst}}}(\Gamma)$.

Now, suppose that $\operatorname{Cone}(\Lambda)$ is a weakly fixed come of
$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$. Pick any multicurve
$\Gamma$ such that $\Lambda=\Lambda_{\varphi}(\Gamma)$. Now, suppose that
$\operatorname{Cone}(\Lambda)$ is a weakly fixed come of
$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$ — the fact that
$\operatorname{Cone}(\Lambda)$ is weakly fixed implied that
$\mu_{*}(\operatorname{\tau}_{2})=\operatorname{\tau}_{1}$. Pick any
multicurve $\Gamma$ such that $\Lambda=\Lambda_{\varphi}(\Gamma)$; for such
$\Gamma$,
$\operatorname{\tau}(\varphi^{*}(\Gamma))=\operatorname{\tau}(\Gamma)$, which
means that there is a homeomorphism $\alpha:S^{2}\to S^{2}$, fixing
$\mathbf{P}$ pointwise, whose mapping class
$[\alpha]\in\operatorname{PMod}(S^{2},\mathbf{P})$ satisfies
$\alpha_{*}(\Gamma)=\varphi^{*}(\Gamma)$. Set
$\varphi^{\prime}=\varphi\circ\alpha$. It is clear that $\varphi^{\prime}$ and
$\varphi$ are Hurwitz equivalent. Also,
$(\varphi^{\prime})^{*}(\Gamma)=\alpha^{*}(\varphi^{*}(\Gamma))=\Gamma$, so
$\Gamma$ is $\varphi^{\prime}$-stable. Finally, by construction,
$\Lambda_{\varphi^{\prime}}(\Gamma)=\Lambda_{\varphi}(\Gamma)=\Lambda$,
proving (2). ∎

###  7.4. Weakly fixed rays

######  Definition 7.10.

A ray of $\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$ is a subset of the
form $\mathbb{R}_{\geq 0}\cdot\Sigma$, where $\Sigma$ is not the one-vertex
tree, or, equivalently, $\Sigma$ is not the cone point of
$\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$). A ray of
$\mathcal{H}^{\operatorname{trop}}(\varphi)$ (resp. of
$\mathcal{H}_{\mathrm{prof}}^{\operatorname{trop}}(\varphi)$) is a subset of
the form $\mathbb{R}_{\geq 0}\cdot(\Sigma_{1},\Sigma_{2},\mathbf{F})$, where
$\Sigma_{1}$ is not the one-vertex tree, or, equivalently,
$(\Sigma_{1},\Sigma_{2},\mathbf{F})$ is not the cone point of
$\mathcal{H}^{\operatorname{trop}}(\varphi)$ (resp. of
$\mathcal{H}_{\mathrm{prof}}^{\operatorname{trop}}(\varphi)$).

The image, under $\pi_{1}^{\operatorname{trop}}$, of a ray of
$\mathcal{H}^{\operatorname{trop}}(\varphi)$ is a ray of
$\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$. The image, under
$\pi_{2}^{\operatorname{trop}}$, of a ray of
$\mathcal{H}^{\operatorname{trop}}(\varphi)$ is either ray of
$\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$ or is the cone point of
$\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$.

######  Definition 7.11.

A weakly fixed ray of the tropical moduli space correspondence is a ray of
$\mathcal{H}^{\operatorname{trop}}(\varphi)$ whose image under
$\pi_{2}^{\operatorname{trop}}$ is contained in its image under
$\pi_{1}^{\operatorname{trop}}$. A weakly fixed ray $R$ of the tropical moduli
space correspondence has associated to it a scaling factor
$\operatorname{\lambda_{\mathrm{scal}}}(R)\in\mathbb{R}_{\geq 0}$; this is
defined to be the factor such that the map
$\pi_{2}^{\operatorname{trop}}\circ(\pi_{1}^{\operatorname{trop}})^{-1}:\pi_{1}(R)\to\pi_{1}(R)$
scales by $\operatorname{\lambda_{\mathrm{scal}}}(R)$.

Suppose that $R=\mathbb{R}_{\geq 0}\cdot(\Sigma_{1},\Sigma_{2},\mathbf{F})$ is
a weakly fixed ray of the moduli space correspondence. Set
$\Lambda=(\operatorname{\tau}_{1},\operatorname{\tau}_{2},\ldots)$ to be the
combinatorial type of $(\Sigma_{1},\Sigma_{2},\mathbf{F})$, so
$\operatorname{Cone}(\Lambda)$ is the smallest cone containing $R$. Then
$\operatorname{\tau}_{1}$ and $\operatorname{\tau}_{2}$ are the underlying
$\mathbf{P}$\- and $\varphi^{-1}(\mathbf{P})$\- marked trees of $\Sigma_{1}$
and $\Sigma_{2}$ respectively. The tropical curve
$\mu^{\operatorname{trop}}(\Sigma_{2})$ is obtained from the tropical curve
$\Sigma_{1}$ by scaling all the edges by
$\operatorname{\lambda_{\mathrm{scal}}}(R)$. Either

  1. (1)

$\operatorname{\lambda_{\mathrm{scal}}}(R)=0$, and
$\mu_{*}(\operatorname{\tau}_{2})$ is the one-vertex tree, or

  2. (2)

$\operatorname{\lambda_{\mathrm{scal}}}(R)\neq 0$, and
$\mu_{*}(\operatorname{\tau}_{2})=\operatorname{\tau}_{1}$.

In either case, we conclude that $\operatorname{Cone}(\Lambda)$ is weakly
fixed by $\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$, and that
$(\Sigma_{1},\Sigma_{2},\mathbf{F})$ is a strictly positive eigenvector for
$\operatorname{MSC}_{\varphi,\Lambda}^{\operatorname{trop}}$. By the Perron-
Frobenius theorem for non-negative matrices,
$\operatorname{\lambda_{\mathrm{scal}}}(R)=\operatorname{\lambda_{\mathrm{dom}}}(\Lambda)$.

Conversely, for
$\Lambda=(\operatorname{\tau}_{1},\operatorname{\tau}_{2},\ldots)$, if
$\operatorname{Cone}(\Lambda)$ is weakly fixed but not the cone point of
$\mathcal{H}^{\operatorname{trop}}(\varphi)$, then we set
$\mathrm{Eigenspace}_{+}(\Lambda)\subset\operatorname{Cone}(\Lambda)$ to be
the intersection with $\operatorname{Cone}(\Lambda)$ of the dominant
eigenspace of $\operatorname{MSC}_{\varphi,\Lambda}^{\operatorname{trop}}$.
Then $\mathrm{Eigenspace}_{+}(\Lambda)\subset\operatorname{Cone}(\Lambda)$ is
a union of weakly fixed rays, all of whose scaling factors equal
$\operatorname{\lambda_{\mathrm{dom}}}(\Lambda)$.

If $\Gamma$ is a $\varphi$-stable multicurve then we set

$\mathrm{FixedRays}_{\varphi}(\Gamma):=\\{R\mbox{ ray }\thickspace|\thickspace
R\subset\mathrm{Eigenspace}_{+}(\Lambda(\Gamma))\\}.$

######  Proposition 7.12.

  1. (1)

If $\Gamma$ is a non-empty $\varphi$-stable multicurve then
$\mathrm{FixedRays}_{\varphi}(\Gamma)$ is non-empty, and for all
$R\in\mathrm{FixedRays}_{\varphi}(\Gamma)$,
$\operatorname{\lambda_{\mathrm{scal}}}(R)=\operatorname{\lambda_{\mathrm{Thurst}}}(\Gamma).$

  2. (2)

If $R$ is a weakly fixed ray of the moduli space correspondence then there
exist $\varphi^{\prime}$ Hurwitz equivalent to $\varphi$ and
$\varphi^{\prime}$-stable multicurve $\Gamma$ such that
$R\in\mathrm{FixedRays}_{\varphi^{\prime}}(\Gamma)$.

###### Proof.

This follows directly from Proposition 7.9. ∎

######  Corollary 7.13.

  1. (1)

If $\Gamma$ is a Thurston obstruction for $\varphi$ then
$\mathrm{FixedRays}_{\varphi}(\Gamma)$ is non-empty, and for all
$R\in\mathrm{FixedRays}_{\varphi}(\Gamma)$,
$\operatorname{\lambda_{\mathrm{scal}}}(R)\geq 1.$

  2. (2)

If $R$ is a weakly fixed ray of the moduli space correspondence with
$\operatorname{\lambda_{\mathrm{scal}}}(R)\geq 1$ then there exist
$\varphi^{\prime}$ Hurwitz equivalent to $\varphi$ and a Thurston obstruction
$\Gamma$ for $\varphi$ such that
$R\in\mathrm{FixedRays}_{\varphi^{\prime}}(\Gamma)$.

######  Remark 7.14.

Suppose that $\mathbb{R}_{\geq 0}\cdot(\Sigma_{1},\Sigma_{2},\mathbf{F})$ is a
weakly-fixed ray of $\operatorname{MSC}^{\operatorname{trop}}_{\varphi}$ with
scaling factor $\operatorname{\lambda_{\mathrm{scal}}}(R)>0$. The
$\mathbf{P}$-marked tropical curves $\mu^{\operatorname{trop}}(\Sigma_{2})$
and $\Sigma_{1}$ have the same underlying tree, and the length of an edge on
$\mu^{\operatorname{trop}}(\Sigma_{2})$ is
$\operatorname{\lambda_{\mathrm{scal}}}(R)$ times the length of the
corresponding edge on $\Sigma_{1}$. There is a “scale by
$\operatorname{\lambda_{\mathrm{scal}}}(R)$” map
$h:\Sigma_{1}\to\mu^{\operatorname{trop}}(\Sigma_{2})$. The composite
$\mathbf{F}\circ h:\Sigma_{1}\to\Sigma_{1}$ is a piecewise-linear self-map
whose dynamics could be of interest. This idea has been mentioned by
Shishikura [Shi22] in the context of $\varphi$-stable multicurves.

######  Remark 7.15.

$\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$ is a cone — set
$\Delta_{\mathbf{P}}$ to be its link; this is the set of rays in
$\mathcal{M}_{0,\mathbf{P}}^{\operatorname{trop}}$. $\Delta_{\mathbf{P}}$ is a
compact polyhedral complex. The multivalued map
$\operatorname{MSC}_{\varphi}^{\operatorname{trop}}$ takes a ray to either a
ray or the cone point. As such it doesn’t quite descend to a multivalued map
on $\Delta_{\mathbf{P}}$. As a (not-quite) dynamical system on
$\Delta_{\mathbf{P}}$, this could be possibly be formulated as a map that
takes weighted finite sums of points on $\Delta_{\mathbf{P}}$ to weighted
finite sums of points on $\Delta_{\mathbf{P}}$. This is currently the topic of
ongoing investigation.

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